From 745585f18b0f5232214e1199ca1d4985f0238836 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Fri, 15 Sep 2006 12:43:52 +0000 Subject: [PATCH] exportation completed!! --- .../LAMBDA-TYPES/Level-1/Base/ext/arith.ma | 75 +- .../Level-1/LambdaDelta/csub3/clear.ma | 73 ++ .../Level-1/LambdaDelta/csub3/defs.ma | 31 + .../Level-1/LambdaDelta/csub3/drop.ma | 805 ++++++++++++ .../Level-1/LambdaDelta/csub3/fwd.ma | 336 +++++ .../Level-1/LambdaDelta/csub3/getl.ma | 390 ++++++ .../Level-1/LambdaDelta/csub3/pc3.ma | 58 + .../Level-1/LambdaDelta/csub3/props.ma | 27 + .../Level-1/LambdaDelta/csub3/ty3.ma | 99 ++ .../Level-1/LambdaDelta/nf2/fwd.ma | 35 +- .../Level-1/LambdaDelta/pc3/dec.ma | 153 +++ .../Level-1/LambdaDelta/pc3/fsubst0.ma | 719 +++++++++++ .../Level-1/LambdaDelta/pc3/fwd.ma | 279 ++++ .../Level-1/LambdaDelta/pc3/left.ma | 109 ++ .../Level-1/LambdaDelta/pc3/pc1.ma | 35 + .../Level-1/LambdaDelta/pc3/subst1.ma | 47 + .../Level-1/LambdaDelta/pc3/wcpr0.ma | 103 ++ .../Level-1/LambdaDelta/theory.ma | 52 + .../Level-1/LambdaDelta/ty3/arity.ma | 191 +++ .../Level-1/LambdaDelta/ty3/arity_props.ma | 78 ++ .../Level-1/LambdaDelta/ty3/dec.ma | 456 +++++++ .../Level-1/LambdaDelta/ty3/defs.ma | 46 + .../Level-1/LambdaDelta/ty3/fsubst0.ma | 978 ++++++++++++++ .../Level-1/LambdaDelta/ty3/fwd.ma | 914 +++++++++++++ .../Level-1/LambdaDelta/ty3/pr3.ma | 732 +++++++++++ .../Level-1/LambdaDelta/ty3/pr3_props.ma | 501 +++++++ .../Level-1/LambdaDelta/ty3/props.ma | 422 ++++++ .../Level-1/LambdaDelta/ty3/subst1.ma | 1149 +++++++++++++++++ .../Level-1/LambdaDelta/ty3/tau0.ma | 634 +++++++++ 29 files changed, 9462 insertions(+), 65 deletions(-) create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/clear.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/defs.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/drop.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/fwd.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/getl.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/pc3.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/props.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/ty3.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fsubst0.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fwd.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/left.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/pc1.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/subst1.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/wcpr0.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity_props.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/defs.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fsubst0.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fwd.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3_props.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/props.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/tau0.ma diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma index 766a34d6f..fc33cc529 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma @@ -43,7 +43,7 @@ False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1))))) (\lambda (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 (n0: nat).(or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: +(\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 @@ -51,11 +51,12 @@ n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat 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 n) \Rightarrow n])) (S n) -(S n0) H2) in (let H4 \def (eq_ind_r nat n0 (\lambda (n0: nat).((eq nat n n0) -\to (\forall (P: Prop).P))) H1 n H3) in (let H5 \def (eq_ind_r nat n0 -(\lambda (n0: nat).(or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: -Prop).P)))) H0 n H3) in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2)))) n1). +(\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). theorem simpl_plus_r: \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n) @@ -152,29 +153,29 @@ n) m) \to P)))) \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 (n: nat).(\lambda (_: (le ? n)).((eq nat n O) \to +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 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 +(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) m) \to P) H3)) H1))]) in (H1 (refl_equal nat -O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P: +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 (n: nat).(\lambda (_: -(le ? n)).((eq nat n 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 m H2) \Rightarrow (\lambda (H3: -(eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e: nat).(match e +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 H3) in (False_ind ((le (S n) m) \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 +\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). theorem le_Sx_x: @@ -293,9 +294,9 @@ p)) \to (lt n m)))) \def \lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus n p) (plus m p))).(plus_lt_reg_l n m p (let H0 \def (eq_ind nat (plus n p) -(\lambda (n: nat).(lt n (plus m p))) H (plus p n) (plus_comm 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_comm m p)) in H1)))))). +(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_comm 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_comm m p)) in H1)))))). theorem minus_x_Sy: \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S @@ -525,8 +526,8 @@ nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O (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 (y1: nat).(\forall (y2: nat).((le O z0) \to ((le O z0) \to ((eq -nat (plus n y1) (plus n y2)) \to (eq nat y2 y1))))))) H_y z0 (minus_n_O z0)) +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) (H2 (plus z0 y2) (plus z0 y1) (le_O_n z0) (le_O_n z0) (f_equal nat nat (plus z0) (plus z0 y2) (plus z0 y1) (sym_equal nat (plus z0 y1) (plus z0 y2) (eq_add_S (plus z0 y1) (plus z0 y2) @@ -537,14 +538,14 @@ 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 (y2: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S -y1)) (plus (minus z0 x3) y2)) \to (eq nat y2 (plus x3 (S y1)))))))) H_y z0 +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 (y2: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus -(minus z0 x3) y2)) \to (eq nat y2 (plus x3 (S y1)))))))) H2 (S (plus z0 y1)) +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 (y2: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus -z0 y1)) (plus (minus z0 x3) y2)) \to (eq nat y2 n)))))) H3 (S (plus x3 y1)) +(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 @@ -574,3 +575,13 @@ 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). +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_plus_r d h n H))) in (le_S d (minus n h) +(le_minus d n h H))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/clear.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/clear.ma new file mode 100644 index 000000000..70eeece9c --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/clear.ma @@ -0,0 +1,73 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/clear". + +include "csub3/defs.ma". + +include "clear/fwd.ma". + +theorem csub3_clear_conf: + \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to +(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) +(\lambda (e2: C).(clear c2 e2)))))))) +\def + \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 +c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c +e1) \to (ex2 C (\lambda (e2: C).(csub3 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).(csub3 g e1 e2)) +(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: +C).(\lambda (H0: (csub3 g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 +e1) \to (ex2 C (\lambda (e2: C).(csub3 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)).((match k in K return (\lambda (k0: K).((clear +(CHead c3 k0 u) e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda +(e2: C).(clear (CHead c4 k0 u) e2))))) with [(Bind b) \Rightarrow (\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).(csub3 g c e2)) (\lambda (e2: +C).(clear (CHead c4 (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g +(CHead c3 (Bind b) u) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)) +(CHead c4 (Bind b) u) (csub3_head g c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) +e1 (clear_gen_bind b c3 e1 u H3))) | (Flat f) \Rightarrow (\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).(csub3 g e1 e2)) (\lambda (e2: +C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: +C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: C).(\lambda (H5: (csub3 g +e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C (\lambda (e2: C).(csub3 g e1 +e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2)) x H5 (clear_flat c4 x +H6 f u))))) H4)))]) H2))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda +(H0: (csub3 g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 +C (\lambda (e2: C).(csub3 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).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) +u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind Void) u1) +e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u2) e2)) (CHead c4 (Bind b) +u2) (csub3_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: (csub3 g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 +e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 +e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c4 u +t)).(\lambda (e1: C).(\lambda (H3: (clear (CHead c3 (Bind Abst) t) +e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda +(e2: C).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) +e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind Abst) t) e2)) +(\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) +u) (csub3_abst g c3 c4 H0 u t H2) (clear_bind Abbr c4 u)) e1 (clear_gen_bind +Abst c3 e1 t H3))))))))))) c1 c2 H)))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/defs.ma new file mode 100644 index 000000000..df8b56ada --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/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 *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/defs". + +include "ty3/defs.ma". + +inductive csub3 (g:G): C \to (C \to Prop) \def +| csub3_sort: \forall (n: nat).(csub3 g (CSort n) (CSort n)) +| csub3_head: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall +(k: K).(\forall (u: T).(csub3 g (CHead c1 k u) (CHead c2 k u)))))) +| csub3_void: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall +(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csub3 g +(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2)))))))) +| csub3_abst: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall +(u: T).(\forall (t: T).((ty3 g c2 u t) \to (csub3 g (CHead c1 (Bind Abst) t) +(CHead c2 (Bind Abbr) u))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/drop.ma new file mode 100644 index 000000000..59141ad9c --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/drop.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 *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/drop". + +include "csub3/defs.ma". + +include "drop/fwd.ma". + +theorem csub3_drop_flat: + \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall +(c2: C).((csub3 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).(csub3 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).((csub3 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).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) +u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 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).(csub3 g c c2)) H +(CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let H2 +\def (match H1 in csub3 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: +(csub3 ? c c0)).((eq C c (CHead d1 (Flat f) u)) \to ((eq C c0 c2) \to (ex2 C +(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 +(Flat f) u))))))))) with [(csub3_sort n0) \Rightarrow (\lambda (H2: (eq C +(CSort n0) (CHead d1 (Flat f) u))).(\lambda (H3: (eq C (CSort n0) c2)).((let +H4 \def (eq_ind C (CSort n0) (\lambda (e: C).(match e in C return (\lambda +(_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow +False])) I (CHead d1 (Flat f) u) H2) in (False_ind ((eq C (CSort n0) c2) \to +(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead +d2 (Flat f) u))))) H4)) H3))) | (csub3_head c0 c3 H2 k u0) \Rightarrow +(\lambda (H3: (eq C (CHead c0 k u0) (CHead d1 (Flat f) u))).(\lambda (H4: (eq +C (CHead c3 k u0) c2)).((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 c0 k u0) (CHead d1 (Flat f) u) H3) 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 c0 k u0) +(CHead d1 (Flat f) u) H3) in ((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 u0) (CHead d1 (Flat f) u) H3) in +(eq_ind C d1 (\lambda (c: C).((eq K k (Flat f)) \to ((eq T u0 u) \to ((eq C +(CHead c3 k u0) c2) \to ((csub3 g c c3) \to (ex2 C (\lambda (d2: C).(csub3 g +d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))))) (\lambda +(H8: (eq K k (Flat f))).(eq_ind K (Flat f) (\lambda (k0: K).((eq T u0 u) \to +((eq C (CHead c3 k0 u0) c2) \to ((csub3 g d1 c3) \to (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) +u)))))))) (\lambda (H9: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C +(CHead c3 (Flat f) t) c2) \to ((csub3 g d1 c3) \to (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) +(\lambda (H10: (eq C (CHead c3 (Flat f) u) c2)).(eq_ind C (CHead c3 (Flat f) +u) (\lambda (c: C).((csub3 g d1 c3) \to (ex2 C (\lambda (d2: C).(csub3 g d1 +d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Flat f) u)))))) (\lambda (H11: +(csub3 g d1 c3)).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: +C).(drop O O (CHead c3 (Flat f) u) (CHead d2 (Flat f) u))) c3 H11 (drop_refl +(CHead c3 (Flat f) u)))) c2 H10)) u0 (sym_eq T u0 u H9))) k (sym_eq K k (Flat +f) H8))) c0 (sym_eq C c0 d1 H7))) H6)) H5)) H4 H2))) | (csub3_void c0 c3 H2 b +H3 u1 u2) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind Void) u1) (CHead d1 +(Flat f) u))).(\lambda (H5: (eq C (CHead c3 (Bind b) u2) c2)).((let H6 \def +(eq_ind C (CHead c0 (Bind Void) u1) (\lambda (e: C).(match e 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) u) +H4) in (False_ind ((eq C (CHead c3 (Bind b) u2) c2) \to ((csub3 g c0 c3) \to +((not (eq B b Void)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) H6)) H5 H2 H3))) | +(csub3_abst c0 c3 H2 u0 t H3) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind +Abst) t) (CHead d1 (Flat f) u))).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) +u0) c2)).((let H6 \def (eq_ind C (CHead c0 (Bind Abst) t) (\lambda (e: +C).(match e 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) u) H4) in (False_ind ((eq C (CHead c3 (Bind Abbr) u0) c2) \to +((csub3 g c0 c3) \to ((ty3 g c3 u0 t) \to (ex2 C (\lambda (d2: C).(csub3 g d1 +d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) H6)) H5 H2 +H3)))]) in (H2 (refl_equal C (CHead d1 (Flat f) u)) (refl_equal C +c2)))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: +C).((csub3 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).(csub3 g d1 d2)) (\lambda +(d2: C).(drop n0 O c2 (CHead d2 (Flat f) u)))))))))))).(\lambda (c1: +C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_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).(csub3 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))).(let H2 \def (match H1 in drop return (\lambda (n2: +nat).(\lambda (n3: nat).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop +n2 n3 c c0)).((eq nat n2 (S n0)) \to ((eq nat n3 O) \to ((eq C c (CSort n1)) +\to ((eq C c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 +d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) +u))))))))))))) with [(drop_refl c) \Rightarrow (\lambda (H2: (eq nat O (S +n0))).(\lambda (H3: (eq nat O O)).(\lambda (H4: (eq C c (CSort n1))).(\lambda +(H5: (eq C c (CHead d1 (Flat f) u))).((let H6 \def (eq_ind nat O (\lambda (e: +nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True +| (S _) \Rightarrow False])) I (S n0) H2) in (False_ind ((eq nat O O) \to +((eq C c (CSort n1)) \to ((eq C c (CHead d1 (Flat f) u)) \to (ex2 C (\lambda +(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 +(Flat f) u))))))) H6)) H3 H4 H5))))) | (drop_drop k h c e H2 u0) \Rightarrow +(\lambda (H3: (eq nat (S h) (S n0))).(\lambda (H4: (eq nat O O)).(\lambda +(H5: (eq C (CHead c k u0) (CSort n1))).(\lambda (H6: (eq C e (CHead d1 (Flat +f) u))).((let H7 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat +return (\lambda (_: nat).nat) with [O \Rightarrow h | (S n2) \Rightarrow +n2])) (S h) (S n0) H3) in (eq_ind nat n0 (\lambda (n2: nat).((eq nat O O) \to +((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Flat f) u)) \to +((drop (r k n2) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u))))))))) (\lambda (_: +(eq nat O O)).(\lambda (H9: (eq C (CHead c k u0) (CSort n1))).(let H10 \def +(eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0 in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow +True])) I (CSort n1) H9) in (False_ind ((eq C e (CHead d1 (Flat f) u)) \to +((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u)))))) H10)))) h +(sym_eq nat h n0 H7))) H4 H5 H6 H2))))) | (drop_skip k h d c e H2 u0) +\Rightarrow (\lambda (H3: (eq nat h (S n0))).(\lambda (H4: (eq nat (S d) +O)).(\lambda (H5: (eq C (CHead c k (lift h (r k d) u0)) (CSort n1))).(\lambda +(H6: (eq C (CHead e k u0) (CHead d1 (Flat f) u))).(eq_ind nat (S n0) (\lambda +(n2: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n2 (r k d) u0)) +(CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop n2 (r +k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop +(S n0) O (CSort n1) (CHead d2 (Flat f) u))))))))) (\lambda (H7: (eq nat (S d) +O)).(let H8 \def (eq_ind nat (S d) (\lambda (e0: nat).(match e0 in nat return +(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) +I O H7) in (False_ind ((eq C (CHead c k (lift (S n0) (r k d) u0)) (CSort n1)) +\to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop (S n0) (r k d) c +e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) +O (CSort n1) (CHead d2 (Flat f) u))))))) H8))) h (sym_eq nat h (S n0) H3) H4 +H5 H6 H2)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal +C (CSort n1)) (refl_equal C (CHead d1 (Flat f) u)))))))) (\lambda (c0: +C).(\lambda (c3: C).(\lambda (H1: (csub3 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).(csub3 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).(csub3 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).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) +(ex2 C (\lambda (d2: C).(csub3 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: +(csub3 g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) +u0))).(ex_intro2 C (\lambda (d2: C).(csub3 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).(csub3 g +d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C +(\lambda (d2: C).(csub3 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: (csub3 g +d1 x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C +(\lambda (d2: C).(csub3 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: (csub3 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).(csub3 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).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) +u))) (ex2 C (\lambda (d2: C).(csub3 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: +(csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2 +C (\lambda (d2: C).(csub3 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: (csub3 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).(csub3 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 c3 u +t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O (CHead c0 +(Bind Abst) t) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g +d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C +(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 +(Bind Abbr) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H5: (csub3 +g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C +(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 +(Bind Abbr) u) (CHead d2 (Flat f) u0))) x H5 (drop_drop (Bind Abbr) n0 c3 +(CHead x (Flat f) u0) H6 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) +c0 (CHead d1 (Flat f) u0) t n0 H4))))))))))))) c1 c2 H0)))))) n))). + +theorem csub3_drop_abbr: + \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 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).(csub3 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).((csub3 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).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) +u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 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).(csub3 g c c2)) H +(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in +(let H2 \def (match H1 in csub3 return (\lambda (c: C).(\lambda (c0: +C).(\lambda (_: (csub3 ? c c0)).((eq C c (CHead d1 (Bind Abbr) u)) \to ((eq C +c0 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O +O c2 (CHead d2 (Bind Abbr) u))))))))) with [(csub3_sort n0) \Rightarrow +(\lambda (H2: (eq C (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (H3: (eq C +(CSort n0) c2)).((let H4 \def (eq_ind C (CSort n0) (\lambda (e: C).(match e +in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ +_ _) \Rightarrow False])) I (CHead d1 (Bind Abbr) u) H2) in (False_ind ((eq C +(CSort n0) c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: +C).(drop O O c2 (CHead d2 (Bind Abbr) u))))) H4)) H3))) | (csub3_head c0 c3 +H2 k u0) \Rightarrow (\lambda (H3: (eq C (CHead c0 k u0) (CHead d1 (Bind +Abbr) u))).(\lambda (H4: (eq C (CHead c3 k u0) c2)).((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 c0 k u0) (CHead d1 +(Bind Abbr) u) H3) 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 c0 k u0) (CHead d1 (Bind Abbr) u) H3) in ((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 +u0) (CHead d1 (Bind Abbr) u) H3) in (eq_ind C d1 (\lambda (c: C).((eq K k +(Bind Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c3 k u0) c2) \to ((csub3 g c +c3) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O +c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (H8: (eq K k (Bind +Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead +c3 k0 u0) c2) \to ((csub3 g d1 c3) \to (ex2 C (\lambda (d2: C).(csub3 g d1 +d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u)))))))) (\lambda +(H9: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c3 (Bind Abbr) t) +c2) \to ((csub3 g d1 c3) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) +(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))) (\lambda (H10: +(eq C (CHead c3 (Bind Abbr) u) c2)).(eq_ind C (CHead c3 (Bind Abbr) u) +(\lambda (c: C).((csub3 g d1 c3) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) +(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u)))))) (\lambda (H11: +(csub3 g d1 c3)).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: +C).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) c3 H11 +(drop_refl (CHead c3 (Bind Abbr) u)))) c2 H10)) u0 (sym_eq T u0 u H9))) k +(sym_eq K k (Bind Abbr) H8))) c0 (sym_eq C c0 d1 H7))) H6)) H5)) H4 H2))) | +(csub3_void c0 c3 H2 b H3 u1 u2) \Rightarrow (\lambda (H4: (eq C (CHead c0 +(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(\lambda (H5: (eq C (CHead c3 +(Bind b) u2) c2)).((let H6 \def (eq_ind C (CHead c0 (Bind Void) u1) (\lambda +(e: C).(match e 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 +((eq C (CHead c3 (Bind b) u2) c2) \to ((csub3 g c0 c3) \to ((not (eq B b +Void)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O +O c2 (CHead d2 (Bind Abbr) u))))))) H6)) H5 H2 H3))) | (csub3_abst c0 c3 H2 +u0 t H3) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind Abst) t) (CHead d1 +(Bind Abbr) u))).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) u0) c2)).((let H6 +\def (eq_ind C (CHead c0 (Bind Abst) t) (\lambda (e: C).(match e 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) H4) in (False_ind ((eq C +(CHead c3 (Bind Abbr) u0) c2) \to ((csub3 g c0 c3) \to ((ty3 g c3 u0 t) \to +(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead +d2 (Bind Abbr) u))))))) H6)) H5 H2 H3)))]) in (H2 (refl_equal C (CHead d1 +(Bind Abbr) u)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: +((\forall (c1: C).(\forall (c2: C).((csub3 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).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 +(Bind Abbr) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0: +(csub3 g c1 c2)).(csub3_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).(csub3 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))).(let H2 +\def (match H1 in drop return (\lambda (n2: nat).(\lambda (n3: nat).(\lambda +(c: C).(\lambda (c0: C).(\lambda (_: (drop n2 n3 c c0)).((eq nat n2 (S n0)) +\to ((eq nat n3 O) \to ((eq C c (CSort n1)) \to ((eq C c0 (CHead d1 (Bind +Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop +(S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))))))) with [(drop_refl c) +\Rightarrow (\lambda (H2: (eq nat O (S n0))).(\lambda (H3: (eq nat O +O)).(\lambda (H4: (eq C c (CSort n1))).(\lambda (H5: (eq C c (CHead d1 (Bind +Abbr) u))).((let H6 \def (eq_ind nat O (\lambda (e: nat).(match e in nat +return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow +False])) I (S n0) H2) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) +\to ((eq C c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g +d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) +u))))))) H6)) H3 H4 H5))))) | (drop_drop k h c e H2 u0) \Rightarrow (\lambda +(H3: (eq nat (S h) (S n0))).(\lambda (H4: (eq nat O O)).(\lambda (H5: (eq C +(CHead c k u0) (CSort n1))).(\lambda (H6: (eq C e (CHead d1 (Bind Abbr) +u))).((let H7 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat +return (\lambda (_: nat).nat) with [O \Rightarrow h | (S n2) \Rightarrow +n2])) (S h) (S n0) H3) in (eq_ind nat n0 (\lambda (n2: nat).((eq nat O O) \to +((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Bind Abbr) u)) \to +((drop (r k n2) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))) (\lambda +(_: (eq nat O O)).(\lambda (H9: (eq C (CHead c k u0) (CSort n1))).(let H10 +\def (eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0 in C return (\lambda +(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow +True])) I (CSort n1) H9) in (False_ind ((eq C e (CHead d1 (Bind Abbr) u)) \to +((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))))) H10)))) h +(sym_eq nat h n0 H7))) H4 H5 H6 H2))))) | (drop_skip k h d c e H2 u0) +\Rightarrow (\lambda (H3: (eq nat h (S n0))).(\lambda (H4: (eq nat (S d) +O)).(\lambda (H5: (eq C (CHead c k (lift h (r k d) u0)) (CSort n1))).(\lambda +(H6: (eq C (CHead e k u0) (CHead d1 (Bind Abbr) u))).(eq_ind nat (S n0) +(\lambda (n2: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n2 (r k d) +u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Bind Abbr) u)) \to +((drop n2 (r k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))) (\lambda +(H7: (eq nat (S d) O)).(let H8 \def (eq_ind nat (S d) (\lambda (e0: +nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow +False | (S _) \Rightarrow True])) I O H7) in (False_ind ((eq C (CHead c k +(lift (S n0) (r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 +(Bind Abbr) u)) \to ((drop (S n0) (r k d) c e) \to (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 +(Bind Abbr) u))))))) H8))) h (sym_eq nat h (S n0) H3) H4 H5 H6 H2)))))]) in +(H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal C (CSort n1)) +(refl_equal C (CHead d1 (Bind Abbr) u)))))))) (\lambda (c0: C).(\lambda (c3: +C).(\lambda (H1: (csub3 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).(csub3 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).(csub3 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).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 +(Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csub3 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: (csub3 g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x +(Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 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).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S +n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csub3 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: (csub3 g d1 x)).(\lambda (H5: (drop (S +n0) O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 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: (csub3 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).(csub3 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).(csub3 +g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u))) (ex2 C +(\lambda (d2: C).(csub3 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: (csub3 +g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abbr) u))).(ex_intro2 C +(\lambda (d2: C).(csub3 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: (csub3 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).(csub3 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 +c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O +(CHead c0 (Bind Abst) t) (CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) +u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) +O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: +C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind +Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: +C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0))) x H5 +(drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abbr) u0) H6 u))))) (H c0 c3 H1 +d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Bind Abbr) u0) t n0 +H4))))))))))))) c1 c2 H0)))))) n)). + +theorem csub3_drop_abst: + \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 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).(csub3 g d1 d2)) (\lambda (d2: +C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n +O c2 (CHead d2 (Bind Abbr) u)))) (\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).((csub3 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).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) +t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda +(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) (\lambda (c1: C).(\lambda +(c2: C).(\lambda (H: (csub3 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).(csub3 g c c2)) H (CHead d1 (Bind Abst) t) +(drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in (let H2 \def (match H1 in +csub3 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c +c0)).((eq C c (CHead d1 (Bind Abst) t)) \to ((eq C c0 c2) \to (or (ex2 C +(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 +(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) +u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) with +[(csub3_sort n0) \Rightarrow (\lambda (H2: (eq C (CSort n0) (CHead d1 (Bind +Abst) t))).(\lambda (H3: (eq C (CSort n0) c2)).((let H4 \def (eq_ind C (CSort +n0) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort +_) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind +Abst) t) H2) in (False_ind ((eq C (CSort n0) c2) \to (or (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) +(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: +C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: +C).(\lambda (u: T).(ty3 g d2 u t)))))) H4)) H3))) | (csub3_head c0 c3 H2 k u) +\Rightarrow (\lambda (H3: (eq C (CHead c0 k u) (CHead d1 (Bind Abst) +t))).(\lambda (H4: (eq C (CHead c3 k u) c2)).((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 c0 k u) (CHead d1 +(Bind Abst) t) H3) 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 c0 k u) (CHead d1 (Bind Abst) t) H3) in ((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 +u) (CHead d1 (Bind Abst) t) H3) in (eq_ind C d1 (\lambda (c: C).((eq K k +(Bind Abst)) \to ((eq T u t) \to ((eq C (CHead c3 k u) c2) \to ((csub3 g c +c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O +O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 +(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) +(\lambda (H8: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0: +K).((eq T u t) \to ((eq C (CHead c3 k0 u) c2) \to ((csub3 g d1 c3) \to (or +(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead +d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 +d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) +u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))) (\lambda +(H9: (eq T u t)).(eq_ind T t (\lambda (t0: T).((eq C (CHead c3 (Bind Abst) +t0) c2) \to ((csub3 g d1 c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) +(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda +(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: +T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: +T).(ty3 g d2 u0 t)))))))) (\lambda (H10: (eq C (CHead c3 (Bind Abst) t) +c2)).(eq_ind C (CHead c3 (Bind Abst) t) (\lambda (c: C).((csub3 g d1 c3) \to +(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c +(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c (CHead d2 +(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))) +(\lambda (H11: (csub3 g d1 c3)).(or_introl (ex2 C (\lambda (d2: C).(csub3 g +d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abst) t) (CHead d2 (Bind +Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) +(\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abst) t) (CHead +d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) +(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O +(CHead c3 (Bind Abst) t) (CHead d2 (Bind Abst) t))) c3 H11 (drop_refl (CHead +c3 (Bind Abst) t))))) c2 H10)) u (sym_eq T u t H9))) k (sym_eq K k (Bind +Abst) H8))) c0 (sym_eq C c0 d1 H7))) H6)) H5)) H4 H2))) | (csub3_void c0 c3 +H2 b H3 u1 u2) \Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind Void) u1) +(CHead d1 (Bind Abst) t))).(\lambda (H5: (eq C (CHead c3 (Bind b) u2) +c2)).((let H6 \def (eq_ind C (CHead c0 (Bind Void) u1) (\lambda (e: C).(match +e 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) t) H4) in (False_ind ((eq C +(CHead c3 (Bind b) u2) c2) \to ((csub3 g c0 c3) \to ((not (eq B b Void)) \to +(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 +(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 +(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H6)) +H5 H2 H3))) | (csub3_abst c0 c3 H2 u t0 H3) \Rightarrow (\lambda (H4: (eq C +(CHead c0 (Bind Abst) t0) (CHead d1 (Bind Abst) t))).(\lambda (H5: (eq C +(CHead c3 (Bind Abbr) u) c2)).((let H6 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t0 | +(CHead _ _ t1) \Rightarrow t1])) (CHead c0 (Bind Abst) t0) (CHead d1 (Bind +Abst) t) H4) in ((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 (Bind Abst) t0) (CHead d1 (Bind Abst) t) H4) in +(eq_ind C d1 (\lambda (c: C).((eq T t0 t) \to ((eq C (CHead c3 (Bind Abbr) u) +c2) \to ((csub3 g c c3) \to ((ty3 g c3 u t0) \to (or (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) +(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: +C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: +C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (H8: (eq T t0 +t)).(eq_ind T t (\lambda (t1: T).((eq C (CHead c3 (Bind Abbr) u) c2) \to +((csub3 g d1 c3) \to ((ty3 g c3 u t1) \to (or (ex2 C (\lambda (d2: C).(csub3 +g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C +T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: +C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: +C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))) (\lambda (H9: (eq C (CHead c3 +(Bind Abbr) u) c2)).(eq_ind C (CHead c3 (Bind Abbr) u) (\lambda (c: +C).((csub3 g d1 c3) \to ((ty3 g c3 u t) \to (or (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) t)))) +(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: +C).(\lambda (u0: T).(drop O O c (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: +C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) (\lambda (H10: (csub3 g d1 +c3)).(\lambda (H11: (ty3 g c3 u t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 +g d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind +Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) +(\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead +d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) +(ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda +(d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind +Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) c3 u H10 +(drop_refl (CHead c3 (Bind Abbr) u)) H11)))) c2 H9)) t0 (sym_eq T t0 t H8))) +c0 (sym_eq C c0 d1 H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal C (CHead d1 +(Bind Abst) t)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: +((\forall (c1: C).(\forall (c2: C).((csub3 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).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 +(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) +u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))).(\lambda +(c1: C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_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).(csub3 g d1 d2)) +(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abst) t)))) (ex3_2 C T +(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u)))) (\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))).(let H2 \def (match H1 in drop return (\lambda (n2: nat).(\lambda +(n3: nat).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop n2 n3 c +c0)).((eq nat n2 (S n0)) \to ((eq nat n3 O) \to ((eq C c (CSort n1)) \to ((eq +C c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 +d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) +(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: +C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) +(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))))) with [(drop_refl +c) \Rightarrow (\lambda (H2: (eq nat O (S n0))).(\lambda (H3: (eq nat O +O)).(\lambda (H4: (eq C c (CSort n1))).(\lambda (H5: (eq C c (CHead d1 (Bind +Abst) t))).((let H6 \def (eq_ind nat O (\lambda (e: nat).(match e in nat +return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow +False])) I (S n0) H2) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) +\to ((eq C c (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 +g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) +t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) +(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H6)) H3 H4 H5))))) | +(drop_drop k h c e H2 u) \Rightarrow (\lambda (H3: (eq nat (S h) (S +n0))).(\lambda (H4: (eq nat O O)).(\lambda (H5: (eq C (CHead c k u) (CSort +n1))).(\lambda (H6: (eq C e (CHead d1 (Bind Abst) t))).((let H7 \def (f_equal +nat nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) +with [O \Rightarrow h | (S n2) \Rightarrow n2])) (S h) (S n0) H3) in (eq_ind +nat n0 (\lambda (n2: nat).((eq nat O O) \to ((eq C (CHead c k u) (CSort n1)) +\to ((eq C e (CHead d1 (Bind Abst) t)) \to ((drop (r k n2) O c e) \to (or +(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O +(CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda +(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O +(CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: +T).(ty3 g d2 u0 t)))))))))) (\lambda (_: (eq nat O O)).(\lambda (H9: (eq C +(CHead c k u) (CSort n1))).(let H10 \def (eq_ind C (CHead c k u) (\lambda +(e0: C).(match e0 in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n1) H9) in +(False_ind ((eq C e (CHead d1 (Bind Abst) t)) \to ((drop (r k n0) O c e) \to +(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O +(CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda +(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O +(CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: +T).(ty3 g d2 u0 t))))))) H10)))) h (sym_eq nat h n0 H7))) H4 H5 H6 H2))))) | +(drop_skip k h d c e H2 u) \Rightarrow (\lambda (H3: (eq nat h (S +n0))).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq C (CHead c k (lift h +(r k d) u)) (CSort n1))).(\lambda (H6: (eq C (CHead e k u) (CHead d1 (Bind +Abst) t))).(eq_ind nat (S n0) (\lambda (n2: nat).((eq nat (S d) O) \to ((eq C +(CHead c k (lift n2 (r k d) u)) (CSort n1)) \to ((eq C (CHead e k u) (CHead +d1 (Bind Abst) t)) \to ((drop n2 (r k d) c e) \to (or (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 +(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 +d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 +(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) +(\lambda (H7: (eq nat (S d) O)).(let H8 \def (eq_ind nat (S d) (\lambda (e0: +nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow +False | (S _) \Rightarrow True])) I O H7) in (False_ind ((eq C (CHead c k +(lift (S n0) (r k d) u)) (CSort n1)) \to ((eq C (CHead e k u) (CHead d1 (Bind +Abst) t)) \to ((drop (S n0) (r k d) c e) \to (or (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 +(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 +d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 +(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) +H8))) h (sym_eq nat h (S n0) H3) H4 H5 H6 H2)))))]) in (H2 (refl_equal nat (S +n0)) (refl_equal nat O) (refl_equal C (CSort n1)) (refl_equal C (CHead d1 +(Bind Abst) t)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 +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).(csub3 g d1 d2)) +(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T +(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\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).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 +(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 +d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 k0 u) (CHead +d2 (Bind Abbr) u0)))) (\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).(csub3 g d1 d2)) (\lambda (d2: C).(drop +n0 O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 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u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda +(x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x +(Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) +(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: +C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind +Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex_intro2 +C (\lambda (d2: C).(csub3 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: (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop +n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g +d2 u0 t))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 +d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) +u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) (or (ex2 C +(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 +(Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda +(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O +(CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda +(u0: T).(ty3 g d2 u0 t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: +(csub3 g d1 x0)).(\lambda (H6: (drop n0 O c3 (CHead x0 (Bind Abbr) +x1))).(\lambda (H7: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) +(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead +c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: +T).(ty3 g d2 u0 t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead +c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: +T).(ty3 g d2 u0 t))) x0 x1 H5 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) +x1) H6 u) H7))))))) H4)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind b) c0 (CHead +d1 (Bind Abst) t) u n0 H3)))))))) (\lambda (f: F).(\lambda (u: T).(\lambda +(d1: C).(\lambda (t: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f) u) +(CHead d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) +(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T +(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda +(u0: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: +C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g +d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind +Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) +(\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead +d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) +(\lambda (H4: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop +(S n0) O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g +d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t))) (or +(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O +(CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop +(S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: +C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (x: C).(\lambda (H5: (csub3 +g d1 x)).(\lambda (H6: (drop (S n0) O c3 (CHead x (Bind Abst) t))).(or_introl +(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O +(CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop +(S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: +C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g +d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind +Abst) t))) x H5 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abst) t) H6 u)))))) +H4)) (\lambda (H4: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 +d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O c3 (CHead d2 (Bind +Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))).(ex3_2_ind +C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: +C).(\lambda (u0: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda +(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) (or (ex2 C (\lambda (d2: C).(csub3 +g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 +(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 +d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) +(CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 +t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (csub3 g d1 +x0)).(\lambda (H6: (drop (S n0) O c3 (CHead x0 (Bind Abbr) x1))).(\lambda +(H7: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) +(\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) +t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda +(d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind +Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) +(ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda +(d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind +Abbr) u0)))) (\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))))))) 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: (csub3 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).(csub3 g d1 d2)) (\lambda +(d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda +(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: +T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\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).(csub3 g d1 d2)) (\lambda (d2: +C).(drop n0 O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop +n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +d2 u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: +C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C +T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: +C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind +Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H5: +(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 +(CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) +(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda +(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) +u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead +c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: +T).(ty3 g d2 u t))))) (\lambda (x: C).(\lambda (H6: (csub3 g d1 x)).(\lambda +(H7: (drop n0 O c3 (CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) +(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead +c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: +T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 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: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda +(d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop +n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 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n0) O (CHead c3 (Bind b) +u2) (CHead d2 (Bind Abbr) u)))) (\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))))))) 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: (csub3 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).(csub3 +g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) +(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: +C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda +(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t: +T).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0: T).(\lambda +(H4: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind Abst) +t0))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop +n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead +d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) +(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O +(CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda +(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: +T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) +(\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (H5: (ex2 C +(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 +(Bind Abst) t0))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))) (or (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) +(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead +c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: +T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H6: (csub3 g d1 +x)).(\lambda (H7: (drop n0 O c3 (CHead x (Bind Abst) t0))).(or_introl (ex2 C +(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 +(Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop +(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: +C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2: C).(csub3 +g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 +(Bind Abst) t0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abst) t0) +H7 u)))))) H5)) (\lambda (H5: (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead +d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 +t0))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) +(\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) +(\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))) (or (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) +(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead +c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: +T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: +(csub3 g d1 x0)).(\lambda (H7: (drop n0 O c3 (CHead x0 (Bind Abbr) +x1))).(\lambda (H8: (ty3 g x0 x1 t0)).(or_intror (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) +(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead +c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: +T).(ty3 g d2 u0 t0)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead +c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: +T).(ty3 g d2 u0 t0))) x0 x1 H6 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind +Abbr) x1) H7 u) H8))))))) H5)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) +c0 (CHead d1 (Bind Abst) t0) t n0 H4))))))))))))) c1 c2 H0)))))) n)). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/fwd.ma new file mode 100644 index 000000000..99b9dae96 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/fwd.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 *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/fwd". + +include "csub3/defs.ma". + +theorem csub3_gen_abbr: + \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csub3 g +(CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 +(Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2))))))) +\def + \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda +(H: (csub3 g (CHead e1 (Bind Abbr) v) c2)).(let H0 \def (match H in csub3 +return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c c0)).((eq C c +(CHead e1 (Bind Abbr) v)) \to ((eq C c0 c2) \to (ex2 C (\lambda (e2: C).(eq C +c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))))) with +[(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind +Abbr) v))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort +n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abbr) +v) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2 C (\lambda (e2: C).(eq C +c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))) H2)) H1))) +| (csub3_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) +(CHead e1 (Bind Abbr) v))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3 +\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) H1) in ((let H4 \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) H1) +in ((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 k u) (CHead e1 (Bind Abbr) v) H1) in (eq_ind C e1 (\lambda (c: +C).((eq K k (Bind Abbr)) \to ((eq T u v) \to ((eq C (CHead c0 k u) c2) \to +((csub3 g c c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) +v))) (\lambda (e2: C).(csub3 g e1 e2)))))))) (\lambda (H6: (eq K k (Bind +Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u v) \to ((eq C (CHead +c0 k0 u) c2) \to ((csub3 g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead +e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2))))))) (\lambda (H7: (eq +T u v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c0 (Bind Abbr) t) c2) \to +((csub3 g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) +v))) (\lambda (e2: C).(csub3 g e1 e2)))))) (\lambda (H8: (eq C (CHead c0 +(Bind Abbr) v) c2)).(eq_ind C (CHead c0 (Bind Abbr) v) (\lambda (c: +C).((csub3 g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) +v))) (\lambda (e2: C).(csub3 g e1 e2))))) (\lambda (H9: (csub3 g e1 c0)).(let +H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind Abbr) v) c)) +H (CHead c0 (Bind Abbr) v) H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead +c0 (Bind Abbr) v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 +e2)) c0 (refl_equal C (CHead c0 (Bind Abbr) v)) H9))) c2 H8)) u (sym_eq T u v +H7))) k (sym_eq K k (Bind Abbr) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 +H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C +(CHead c1 (Bind Void) u1) (CHead e1 (Bind Abbr) v))).(\lambda (H3: (eq C +(CHead c0 (Bind b) u2) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Void) u1) +(\lambda (e: C).(match e 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) H2) in (False_ind ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) +\to ((not (eq B b Void)) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind +Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))) H4)) H3 H0 H1))) | +(csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind +Abst) t) (CHead e1 (Bind Abbr) v))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) +u) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: +C).(match e 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) H2) in (False_ind +((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g c1 c0) \to ((ty3 g c0 u t) +\to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: +C).(csub3 g e1 e2)))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead e1 +(Bind Abbr) v)) (refl_equal C c2))))))). + +theorem csub3_gen_abst: + \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csub3 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).(csub3 g e1 e2))) (ex3_2 C T (\lambda +(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: +C).(\lambda (_: T).(csub3 g e1 e2))) (\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: (csub3 g (CHead e1 (Bind Abst) v1) c2)).(let H0 \def (match H in csub3 +return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c c0)).((eq C c +(CHead e1 (Bind Abst) v1)) \to ((eq C c0 c2) \to (or (ex2 C (\lambda (e2: +C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) +(ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) +v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: +C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))) with [(csub3_sort n) +\Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind Abst) +v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort n) +(\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abst) +v1) H0) in (False_ind ((eq C (CSort n) c2) \to (or (ex2 C (\lambda (e2: +C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) +(ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) +v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: +C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) H2)) H1))) | (csub3_head c1 c0 H0 k +u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind Abst) +v1))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3 \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) H1) in ((let H4 \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) H1) in ((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 k +u) (CHead e1 (Bind Abst) v1) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k +(Bind Abst)) \to ((eq T u v1) \to ((eq C (CHead c0 k u) c2) \to ((csub3 g c +c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) +(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: +T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: +T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 +v1)))))))))) (\lambda (H6: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) +(\lambda (k0: K).((eq T u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 g e1 +c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) +(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: +T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: +T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 +v1))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C +(CHead c0 (Bind Abst) t) c2) \to ((csub3 g e1 c0) \to (or (ex2 C (\lambda +(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 +e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind +Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: +C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (H8: (eq C (CHead c0 +(Bind Abst) v1) c2)).(eq_ind C (CHead c0 (Bind Abst) v1) (\lambda (c: +C).((csub3 g e1 c0) \to (or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind +Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: +C).(\lambda (v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: +C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g +e2 v2 v1))))))) (\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r C c2 +(\lambda (c: C).(csub3 g (CHead e1 (Bind Abst) v1) c)) H (CHead c0 (Bind +Abst) v1) H8) in (or_introl (ex2 C (\lambda (e2: C).(eq C (CHead c0 (Bind +Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) +(ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abst) v1) +(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 +e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex_intro2 C +(\lambda (e2: C).(eq C (CHead c0 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) +(\lambda (e2: C).(csub3 g e1 e2)) c0 (refl_equal C (CHead c0 (Bind Abst) v1)) +H9)))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind Abst) H6))) c1 +(sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) +\Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind +Abst) v1))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def +(eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match e 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) H2) in (False_ind ((eq +C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) +\to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda +(e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C +c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 +e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) H4)) H3 H0 +H1))) | (csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C (CHead +c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1))).(\lambda (H3: (eq C (CHead c0 +(Bind Abbr) u) c2)).((let H4 \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) H2) in +((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 Abst) t) (CHead e1 (Bind Abst) v1) H2) in (eq_ind C e1 (\lambda (c: +C).((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g c c0) +\to ((ty3 g c0 u t) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind +Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: +C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: +C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g +e2 v2 v1)))))))))) (\lambda (H6: (eq T t v1)).(eq_ind T v1 (\lambda (t0: +T).((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u +t0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) +(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: +T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: +T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 +v1))))))))) (\lambda (H7: (eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C +(CHead c0 (Bind Abbr) u) (\lambda (c: C).((csub3 g e1 c0) \to ((ty3 g c0 u +v1) \to (or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abst) v1))) +(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: +T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: +T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 +v1)))))))) (\lambda (H8: (csub3 g e1 c0)).(\lambda (H9: (ty3 g c0 u v1)).(let +H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind Abst) v1) +c)) H (CHead c0 (Bind Abbr) u) H7) in (or_intror (ex2 C (\lambda (e2: C).(eq +C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda (e2: +C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C +(CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: +C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g +e2 v2 v1)))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead +c0 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: +T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c0 +u (refl_equal C (CHead c0 (Bind Abbr) u)) H8 H9))))) c2 H7)) t (sym_eq T t v1 +H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead +e1 (Bind Abst) v1)) (refl_equal C c2))))))). + +theorem csub3_gen_bind: + \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall +(v1: T).((csub3 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).(csub3 g e1 e2)))))))))) +\def + \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda +(v1: T).(\lambda (H: (csub3 g (CHead e1 (Bind b1) v1) c2)).(let H0 \def +(match H in csub3 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 +? c c0)).((eq C c (CHead e1 (Bind b1) v1)) \to ((eq C c0 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).(csub3 g e1 +e2)))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) +(CHead e1 (Bind b1) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def +(eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow +False])) I (CHead e1 (Bind b1) v1) H0) in (False_ind ((eq C (CSort n) 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).(csub3 g e1 e2)))))) H2)) H1))) | (csub3_head c1 c0 H0 k u) \Rightarrow +(\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2: +(eq C (CHead c0 k u) c2)).((let H3 \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) H1) in ((let H4 \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) H1) in ((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 k u) (CHead e1 (Bind b1) v1) H1) in +(eq_ind C e1 (\lambda (c: C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((eq C +(CHead c0 k u) c2) \to ((csub3 g c c0) \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).(csub3 g e1 e2)))))))))) +(\lambda (H6: (eq K k (Bind b1))).(eq_ind K (Bind b1) (\lambda (k0: K).((eq T +u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 g e1 c0) \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).(csub3 g e1 +e2))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C +(CHead c0 (Bind b1) t) c2) \to ((csub3 g e1 c0) \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).(csub3 g e1 +e2)))))))) (\lambda (H8: (eq C (CHead c0 (Bind b1) v1) c2)).(eq_ind C (CHead +c0 (Bind b1) v1) (\lambda (c: C).((csub3 g e1 c0) \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).(csub3 g e1 e2))))))) +(\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r C c2 (\lambda (c: +C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b1) v1) H8) in +(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C +(CHead c0 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda +(e2: C).(\lambda (_: T).(csub3 g e1 e2)))) b1 c0 v1 (refl_equal C (CHead c0 +(Bind b1) v1)) H9))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind b1) +H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 +u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead e1 +(Bind b1) v1))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \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) H2) 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 b0) \Rightarrow b0 | (Flat _) \Rightarrow Void])])) (CHead c1 (Bind +Void) u1) (CHead e1 (Bind b1) v1) H2) in ((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 Void) u1) (CHead e1 (Bind +b1) v1) H2) in (eq_ind C e1 (\lambda (c: C).((eq B Void b1) \to ((eq T u1 v1) +\to ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c c0) \to ((not (eq B b +Void)) \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).(csub3 g e1 e2))))))))))) (\lambda (H7: (eq B Void +b1)).(eq_ind B Void (\lambda (_: B).((eq T u1 v1) \to ((eq C (CHead c0 (Bind +b) u2) c2) \to ((csub3 g e1 c0) \to ((not (eq B b Void)) \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).(csub3 g e1 +e2)))))))))) (\lambda (H8: (eq T u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq C +(CHead c0 (Bind b) u2) c2) \to ((csub3 g e1 c0) \to ((not (eq B b Void)) \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).(csub3 g e1 e2))))))))) (\lambda (H9: (eq C (CHead c0 (Bind b) u2) +c2)).(eq_ind C (CHead c0 (Bind b) u2) (\lambda (c: C).((csub3 g e1 c0) \to +((not (eq B b Void)) \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).(csub3 g e1 e2)))))))) (\lambda (H10: +(csub3 g e1 c0)).(\lambda (_: (not (eq B b Void))).(let H12 \def (eq_ind_r C +c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b) +u2) H9) in (let H13 \def (eq_ind_r B b1 (\lambda (b0: B).(csub3 g (CHead e1 +(Bind b0) v1) (CHead c0 (Bind b) u2))) H12 Void H7) in (ex2_3_intro B C T +(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind b) +u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csub3 g e1 e2)))) b c0 u2 (refl_equal C (CHead c0 (Bind b) u2)) H10))))) +c2 H9)) u1 (sym_eq T u1 v1 H8))) b1 H7)) c1 (sym_eq C c1 e1 H6))) H5)) H4)) +H3 H0 H1))) | (csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C +(CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(\lambda (H3: (eq C (CHead +c0 (Bind Abbr) u) c2)).((let H4 \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) H2) in +((let H5 \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) H2) in +((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) H2) in (eq_ind C e1 (\lambda (c: +C).((eq B Abst b1) \to ((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) +\to ((csub3 g c c0) \to ((ty3 g c0 u t) \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).(csub3 g e1 e2))))))))))) +(\lambda (H7: (eq B Abst b1)).(eq_ind B Abst (\lambda (_: B).((eq T t v1) \to +((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u t) +\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).(csub3 g e1 e2)))))))))) (\lambda (H8: (eq T t v1)).(eq_ind T v1 (\lambda +(t0: T).((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g +c0 u t0) \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).(csub3 g e1 e2))))))))) (\lambda (H9: (eq C (CHead c0 +(Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: +C).((csub3 g e1 c0) \to ((ty3 g c0 u 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).(csub3 g e1 e2)))))))) +(\lambda (H10: (csub3 g e1 c0)).(\lambda (_: (ty3 g c0 u v1)).(let H12 \def +(eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead +c0 (Bind Abbr) u) H9) in (let H13 \def (eq_ind_r B b1 (\lambda (b: B).(csub3 +g (CHead e1 (Bind b) v1) (CHead c0 (Bind Abbr) u))) H12 Abst H7) in +(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C +(CHead c0 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda +(e2: C).(\lambda (_: T).(csub3 g e1 e2)))) Abbr c0 u (refl_equal C (CHead c0 +(Bind Abbr) u)) H10))))) c2 H9)) t (sym_eq T t v1 H8))) b1 H7)) c1 (sym_eq C +c1 e1 H6))) H5)) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead e1 (Bind b1) +v1)) (refl_equal C c2)))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/getl.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/getl.ma new file mode 100644 index 000000000..68f1d825e --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/getl.ma @@ -0,0 +1,390 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/getl". + +include "csub3/fwd.ma". + +include "csub3/clear.ma". + +include "csub3/drop.ma". + +include "getl/clear.ma". + +theorem csub3_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).((csub3 g +c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 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).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 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))).((match x in C return (\lambda (c: C).((drop n O c1 c) \to ((clear +c (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C +(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind +Abbr) u))))))))) with [(CSort n0) \Rightarrow (\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).((csub3 +g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl +n c2 (CHead d2 (Bind Abbr) u))))))))) | (CHead c k t) \Rightarrow (\lambda +(H3: (drop n O c1 (CHead c k t))).(\lambda (H4: (clear (CHead c k t) (CHead +d1 (Bind Abbr) u))).((match k in K return (\lambda (k0: K).((drop n O c1 +(CHead c k0 t)) \to ((clear (CHead c k0 t) (CHead d1 (Bind Abbr) u)) \to +(\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 +d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))))))))) with [(Bind +b) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Bind b) t))).(\lambda +(H6: (clear (CHead c (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 c0 _ _) \Rightarrow c0])) (CHead d1 (Bind +Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (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 c (Bind b) t) (clear_gen_bind b c (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 c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind +Abbr) u) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 +c)).(\lambda (c2: C).(\lambda (H12: (csub3 g c1 c2)).(let H13 \def (eq_ind_r +T t (\lambda (t0: T).(drop n O c1 (CHead c (Bind b) t0))) H5 u H9) in (let +H14 \def (eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead c (Bind b0) u))) +H13 Abbr H10) in (let H15 \def (eq_ind_r C c (\lambda (c0: C).(drop n O c1 +(CHead c0 (Bind Abbr) u))) H14 d1 H11) in (ex2_ind C (\lambda (d2: C).(csub3 +g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (ex2 C +(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind +Abbr) u)))) (\lambda (x0: C).(\lambda (H16: (csub3 g d1 x0)).(\lambda (H17: +(drop n O c2 (CHead x0 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csub3 +g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) x0 H16 +(getl_intro n c2 (CHead x0 (Bind Abbr) u) (CHead x0 (Bind Abbr) u) H17 +(clear_bind Abbr x0 u)))))) (csub3_drop_abbr g n c1 c2 H12 d1 u H15)))))))))) +H8)) H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Flat +f) t))).(\lambda (H6: (clear (CHead c (Flat f) t) (CHead d1 (Bind Abbr) +u))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop n O c0 (CHead c +(Flat f) t)) \to (\forall (c2: C).((csub3 g c0 c2) \to (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) +u)))))))) (nat_ind (\lambda (n0: nat).(\forall (x0: C).((drop n0 O x0 (CHead +c (Flat f) t)) \to (\forall (c2: C).((csub3 g x0 c2) \to (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) +u))))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) +t))).(\lambda (c2: C).(\lambda (H9: (csub3 g x0 c2)).(let H10 \def (eq_ind C +x0 (\lambda (c0: C).(csub3 g c0 c2)) H9 (CHead c (Flat f) t) (drop_gen_refl +x0 (CHead c (Flat f) t) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind +Abbr) u) (clear_gen_flat f c (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 +\def (csub3_clear_conf g (CHead c (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u) +H_y) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead d1 (Bind Abbr) u) e2)) +(\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) +(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x1: +C).(\lambda (H12: (csub3 g (CHead d1 (Bind Abbr) u) x1)).(\lambda (H13: +(clear c2 x1)).(let H14 \def (csub3_gen_abbr g d1 x1 u H12) in (ex2_ind C +(\lambda (e2: C).(eq C x1 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csub3 +g d1 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2: C).(\lambda (H15: (eq C x1 +(CHead x2 (Bind Abbr) u))).(\lambda (H16: (csub3 g d1 x2)).(let H17 \def +(eq_ind C x1 (\lambda (c0: C).(clear c2 c0)) H13 (CHead x2 (Bind Abbr) u) +H15) in (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl +O c2 (CHead d2 (Bind Abbr) u))) x2 H16 (getl_intro O c2 (CHead x2 (Bind Abbr) +u) c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: +nat).(\lambda (H8: ((\forall (x0: C).((drop n0 O x0 (CHead c (Flat f) t)) \to +(\forall (c2: C).((csub3 g x0 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 +d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda +(x0: C).(\lambda (H9: (drop (S n0) O x0 (CHead c (Flat f) t))).(\lambda (c2: +C).(\lambda (H10: (csub3 g x0 c2)).(let H11 \def (drop_clear x0 (CHead c +(Flat f) t) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: +C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) (\lambda (_: +B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead c (Flat f) t))))) (ex2 +C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead +d2 (Bind Abbr) u)))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: +T).(\lambda (H12: (clear x0 (CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n0 +O x2 (CHead c (Flat f) t))).(let H14 \def (csub3_clear_conf g x0 c2 H10 +(CHead x2 (Bind x1) x3) H12) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead +x2 (Bind x1) x3) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) +u)))) (\lambda (x4: C).(\lambda (H15: (csub3 g (CHead x2 (Bind x1) x3) +x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csub3_gen_bind g x1 x2 x4 +x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: +T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csub3 g x2 e2)))) (ex2 C (\lambda (d2: C).(csub3 g d1 +d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda +(x5: B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 +(Bind x5) x7))).(\lambda (H19: (csub3 g x2 x6)).(let H20 \def (eq_ind C x4 +(\lambda (c0: C).(clear c2 c0)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 +\def (H8 x2 H13 x6 H19) in (ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) +(\lambda (d2: C).(getl n0 x6 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) +u)))) (\lambda (x8: C).(\lambda (H22: (csub3 g d1 x8)).(\lambda (H23: (getl +n0 x6 (CHead x8 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 +d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x8 H22 +(getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abbr) u) n0 H23))))) +H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))]) H3 H4)))]) H1 H2)))) +H0))))))). + +theorem csub3_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).((csub3 g +c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: +C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n +c2 (CHead d2 (Bind Abbr) u)))) (\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).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 +d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T +(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\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))).((match x in C return +(\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 (Bind Abst) t)) \to +(\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 +d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T +(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: +T).(ty3 g d2 u t)))))))))) with [(CSort n0) \Rightarrow (\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).((csub3 +g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: +C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u +t)))))))))) | (CHead c k t0) \Rightarrow (\lambda (H3: (drop n O c1 (CHead c +k t0))).(\lambda (H4: (clear (CHead c k t0) (CHead d1 (Bind Abst) +t))).((match k in K return (\lambda (k0: K).((drop n O c1 (CHead c k0 t0)) +\to ((clear (CHead c k0 t0) (CHead d1 (Bind Abst) t)) \to (\forall (c2: +C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u +t)))))))))) with [(Bind b) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c +(Bind b) t0))).(\lambda (H6: (clear (CHead c (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 c0 _ _) +\Rightarrow c0])) (CHead d1 (Bind Abst) t) (CHead c (Bind b) t0) +(clear_gen_bind b c (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 c (Bind b) t0) +(clear_gen_bind b c (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 c (Bind b) t0) (clear_gen_bind b c (CHead d1 (Bind Abst) t) +t0 H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 c)).(\lambda +(c2: C).(\lambda (H12: (csub3 g c1 c2)).(let H13 \def (eq_ind_r T t0 (\lambda +(t1: T).(drop n O c1 (CHead c (Bind b) t1))) H5 t H9) in (let H14 \def +(eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead c (Bind b0) t))) H13 Abst +H10) in (let H15 \def (eq_ind_r C c (\lambda (c0: C).(drop n O c1 (CHead c0 +(Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 +d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T +(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: +T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u +t))))) (\lambda (H16: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: +C).(drop n O c2 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))) +(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 +(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 +(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (x0: C).(\lambda (H17: (csub3 g d1 x0)).(\lambda (H18: (drop n O c2 +(CHead x0 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 +d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T +(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: +T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(getl n c2 (CHead d2 (Bind Abst) t))) x0 H17 (getl_intro n c2 (CHead +x0 (Bind Abst) t) (CHead x0 (Bind Abst) t) H18 (clear_bind Abst x0 t))))))) +H16)) (\lambda (H16: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) +u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T +(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: +T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u +t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H17: (csub3 g d1 +x0)).(\lambda (H18: (drop n O c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H19: +(ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u +t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) +(\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) +(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x0 x1 H17 (getl_intro n c2 +(CHead x0 (Bind Abbr) x1) (CHead x0 (Bind Abbr) x1) H18 (clear_bind Abbr x0 +x1)) H19))))))) H16)) (csub3_drop_abst g n c1 c2 H12 d1 t H15)))))))))) H8)) +H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Flat f) +t0))).(\lambda (H6: (clear (CHead c (Flat f) t0) (CHead d1 (Bind Abst) +t))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop n O c0 (CHead c +(Flat f) t0)) \to (\forall (c2: C).((csub3 g c0 c2) \to (or (ex2 C (\lambda +(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) +t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda +(d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: +C).(\lambda (u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: +nat).(\forall (x0: C).((drop n0 O x0 (CHead c (Flat f) t0)) \to (\forall (c2: +C).((csub3 g x0 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(getl n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl +n0 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 +u t)))))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) +t0))).(\lambda (c2: C).(\lambda (H9: (csub3 g x0 c2)).(let H10 \def (eq_ind C +x0 (\lambda (c0: C).(csub3 g c0 c2)) H9 (CHead c (Flat f) t0) (drop_gen_refl +x0 (CHead c (Flat f) t0) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind +Abst) t) (clear_gen_flat f c (CHead d1 (Bind Abst) t) t0 H6) f t0) in (let +H11 \def (csub3_clear_conf g (CHead c (Flat f) t0) c2 H10 (CHead d1 (Bind +Abst) t) H_y) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead d1 (Bind Abst) +t) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csub3 g +d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T +(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: +T).(ty3 g d2 u t))))) (\lambda (x1: C).(\lambda (H12: (csub3 g (CHead d1 +(Bind Abst) t) x1)).(\lambda (H13: (clear c2 x1)).(let H14 \def +(csub3_gen_abst g d1 x1 t H12) in (or_ind (ex2 C (\lambda (e2: C).(eq C x1 +(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2))) (ex3_2 C T +(\lambda (e2: C).(\lambda (v2: T).(eq C x1 (CHead e2 (Bind Abbr) v2)))) +(\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda +(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) +(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda +(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: +T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: +T).(ty3 g d2 u t))))) (\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x1 (CHead +e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2)))).(ex2_ind C (\lambda +(e2: C).(eq C x1 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2)) +(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 +(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 +(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (x2: C).(\lambda (H16: (eq C x1 (CHead x2 (Bind Abst) t))).(\lambda +(H17: (csub3 g d1 x2)).(let H18 \def (eq_ind C x1 (\lambda (c0: C).(clear c2 +c0)) H13 (CHead x2 (Bind Abst) t) H16) in (or_introl (ex2 C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) +(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: +C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: +C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g +d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x2 H17 +(getl_intro O c2 (CHead x2 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) +(\lambda (H15: (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x1 (CHead +e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2))) +(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))))).(ex3_2_ind C T (\lambda +(e2: C).(\lambda (v2: T).(eq C x1 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: +C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g +e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: +C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O +c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u +t))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (H16: (eq C x1 (CHead x2 +(Bind Abbr) x3))).(\lambda (H17: (csub3 g d1 x2)).(\lambda (H18: (ty3 g x2 x3 +t)).(let H19 \def (eq_ind C x1 (\lambda (c0: C).(clear c2 c0)) H13 (CHead x2 +(Bind Abbr) x3) H16) in (or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) +(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda +(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: +T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: +T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 +g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) +u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x2 x3 H17 (getl_intro +O c2 (CHead x2 (Bind Abbr) x3) c2 (drop_refl c2) H19) H18)))))))) H15)) +H14))))) H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x0: +C).((drop n0 O x0 (CHead c (Flat f) t0)) \to (\forall (c2: C).((csub3 g x0 +c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl +n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 +(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u +t))))))))))).(\lambda (x0: C).(\lambda (H9: (drop (S n0) O x0 (CHead c (Flat +f) t0))).(\lambda (c2: C).(\lambda (H10: (csub3 g x0 c2)).(let H11 \def +(drop_clear x0 (CHead c (Flat f) t0) n0 H9) in (ex2_3_ind B C T (\lambda (b: +B).(\lambda (e: C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) +(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead c (Flat +f) t0))))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: +C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl +(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +d2 u t))))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: +(clear x0 (CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n0 O x2 (CHead c +(Flat f) t0))).(let H14 \def (csub3_clear_conf g x0 c2 H10 (CHead x2 (Bind +x1) x3) H12) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead x2 (Bind x1) x3) +e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csub3 g d1 +d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T +(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t))))) (\lambda (x4: C).(\lambda (H15: (csub3 g (CHead x2 +(Bind x1) x3) x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def +(csub3_gen_bind g x1 x2 x4 x3 H15) in (ex2_3_ind B C T (\lambda (b2: +B).(\lambda (e2: C).(\lambda (v2: T).(eq C x4 (CHead e2 (Bind b2) v2))))) +(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g x2 e2)))) (or (ex2 +C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead +d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) +u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x5: +B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind +x5) x7))).(\lambda (H19: (csub3 g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda +(c0: C).(clear c2 c0)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 +x2 H13 x6 H19) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(getl n0 x6 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl +n0 x6 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 +u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl +(S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda +(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 +(CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u +t))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: +C).(getl n0 x6 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: +C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 x6 (CHead d2 (Bind Abst) t))) +(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 +(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: +T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead +d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) +(\lambda (x8: C).(\lambda (H23: (csub3 g d1 x8)).(\lambda (H24: (getl n0 x6 +(CHead x8 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 +d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T +(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda +(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda +(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) +(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x8 H23 +(getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abst) t) n0 H24)))))) H22)) +(\lambda (H22: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) +(\lambda (d2: C).(\lambda (u: T).(getl n0 x6 (CHead d2 (Bind Abbr) u)))) +(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda +(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: +T).(getl n0 x6 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: +T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl +(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +d2 u t))))) (\lambda (x8: C).(\lambda (x9: T).(\lambda (H23: (csub3 g d1 +x8)).(\lambda (H24: (getl n0 x6 (CHead x8 (Bind Abbr) x9))).(\lambda (H25: +(ty3 g x8 x9 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda +(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl +(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g +d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 +d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) +u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x8 x9 H23 +(getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abbr) x9) n0 H24) H25))))))) +H22)) H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))]) H3 H4)))]) H1 +H2)))) H0))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/pc3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/pc3.ma new file mode 100644 index 000000000..980672598 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/pc3.ma @@ -0,0 +1,58 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/pc3". + +include "csub3/getl.ma". + +include "pc3/left.ma". + +theorem csub3_pr2: + \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 +t1 t2) \to (\forall (c2: C).((csub3 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).((csub3 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 (_: (csub3 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: (csub3 g c c2)).(let H4 \def (csub3_getl_abbr g c d u i H0 +c2 H3) in (ex2_ind C (\lambda (d2: C).(csub3 g d d2)) (\lambda (d2: C).(getl +i c2 (CHead d2 (Bind Abbr) u))) (pr2 c2 t3 t) (\lambda (x: C).(\lambda (_: +(csub3 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))))). + +theorem csub3_pc3: + \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 +t1 t2) \to (\forall (c2: C).((csub3 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).((csub3 g c1 c2) \to (pc3 c2 t t0))))) (\lambda (t: +T).(\lambda (c2: C).(\lambda (_: (csub3 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).((csub3 g c1 +c2) \to (pc3 c2 t3 t4))))).(\lambda (c2: C).(\lambda (H3: (csub3 g c1 +c2)).(pc3_pr2_u c2 t3 t0 (csub3_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).((csub3 g c1 c2) \to (pc3 c2 t0 t4))))).(\lambda (c2: C).(\lambda +(H3: (csub3 g c1 c2)).(pc3_t t0 c2 t3 (pc3_pr2_x c2 t3 t0 (csub3_pr2 g c1 t0 +t3 H0 c2 H3)) t4 (H2 c2 H3)))))))))) t1 t2 H))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/props.ma new file mode 100644 index 000000000..3258cda57 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/props.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 *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/props". + +include "csub3/defs.ma". + +theorem csub3_refl: + \forall (g: G).(\forall (c: C).(csub3 g c c)) +\def + \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csub3 g c0 c0)) +(\lambda (n: nat).(csub3_sort g n)) (\lambda (c0: C).(\lambda (H: (csub3 g c0 +c0)).(\lambda (k: K).(\lambda (t: T).(csub3_head g c0 c0 H k t))))) c)). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/ty3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/ty3.ma new file mode 100644 index 000000000..26ad14f35 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/ty3.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 *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/csub3/ty3". + +include "csub3/pc3.ma". + +include "csub3/props.ma". + +theorem csub3_ty3: + \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 +t1 t2) \to (\forall (c2: C).((csub3 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).((csub3 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).((csub3 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).((csub3 g c c2) \to (ty3 g c2 u t3))))).(\lambda (H4: (pc3 c t3 +t0)).(\lambda (c2: C).(\lambda (H5: (csub3 g c c2)).(ty3_conv g c2 t0 t (H1 +c2 H5) u t3 (H3 c2 H5) (csub3_pc3 g c t3 t0 H4 c2 H5)))))))))))))) (\lambda +(c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (csub3 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).((csub3 g +d c2) \to (ty3 g c2 u t))))).(\lambda (c2: C).(\lambda (H3: (csub3 g c +c2)).(let H4 \def (csub3_getl_abbr g c d u n H0 c2 H3) in (ex2_ind C (\lambda +(d2: C).(csub3 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: +(csub3 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).((csub3 g d c2) \to (ty3 g c2 u t))))).(\lambda (c2: +C).(\lambda (H3: (csub3 g c c2)).(let H4 \def (csub3_getl_abst g c d u n H0 +c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d d2)) (\lambda (d2: +C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex3_2 C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n +c2 (CHead d2 (Bind Abbr) u0)))) (\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).(csub3 g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) +u))))).(ex2_ind C (\lambda (d2: C).(csub3 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: (csub3 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: (ex3_2 +C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d d2))) (\lambda (d2: +C).(\lambda (u0: T).(getl n c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: +C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex3_2_ind C T (\lambda (d2: +C).(\lambda (_: T).(csub3 g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n +c2 (CHead d2 (Bind Abbr) u0)))) (\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 (_: (csub3 g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind +Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 u)).(ty3_abbr g n c2 x0 x1 H7 u +H8)))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t: +T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2: C).((csub3 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).((csub3 g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0 t3))))).(\lambda +(t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3 t4)).(\lambda (H5: +((\forall (c2: C).((csub3 g (CHead c (Bind b) u) c2) \to (ty3 g c2 t3 +t4))))).(\lambda (c2: C).(\lambda (H6: (csub3 g c c2)).(ty3_bind g c2 u t (H1 +c2 H6) b t0 t3 (H3 (CHead c2 (Bind b) u) (csub3_head g c c2 H6 (Bind b) u)) +t4 (H5 (CHead c2 (Bind b) u) (csub3_head g c c2 H6 (Bind b) +u)))))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda +(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((csub3 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).((csub3 g c c2) \to (ty3 g +c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csub3 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).((csub3 g c c2) \to (ty3 g c2 t0 t3))))).(\lambda (t4: +T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c2: C).((csub3 g c +c2) \to (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (csub3 g c +c2)).(ty3_cast g c2 t0 t3 (H1 c2 H4) t4 (H3 c2 H4)))))))))))) c1 t1 t2 H))))). + +theorem csub3_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)).(csub3_ty3 g (CHead c (Bind Abst) v) t1 t2 H0 (CHead +c (Bind Abbr) u) (csub3_abst g c c (csub3_refl g c) u v H))))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/fwd.ma index a9ad986e1..d9081db2a 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/nf2/fwd.ma @@ -20,38 +20,7 @@ include "nf2/defs.ma". include "pr2/clen.ma". -theorem nf2_gen_base__aux: - \forall (k: K).(\forall (t: T).(\forall (u: T).((eq T (THead k u t) t) \to -(\forall (P: Prop).P)))) -\def - \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).((eq -T (THead k u t0) t0) \to (\forall (P: Prop).P)))) (\lambda (n: nat).(\lambda -(u: T).(\lambda (H: (eq T (THead k u (TSort n)) (TSort n))).(\lambda (P: -Prop).(let H0 \def (eq_ind T (THead k u (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 (H: (eq T -(THead k u (TLRef n)) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind T -(THead k u (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 (u: T).((eq T (THead -k u t0) t0) \to (\forall (P: Prop).P))))).(\lambda (t1: T).(\lambda (H0: -((\forall (u: T).((eq T (THead k u t1) t1) \to (\forall (P: -Prop).P))))).(\lambda (u: T).(\lambda (H1: (eq T (THead k u (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 u (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 u | -(TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2])) (THead k u (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 u (THead k0 t0 t1)) (THead k0 t0 t1) H1) in -(\lambda (_: (eq T u t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind K k -(\lambda (k1: K).(\forall (u0: T).((eq T (THead k1 u0 t1) t1) \to (\forall -(P0: Prop).P0)))) H0 k0 H6) in (H7 t0 H4 P))))) H3)) H2)))))))))) t)). +include "T/props.ma". theorem nf2_gen_lref: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c @@ -92,7 +61,7 @@ theorem nf2_gen_cast: 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).(nf2_gen_base__aux (Flat Cast) t u (H t +(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_epsilon t t (pr0_refl t) u))) P))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec.ma new file mode 100644 index 000000000..761030b2a --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec.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 *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec". + +include "ty3/arity_props.ma". + +include "ty3/pr3.ma". + +include "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 (\forall (P: Prop).P)))))))))) +\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 (\forall (P: Prop).P))) (\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 (\forall (P: Prop).P))) (\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 (\forall (P: +Prop).P))) (\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 +(\forall (P: Prop).P)) (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 (\forall (P: Prop).P)) (\lambda (H9: (pc3 c u1 u2)).(\lambda (P: +Prop).(let H10 \def H9 in (ex2_ind T (\lambda (t: T).(pr3 c u1 t)) (\lambda +(t: T).(pr3 c u2 t)) P (\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)) P (\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)) P))))))) H13)))))) H10)))))) H7)))))) H4)))))) +H1)))))))))))). + +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 (\forall (P: Prop).P))))))))))) +\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 (\forall (P: Prop).P)))) (\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 (\forall (P: Prop).P)))) (\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 (\forall (P: Prop).P)))) (\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 (\forall (P: Prop).P)))) +(\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 (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 (\forall (P: Prop).P))) +(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))) (ty3_sred_pr3 c u1 (THead (Bind +Abst) x0 x1) H13 g t1 H) 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 (\forall (P: Prop).P))) (\lambda (u: +T).(\lambda (H11: (pc3 c u1 (THead (Bind Abst) u2 u))).(\lambda (P: +Prop).(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)) P (\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)) P (\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))))) P (\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))))) P (\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 (P0: Prop).P0)))) 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 (and_ind (nf2 c x5) +(nf2 (CHead c (Bind Abst) x5) x6) P (\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_equal 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))) P)))) H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x +H4))))) H12))))))) H9)))))) H6)))))) H3)))))))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fsubst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fsubst0.ma new file mode 100644 index 000000000..675efe302 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fsubst0.ma @@ -0,0 +1,719 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fsubst0". + +include "pc3/left.ma". + +include "fsubst0/defs.ma". + +include "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)))). + +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)))). + +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)))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fwd.ma new file mode 100644 index 000000000..a393d0f18 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fwd.ma @@ -0,0 +1,279 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/fwd". + +include "pc3/props.ma". + +include "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))))). + +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))))))). + +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))))))))). + +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). + +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)))))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/left.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/left.ma new file mode 100644 index 000000000..5caeee043 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/left.ma @@ -0,0 +1,109 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/left". + +include "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)))). + +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)))). + +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)))). + +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))))). + +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_pr2_u c t3 t0 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)))). + +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))))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/pc1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/pc1.ma new file mode 100644 index 000000000..6bc140307 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/pc1.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 *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/pc1". + +include "pc3/defs.ma". + +include "pc1/defs.ma". + +include "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))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/subst1.ma new file mode 100644 index 000000000..c2cbcd118 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/subst1.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 *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/subst1". + +include "pc3/props.ma". + +include "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))))))))))))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/wcpr0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/wcpr0.ma new file mode 100644 index 000000000..5a8fb4625 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/wcpr0.ma @@ -0,0 +1,103 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/wcpr0". + +include "pc3/props.ma". + +include "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 (t0: T).(\lambda (t3: T).(\lambda (H1: (pr2 (CHead c1 k u) t3 +t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead c1 k u) t0 t4)).(\lambda (H3: +(pc3 (CHead c2 k u) t0 t4)).(pc3_t t0 (CHead c2 k u) t3 (let H4 \def (match +H1 in pr2 return (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).(\lambda +(_: (pr2 c t t5)).((eq C c (CHead c1 k u)) \to ((eq T t t3) \to ((eq T t5 t0) +\to (pc3 (CHead c2 k u) t3 t0)))))))) with [(pr2_free c t5 t6 H4) \Rightarrow +(\lambda (H5: (eq C c (CHead c1 k u))).(\lambda (H6: (eq T t5 t3)).(\lambda +(H7: (eq T t6 t0)).(eq_ind C (CHead c1 k u) (\lambda (_: C).((eq T t5 t3) \to +((eq T t6 t0) \to ((pr0 t5 t6) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda +(H8: (eq T t5 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t6 t0) \to ((pr0 t t6) +\to (pc3 (CHead c2 k u) t3 t0)))) (\lambda (H9: (eq T t6 t0)).(eq_ind T t0 +(\lambda (t: T).((pr0 t3 t) \to (pc3 (CHead c2 k u) t3 t0))) (\lambda (H10: +(pr0 t3 t0)).(pc3_pr2_r (CHead c2 k u) t3 t0 (pr2_free (CHead c2 k u) t3 t0 +H10))) t6 (sym_eq T t6 t0 H9))) t5 (sym_eq T t5 t3 H8))) c (sym_eq C c (CHead +c1 k u) H5) H6 H7 H4)))) | (pr2_delta c d u0 i H4 t5 t6 H5 t H6) \Rightarrow +(\lambda (H7: (eq C c (CHead c1 k u))).(\lambda (H8: (eq T t5 t3)).(\lambda +(H9: (eq T t t0)).(eq_ind C (CHead c1 k u) (\lambda (c0: C).((eq T t5 t3) \to +((eq T t t0) \to ((getl i c0 (CHead d (Bind Abbr) u0)) \to ((pr0 t5 t6) \to +((subst0 i u0 t6 t) \to (pc3 (CHead c2 k u) t3 t0))))))) (\lambda (H10: (eq T +t5 t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t t0) \to ((getl i (CHead c1 k +u) (CHead d (Bind Abbr) u0)) \to ((pr0 t7 t6) \to ((subst0 i u0 t6 t) \to +(pc3 (CHead c2 k u) t3 t0)))))) (\lambda (H11: (eq T t t0)).(eq_ind T t0 +(\lambda (t7: T).((getl i (CHead c1 k u) (CHead d (Bind Abbr) u0)) \to ((pr0 +t3 t6) \to ((subst0 i u0 t6 t7) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda +(H12: (getl i (CHead c1 k u) (CHead d (Bind Abbr) u0))).(\lambda (H13: (pr0 +t3 t6)).(\lambda (H14: (subst0 i u0 t6 t0)).(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) t3 t0) (\lambda (x0: C).(\lambda (x1: +T).(\lambda (H15: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda +(_: (wcpr0 d x0)).(\lambda (H17: (pr0 u0 x1)).(ex2_ind T (\lambda (t7: +T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t0 t7)) (pc3 (CHead c2 k u) t3 +t0) (\lambda (x: T).(\lambda (H18: (subst0 i x1 t6 x)).(\lambda (H19: (pr0 t0 +x)).(pc3_pr2_u (CHead c2 k u) x t3 (pr2_delta (CHead c2 k u) x0 x1 i H15 t3 +t6 H13 x H18) t0 (pc3_pr2_x (CHead c2 k u) x t0 (pr2_free (CHead c2 k u) t0 x +H19)))))) (pr0_subst0_fwd u0 t6 t0 i H14 x1 H17))))))) (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) H12))))) t (sym_eq T t t0 H11))) t5 (sym_eq T t5 t3 H10))) c (sym_eq C +c (CHead c1 k u) H7) H8 H9 H4 H5 H6))))]) in (H4 (refl_equal C (CHead c1 k +u)) (refl_equal T t3) (refl_equal T t0))) t4 H3))))))) t1 t2 H0)))))))). + +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))). + +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))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma index 7f7761df1..2c1ae46cc 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma @@ -324,3 +324,55 @@ include "pc3/defs.ma". include "pc3/props.ma". +include "pc3/pc1.ma". + +include "pc3/wcpr0.ma". + +include "pc3/left.ma". + +include "pc3/fwd.ma". + +include "pc3/fsubst0.ma". + +include "pc3/subst1.ma". + +include "ty3/defs.ma". + +include "ty3/fwd.ma". + +include "ty3/props.ma". + +include "ty3/fsubst0.ma". + +include "ty3/subst1.ma". + +include "csub3/defs.ma". + +include "csub3/fwd.ma". + +include "csub3/props.ma". + +include "csub3/clear.ma". + +include "csub3/drop.ma". + +include "csub3/getl.ma". + +include "csub3/pc3.ma". + +include "csub3/ty3.ma". + +include "ty3/pr3.ma". + +include "ty3/pr3_props.ma". + +include "ty3/tau0.ma". + +include "ty3/arity.ma". + +include "ty3/arity_props.ma". + +include "pc3/dec.ma". + +include "ty3/dec.ma". + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity.ma new file mode 100644 index 000000000..23f81bacb --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity.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 *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity". + +include "ty3/defs.ma". + +include "arity/pr3.ma". + +include "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_lift g d u (asucc g x0) (arity_repl g d u x +H4 (asucc g x0) H7) c0 (S n) O (getl_drop Abst c0 d u n H0))))) 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))))).(\lambda (t0: T).(\lambda (H4: (ty3 g (CHead c0 (Bind b) u) t4 +t0)).(\lambda (H5: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 +a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t0 (asucc g a1))))).(let +H6 \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 (H7: (arity g c0 u x)).(\lambda (_: (arity +g c0 t (asucc g x))).(let H9 \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 (H10: (arity g (CHead c0 (Bind b) u) t3 +x0)).(\lambda (H11: (arity g (CHead c0 (Bind b) u) t4 (asucc g x0))).(let H_x +\def (leq_asucc g x) in (let H12 \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 (H13: (leq g x (asucc g x1))).((match b in B return +(\lambda (b0: B).((ty3 g (CHead c0 (Bind b0) u) t4 t0) \to ((ex2 A (\lambda +(a1: A).(arity g (CHead c0 (Bind b0) u) t4 a1)) (\lambda (a1: A).(arity g +(CHead c0 (Bind b0) u) t0 (asucc g a1)))) \to ((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))))))))) with [Abbr +\Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind Abbr) u) t4 t0)).(\lambda (_: +(ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Abbr) u) t4 a1)) (\lambda +(a1: A).(arity g (CHead c0 (Bind Abbr) u) t0 (asucc g a1))))).(\lambda (H16: +(arity g (CHead c0 (Bind Abbr) u) t3 x0)).(\lambda (H17: (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 H7 t3 x0 H16) +(arity_bind g Abbr not_abbr_abst c0 u x H7 t4 (asucc g x0) H17)))))) | Abst +\Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind Abst) u) t4 t0)).(\lambda (_: +(ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Abst) u) t4 a1)) (\lambda +(a1: A).(arity g (CHead c0 (Bind Abst) u) t0 (asucc g a1))))).(\lambda (H16: +(arity g (CHead c0 (Bind Abst) u) t3 x0)).(\lambda (H17: (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 +H7 (asucc g x1) H13) t3 x0 H16) (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 H7 (asucc +g x1) H13) t4 (asucc g x0) H17) (asucc g (AHead x1 x0)) (leq_refl g (asucc g +(AHead x1 x0))))))))) | Void \Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind +Void) u) t4 t0)).(\lambda (_: (ex2 A (\lambda (a1: A).(arity g (CHead c0 +(Bind Void) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind Void) u) t0 +(asucc g a1))))).(\lambda (H16: (arity g (CHead c0 (Bind Void) u) t3 +x0)).(\lambda (H17: (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 not_void_abst c0 u x H7 t3 x0 H16) (arity_bind g Void +not_void_abst c0 u x H7 t4 (asucc g x0) H17))))))]) H4 H5 H10 H11))) +H12)))))) H9))))) H6))))))))))))))) (\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_equal 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 (_: (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 t4 (asucc g +a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3 x)).(\lambda (H6: (arity +g c0 t4 (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat +Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) x +(arity_cast g c0 t4 x H6 t3 H5) H6)))) H4)))))))))) c t1 t2 H))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity_props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity_props.ma new file mode 100644 index 000000000..098a32c0b --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity_props.ma @@ -0,0 +1,78 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity_props". + +include "ty3/arity.ma". + +include "ty3/fwd.ma". + +include "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 (ex4_3_ind T T T (\lambda (t2: +T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) v t2) u)))) +(\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c v t0)))) (\lambda +(t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) v) t +t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead c +(Bind Abst) v) t2 t1)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: +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)).(\lambda (_: (ty3 g +(CHead c (Bind Abst) v) x0 x2)).(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 H6 \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 (H7: (arity g c (THead (Bind Abst) v t) x)).(\lambda +(H8: (arity g c v (asucc g x))).(let H9 \def (arity_gen_abst g c v t x H7) 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 (x3: +A).(\lambda (x4: A).(\lambda (H10: (eq A x (AHead x3 x4))).(\lambda (H11: +(arity g c v (asucc g x3))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t +x4)).(let H13 \def (eq_ind A x (\lambda (a: A).(arity g c v (asucc g a))) H8 +(AHead x3 x4) H10) in (leq_ahead_asucc_false g x3 (asucc g x4) (arity_mono g +c v (asucc g (AHead x3 x4)) H13 (asucc g x3) H11) P))))))) H9))))) +H6))))))))))) (ty3_gen_bind g Abst c v t u H1)))))))))). + +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)))))))))). + +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))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec.ma new file mode 100644 index 000000000..efbd85e19 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec.ma @@ -0,0 +1,456 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/dec". + +include "ty3/pr3_props.ma". + +include "pc3/dec.ma". + +include "getl/flt.ma". + +include "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 (\forall (P: +Prop).P)))))) +\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 (\forall (P: Prop).P)))))) (\lambda (c2: C).(\lambda +(t2: T).(match t2 in T return (\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 (\forall (P: Prop).P)))))))) \to (or +(ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) +\to (\forall (P: Prop).P)))))) with [(TSort n) \Rightarrow (\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 +(\forall (P: Prop).P))))))))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 +(TSort n) t3))) (\forall (t3: T).((ty3 g c2 (TSort n) t3) \to (\forall (P: +Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TSort n) t3)) (TSort (next +g n)) (ty3_sort g c2 n)))) | (TLRef n) \Rightarrow (\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 (\forall +(P: Prop).P))))))))).(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 (\forall (P: Prop).P)))) (\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 (\forall (P: +Prop).P)))) (\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 (\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 +(\forall (P: Prop).P)))) (\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 +(\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H5: (ty3 g x0 x2 +x)).((match x1 in B return (\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 (\forall (P: Prop).P)))))) with [Abbr +\Rightarrow (\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 (\forall (P: Prop).P))) (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)))) | Abst +\Rightarrow (\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 (\forall (P: Prop).P))) (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)))) | Void +\Rightarrow (\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 (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H7: (ty3 +g c2 (TLRef n) t3)).(\lambda (P: Prop).(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))))) P (\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)))) P (\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 P +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)))) P (\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 P +H13))))))))) H8)) (ty3_gen_lref g c2 t3 n H7)))))))]) H2))) H4)) (\lambda +(H4: ((\forall (t3: T).((ty3 g x0 x2 t3) \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 (\forall (P: Prop).P))) +(\lambda (t3: T).(\lambda (H5: (ty3 g c2 (TLRef n) t3)).(\lambda (P: +Prop).(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))))) P (\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)))) P (\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 P)))))))) 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)))) P (\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 P))))))))) 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 (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H2: (ty3 g c2 (TLRef +n) t3)).(\lambda (P: Prop).(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))))) P (\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)))) P (\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 +P))))))) 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)))) P (\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 +P))))))) H3)) (ty3_gen_lref g c2 t3 n H2))))))) H0)))) | (THead k t t0) +\Rightarrow (\lambda (H: ((\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 (\forall (P: Prop).P))))))))).((match k in K +return (\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 (\forall (P: Prop).P)))))))) \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 (\forall (P: Prop).P)))))) with [(Bind b) \Rightarrow +(\lambda (H0: ((\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 (\forall (P: Prop).P))))))))).(let H1 \def (H0 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 (\forall (P: Prop).P))) (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 (\forall (P: Prop).P)))) (\lambda +(H2: (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 (\forall +(P: Prop).P)))) (\lambda (x: T).(\lambda (H3: (ty3 g c2 t x)).(let H4 \def +(H0 (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 (\forall (P: Prop).P))) (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 (\forall (P: Prop).P)))) (\lambda (H5: (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 (\forall (P: Prop).P)))) (\lambda (x0: T).(\lambda (H6: (ty3 g +(CHead c2 (Bind b) t) t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 +(Bind b) t) x0 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 (\forall +(P: Prop).P)))) (\lambda (x1: T).(\lambda (H7: (ty3 g (CHead c2 (Bind b) t) +x0 x1)).(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 (\forall (P: +Prop).P))) (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 H3 b t0 x0 H6 x1 H7))))) +(ty3_correct g (CHead c2 (Bind b) t) t0 x0 H6)))) H5)) (\lambda (H5: +((\forall (t3: T).((ty3 g (CHead c2 (Bind b) t) t0 t3) \to (\forall (P: +Prop).P))))).(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 (\forall +(P: Prop).P))) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Bind b) t t0) +t3)).(\lambda (P: Prop).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: +T).(\lambda (_: T).(pc3 c2 (THead (Bind b) t t4) t3)))) (\lambda (_: +T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c2 t t5)))) (\lambda (t4: +T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4)))) +(\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c2 (Bind b) +t) t4 t6)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda +(_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda (_: (ty3 g c2 t x1)).(\lambda +(H9: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(\lambda (_: (ty3 g (CHead c2 (Bind +b) t) x0 x2)).(H5 x0 H9 P)))))))) (ty3_gen_bind g b c2 t t0 t3 H6))))))) +H4)))) H2)) (\lambda (H2: ((\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: +Prop).P))))).(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 (\forall +(P: Prop).P))) (\lambda (t3: T).(\lambda (H3: (ty3 g c2 (THead (Bind b) t t0) +t3)).(\lambda (P: Prop).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: +T).(\lambda (_: T).(pc3 c2 (THead (Bind b) t t4) t3)))) (\lambda (_: +T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c2 t t5)))) (\lambda (t4: +T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4)))) +(\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c2 (Bind b) +t) t4 t6)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda +(_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda (H5: (ty3 g c2 t +x1)).(\lambda (_: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(\lambda (_: (ty3 g +(CHead c2 (Bind b) t) x0 x2)).(H2 x1 H5 P)))))))) (ty3_gen_bind g b c2 t t0 +t3 H3))))))) H1))) | (Flat f) \Rightarrow (\lambda (H0: ((\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 +(\forall (P: Prop).P))))))))).((match f in F return (\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 (\forall (P: Prop).P)))))))) \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 (\forall (P: Prop).P)))))) with [Appl \Rightarrow (\lambda (H1: +((\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 (\forall (P: Prop).P))))))))).(let H2 \def (H1 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 (\forall (P: Prop).P))) (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 (\forall (P: Prop).P)))) +(\lambda (H3: (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 (\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H4: (ty3 g c2 t +x)).(let H5 \def (H1 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 +(\forall (P: Prop).P))) (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 (\forall (P: Prop).P)))) (\lambda (H6: (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 (\forall (P: Prop).P)))) (\lambda (x0: +T).(\lambda (H7: (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 (\forall (P: +Prop).P)))) (\lambda (x1: T).(\lambda (H8: (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 (\forall (P: Prop).P)))) (\lambda (x2: T).(\lambda (H9: (ty3 g c2 x +x2)).(let H10 \def (ty3_sn3 g c2 x x2 H9) in (let H_x \def (nf2_sn3 c2 x H10) +in (let H11 \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 (\forall +(P: Prop).P)))) (\lambda (x3: T).(\lambda (H12: (pr3 c2 x x3)).(\lambda (H13: +(nf2 c2 x3)).(let H14 \def (ty3_sred_pr3 c2 x x3 H12 g x2 H9) in (let H_x0 +\def (pc3_abst_dec g c2 x0 x1 H8 x3 x2 H14) in (let H15 \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 (\forall (P: Prop).P))) (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 (\forall (P: Prop).P)))) (\lambda (H16: (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 (\forall (P: Prop).P)))) (\lambda (x4: T).(\lambda (x5: +T).(\lambda (H17: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H18: (ty3 +g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H19: (pr3 c2 x3 x5)).(\lambda +(_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H19 H13) in (let H21 +\def (eq_ind_r T x5 (\lambda (t3: T).(pr3 c2 x3 t3)) H19 x3 H_y) in (let H22 +\def (eq_ind_r T x5 (\lambda (t3: T).(ty3 g c2 (THead (Bind Abst) t3 x4) x1)) +H18 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 +(\forall (P: Prop).P))) (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 H4 x3 H12) t0 x4 (ty3_conv g c2 (THead (Bind Abst) +x3 x4) x1 H22 t0 x0 H7 H17))))))))))))) H16)) (\lambda (H16: ((\forall (u: +T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to (\forall (P: +Prop).P))))).(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 +(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H17: (ty3 g c2 (THead +(Flat Appl) t t0) t3)).(\lambda (P: Prop).(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))) P (\lambda (x4: +T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind +Abst) x4 x5)) t3)).(\lambda (H19: (ty3 g c2 t0 (THead (Bind Abst) x4 +x5))).(\lambda (H20: (ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H20 +x H4) in (let H_y0 \def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H19 x0 +H7) in (H16 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 H12)) (Bind Abst) x5)) P)))))))) +(ty3_gen_appl g c2 t t0 t3 H17))))))) H15))))))) H11)))))) (ty3_correct g c2 +t x H4)))) (ty3_correct g c2 t0 x0 H7)))) H6)) (\lambda (H6: ((\forall (t3: +T).((ty3 g c2 t0 t3) \to (\forall (P: Prop).P))))).(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 (\forall (P: Prop).P))) (\lambda (t3: +T).(\lambda (H7: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(\lambda (P: +Prop).(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))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat +Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H9: (ty3 g c2 t0 (THead +(Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H6 (THead (Bind Abst) x0 +x1) H9 P)))))) (ty3_gen_appl g c2 t t0 t3 H7))))))) H5)))) H3)) (\lambda (H3: +((\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))))).(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 (\forall (P: Prop).P))) +(\lambda (t3: T).(\lambda (H4: (ty3 g c2 (THead (Flat Appl) t t0) +t3)).(\lambda (P: Prop).(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))) P (\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 (H7: (ty3 +g c2 t x0)).(H3 x0 H7 P)))))) (ty3_gen_appl g c2 t t0 t3 H4))))))) H2))) | +Cast \Rightarrow (\lambda (H1: ((\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 (\forall (P: +Prop).P))))))))).(let H2 \def (H1 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 (\forall (P: Prop).P))) (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 (\forall (P: Prop).P)))) (\lambda (H3: (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 (\forall (P: Prop).P)))) (\lambda (x: +T).(\lambda (H4: (ty3 g c2 t x)).(let H5 \def (H1 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 (\forall (P: Prop).P))) (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 (\forall (P: Prop).P)))) (\lambda (H6: (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 (\forall (P: +Prop).P)))) (\lambda (x0: T).(\lambda (H7: (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 (\forall (P: Prop).P)))) (\lambda (x1: T).(\lambda (H8: (ty3 g +c2 x0 x1)).(let H_x \def (pc3_dec g c2 x0 x1 H8 t x H4) in (let H9 \def H_x +in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0 t) \to (\forall (P: Prop).P)) (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 (\forall (P: Prop).P)))) +(\lambda (H10: (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 (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 +(THead (Flat Cast) t t0) t3)) t (ty3_cast g c2 t0 t (ty3_conv g c2 t x H4 t0 +x0 H7 H10) x H4)))) (\lambda (H10: (((pc3 c2 x0 t) \to (\forall (P: +Prop).P)))).(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 +(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H11: (ty3 g c2 (THead +(Flat Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 +t) P (\lambda (_: (pc3 c2 t t3)).(\lambda (H13: (ty3 g c2 t0 t)).(let H_y +\def (ty3_unique g c2 t0 t H13 x0 H7) in (H10 (pc3_s c2 x0 t H_y) P)))) +(ty3_gen_cast g c2 t0 t t3 H11))))))) H9))))) (ty3_correct g c2 t0 x0 H7)))) +H6)) (\lambda (H6: ((\forall (t3: T).((ty3 g c2 t0 t3) \to (\forall (P: +Prop).P))))).(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 +(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H7: (ty3 g c2 (THead (Flat +Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P +(\lambda (_: (pc3 c2 t t3)).(\lambda (H9: (ty3 g c2 t0 t)).(H6 t H9 P))) +(ty3_gen_cast g c2 t0 t t3 H7))))))) H5)))) H3)) (\lambda (H3: ((\forall (t3: +T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))))).(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 (\forall (P: Prop).P))) (\lambda (t3: +T).(\lambda (H4: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(\lambda (P: +Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P (\lambda (_: (pc3 c2 t +t3)).(\lambda (H6: (ty3 g c2 t0 t)).(ex_ind T (\lambda (t4: T).(ty3 g c2 t +t4)) P (\lambda (x: T).(\lambda (H7: (ty3 g c2 t x)).(H3 x H7 P))) +(ty3_correct g c2 t0 t H6)))) (ty3_gen_cast g c2 t0 t t3 H4))))))) H2)))]) +H0))]) H))]))) c t1))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/defs.ma new file mode 100644 index 000000000..1eecab2ff --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/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 *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/defs". + +include "G/defs.ma". + +include "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 (\forall (t0: T).((ty3 g (CHead c (Bind b) u) t2 t0) \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) +t2)))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fsubst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fsubst0.ma new file mode 100644 index 000000000..3a7c41e39 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fsubst0.ma @@ -0,0 +1,978 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fsubst0". + +include "ty3/props.ma". + +include "pc3/fsubst0.ma". + +include "csubst0/props.ma". + +include "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))).(and_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))).(and_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))).(and_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))).(and_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 (_: +(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 (t4: T).(\lambda (H4: (ty3 +g (CHead c (Bind b) u) t3 t4)).(\lambda (_: ((\forall (i: nat).(\forall (u0: +T).(\forall (c2: C).(\forall (t5: T).((fsubst0 i u0 (CHead c (Bind b) u) t3 +c2 t5) \to (\forall (e: C).((getl i (CHead c (Bind b) u) (CHead e (Bind Abbr) +u0)) \to (ty3 g c2 t5 t4)))))))))).(\lambda (i: nat).(\lambda (u0: +T).(\lambda (c2: C).(\lambda (t5: T).(\lambda (H6: (fsubst0 i u0 c (THead +(Bind b) u t2) c2 t5)).(fsubst0_ind i u0 c (THead (Bind b) u t2) (\lambda +(c0: C).(\lambda (t6: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) +\to (ty3 g c0 t6 (THead (Bind b) u t3)))))) (\lambda (t6: T).(\lambda (H7: +(subst0 i u0 (THead (Bind b) u t2) t6)).(\lambda (e: C).(\lambda (H8: (getl i +c (CHead e (Bind Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t6 (THead +(Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t7: +T).(eq T t6 (THead (Bind b) u t7))) (\lambda (t7: T).(subst0 (s (Bind b) i) +u0 t2 t7))) (ex3_2 T T (\lambda (u2: T).(\lambda (t7: T).(eq T t6 (THead +(Bind b) u2 t7)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) +(\lambda (_: T).(\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7)))) (ty3 g c +t6 (THead (Bind b) u t3)) (\lambda (H9: (ex2 T (\lambda (u2: T).(eq T t6 +(THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T +(\lambda (u2: T).(eq T t6 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i +u0 u u2)) (ty3 g c t6 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H10: +(eq T t6 (THead (Bind b) x t2))).(\lambda (H11: (subst0 i u0 u x)).(eq_ind_r +T (THead (Bind b) x t2) (\lambda (t7: T).(ty3 g c t7 (THead (Bind b) u t3))) +(ex_ind T (\lambda (t7: T).(ty3 g (CHead c (Bind b) u) t4 t7)) (ty3 g c +(THead (Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H12: +(ty3 g (CHead c (Bind b) u) t4 x0)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead +c (Bind b) x) t3 t7)) (ty3 g c (THead (Bind b) x t2) (THead (Bind b) u t3)) +(\lambda (x1: T).(\lambda (H13: (ty3 g (CHead c (Bind b) x) t3 x1)).(ty3_conv +g c (THead (Bind b) u t3) (THead (Bind b) u t4) (ty3_bind g c u t0 H0 b t3 t4 +H4 x0 H12) (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 H11) e H8) 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 H11 c)) e (getl_head (Bind b) i c (CHead e +(Bind Abbr) u0) H8 u)) x1 H13) (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 H11 t3 (Bind b))) e H8)))) (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 H11 c)) e +(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H8 u)))))) (ty3_correct g +(CHead c (Bind b) u) t3 t4 H4)) t6 H10)))) H9)) (\lambda (H9: (ex2 T (\lambda +(t7: T).(eq T t6 (THead (Bind b) u t7))) (\lambda (t7: T).(subst0 (s (Bind b) +i) u0 t2 t7)))).(ex2_ind T (\lambda (t7: T).(eq T t6 (THead (Bind b) u t7))) +(\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7)) (ty3 g c t6 (THead (Bind +b) u t3)) (\lambda (x: T).(\lambda (H10: (eq T t6 (THead (Bind b) u +x))).(\lambda (H11: (subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead (Bind +b) u x) (\lambda (t7: T).(ty3 g c t7 (THead (Bind b) u t3))) (ex_ind T +(\lambda (t7: T).(ty3 g (CHead c (Bind b) u) t3 t7)) (ty3 g c (THead (Bind b) +u x) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H12: (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 H11) e (getl_head +(Bind b) i c (CHead e (Bind Abbr) u0) H8 u)) x0 H12))) (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 H11) e (getl_head (Bind b) i c (CHead e (Bind Abbr) +u0) H8 u)))) t6 H10)))) H9)) (\lambda (H9: (ex3_2 T T (\lambda (u2: +T).(\lambda (t7: T).(eq T t6 (THead (Bind b) u2 t7)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t7: +T).(subst0 (s (Bind b) i) u0 t2 t7))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t7: T).(eq T t6 (THead (Bind b) u2 t7)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t7: +T).(subst0 (s (Bind b) i) u0 t2 t7))) (ty3 g c t6 (THead (Bind b) u t3)) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T t6 (THead (Bind b) x0 +x1))).(\lambda (H11: (subst0 i u0 u x0)).(\lambda (H12: (subst0 (s (Bind b) +i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t7: T).(ty3 g c t7 +(THead (Bind b) u t3))) (ex_ind T (\lambda (t7: T).(ty3 g (CHead c (Bind b) +u) t4 t7)) (ty3 g c (THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda +(x: T).(\lambda (H13: (ty3 g (CHead c (Bind b) u) t4 x)).(ex_ind T (\lambda +(t7: T).(ty3 g (CHead c (Bind b) x0) t3 t7)) (ty3 g c (THead (Bind b) x0 x1) +(THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (H14: (ty3 g (CHead c (Bind +b) x0) t3 x2)).(ty3_conv g c (THead (Bind b) u t3) (THead (Bind b) u t4) +(ty3_bind g c u t0 H0 b t3 t4 H4 x H13) (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 H11) e +H8) 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 H12 (CHead c (Bind b) x0) (csubst0_snd_bind b i u0 +u x0 H11 c)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H8 u)) x2 +H14) (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 H11 t3 +(Bind b))) e H8)))) (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 +H12 (CHead c (Bind b) x0) (csubst0_snd_bind b i u0 u x0 H11 c)) e (getl_head +(Bind b) i c (CHead e (Bind Abbr) u0) H8 u)))))) (ty3_correct g (CHead c +(Bind b) u) t3 t4 H4)) t6 H10)))))) H9)) (subst0_gen_head (Bind b) u0 u t2 t6 +i H7)))))) (\lambda (c3: C).(\lambda (H7: (csubst0 i u0 c c3)).(\lambda (e: +C).(\lambda (H8: (getl i c (CHead e (Bind Abbr) u0))).(ex_ind T (\lambda (t6: +T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead (Bind b) u t2) +(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: (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 H7) e +H8) 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 +H7 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H8 u)) x H9))) +(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 H7 u)) e (getl_head (Bind b) i c (CHead e (Bind +Abbr) u0) H8 u)))))))) (\lambda (t6: T).(\lambda (H7: (subst0 i u0 (THead +(Bind b) u t2) t6)).(\lambda (c3: C).(\lambda (H8: (csubst0 i u0 c +c3)).(\lambda (e: C).(\lambda (H9: (getl i c (CHead e (Bind Abbr) +u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t6 (THead (Bind b) u2 t2))) +(\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t7: T).(eq T t6 (THead +(Bind b) u t7))) (\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7))) (ex3_2 T +T (\lambda (u2: T).(\lambda (t7: T).(eq T t6 (THead (Bind b) u2 t7)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: +T).(\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7)))) (ty3 g c3 t6 (THead +(Bind b) u t3)) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t6 (THead (Bind +b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: +T).(eq T t6 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)) +(ty3 g c3 t6 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (eq T t6 +(THead (Bind b) x t2))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead +(Bind b) x t2) (\lambda (t7: T).(ty3 g c3 t7 (THead (Bind b) u t3))) (ex_ind +T (\lambda (t7: T).(ty3 g (CHead c3 (Bind b) u) t3 t7)) (ty3 g c3 (THead +(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H13: (ty3 g +(CHead c3 (Bind b) u) t3 x0)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c3 +(Bind b) u) x0 t7)) (ty3 g c3 (THead (Bind b) x t2) (THead (Bind b) u t3)) +(\lambda (x1: T).(\lambda (H14: (ty3 g (CHead c3 (Bind b) u) x0 x1)).(ex_ind +T (\lambda (t7: T).(ty3 g (CHead c3 (Bind b) x) t3 t7)) (ty3 g c3 (THead +(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (H15: (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 H8) +e H9) b t3 x0 H13 x1 H14) (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 H12 c3 H8) e H9) 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 H12 c c3 +H8)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H9 u)) x2 H15) +(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 H12 t3 (Bind b)) c3 +H8) e H9)))) (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 H12 c c3 H8)) e (getl_head (Bind b) +i c (CHead e (Bind Abbr) u0) H9 u)))))) (ty3_correct g (CHead c3 (Bind b) u) +t3 x0 H13)))) (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 H8 u)) e (getl_head (Bind b) i c +(CHead e (Bind Abbr) u0) H9 u)))) t6 H11)))) H10)) (\lambda (H10: (ex2 T +(\lambda (t7: T).(eq T t6 (THead (Bind b) u t7))) (\lambda (t7: T).(subst0 (s +(Bind b) i) u0 t2 t7)))).(ex2_ind T (\lambda (t7: T).(eq T t6 (THead (Bind b) +u t7))) (\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7)) (ty3 g c3 t6 +(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (eq T t6 (THead (Bind +b) u x))).(\lambda (H12: (subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead +(Bind b) u x) (\lambda (t7: T).(ty3 g c3 t7 (THead (Bind b) u t3))) (ex_ind T +(\lambda (t7: T).(ty3 g (CHead c3 (Bind b) u) t3 t7)) (ty3 g c3 (THead (Bind +b) u x) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H13: (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 H8) e H9) 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 H12 (CHead c3 (Bind b) u) (csubst0_fst_bind b +i c c3 u0 H8 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H9 u)) x0 +H13))) (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 H12 (CHead c3 (Bind +b) u) (csubst0_fst_bind b i c c3 u0 H8 u)) e (getl_head (Bind b) i c (CHead e +(Bind Abbr) u0) H9 u)))) t6 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda +(u2: T).(\lambda (t7: T).(eq T t6 (THead (Bind b) u2 t7)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t7: +T).(subst0 (s (Bind b) i) u0 t2 t7))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t7: T).(eq T t6 (THead (Bind b) u2 t7)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t7: +T).(subst0 (s (Bind b) i) u0 t2 t7))) (ty3 g c3 t6 (THead (Bind b) u t3)) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t6 (THead (Bind b) x0 +x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Bind b) +i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t7: T).(ty3 g c3 +t7 (THead (Bind b) u t3))) (ex_ind T (\lambda (t7: T).(ty3 g (CHead c3 (Bind +b) u) t3 t7)) (ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3)) +(\lambda (x: T).(\lambda (H14: (ty3 g (CHead c3 (Bind b) u) t3 x)).(ex_ind T +(\lambda (t7: T).(ty3 g (CHead c3 (Bind b) u) x t7)) (ty3 g c3 (THead (Bind +b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (H15: (ty3 g +(CHead c3 (Bind b) u) x x2)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c3 +(Bind b) x0) t3 t7)) (ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3)) +(\lambda (x3: T).(\lambda (H16: (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 H8) e H9) b t3 x H14 x2 H15) +(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 H12 c3 H8) e H9) 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 H13 +(CHead c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H12 c c3 H8)) e +(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H9 u)) x3 H16) (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 H12 t3 (Bind b)) c3 H8) e +H9)))) (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 H13 (CHead +c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H12 c c3 H8)) e (getl_head +(Bind b) i c (CHead e (Bind Abbr) u0) H9 u)))))) (ty3_correct g (CHead c3 +(Bind b) u) t3 x H14)))) (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 H8 u)) e (getl_head (Bind +b) i c (CHead e (Bind Abbr) u0) H9 u)))) t6 H11)))))) H10)) (subst0_gen_head +(Bind b) u0 u t2 t6 i H7)))))))) c2 t5 H6))))))))))))))))))) (\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)).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c +(THead (Bind Abst) u t4) x0)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: +T).(ty3 g c u t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g +(CHead c (Bind Abst) u) t0 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda +(t6: T).(ty3 g (CHead c (Bind Abst) u) t4 t6)))) (ty3 g c (THead (Flat Appl) +x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x1: +T).(\lambda (x2: T).(\lambda (x3: 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)).(\lambda (H14: (ty3 g (CHead c (Bind Abst) u) x1 +x3)).(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 (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 x1)) (ty3_appl g c w u H0 +(THead (Bind Abst) u t0) x1 (ty3_bind g c u x4 H15 Abst t0 x1 H13 x3 H14)) +(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)).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: +T).(\lambda (_: T).(pc3 c (THead (Bind Abst) u t4) x)))) (\lambda (_: +T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c u t5)))) (\lambda (t4: +T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0 t4)))) +(\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c (Bind Abst) +u) t4 t6)))) (ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead +(Bind Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: +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)).(\lambda (H15: +(ty3 g (CHead c (Bind Abst) u) x2 x4)).(ex_ind T (\lambda (t4: T).(ty3 g c u +t4)) 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(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)).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: +T).(\lambda (_: T).(pc3 c3 (THead (Bind Abst) u t4) x0)))) (\lambda (_: +T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c3 u t5)))) (\lambda (t4: +T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c3 (Bind Abst) u) t0 t4)))) +(\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c3 (Bind +Abst) u) t4 t6)))) (ty3 g c3 (THead (Flat Appl) x v) (THead (Flat Appl) w +(THead (Bind Abst) u t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: +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)).(\lambda +(H15: (ty3 g (CHead c3 (Bind Abst) u) x1 x3)).(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 x3 H15)) +(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 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(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)).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: T).(\lambda +(_: T).(pc3 c3 (THead (Bind Abst) u t4) x)))) (\lambda (_: T).(\lambda (t5: +T).(\lambda (_: T).(ty3 g c3 u t5)))) (\lambda (t4: T).(\lambda (_: +T).(\lambda (_: T).(ty3 g (CHead c3 (Bind Abst) u) t0 t4)))) (\lambda (t4: +T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c3 (Bind Abst) u) t4 t6)))) +(ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind Abst) u +t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 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)).(\lambda (H16: (ty3 g (CHead c3 (Bind +Abst) u) x2 x4)).(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 +(x5: T).(\lambda (H17: (ty3 g c3 u x5)).(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 x5 H17 Abst t0 x2 H15 x4 H16)) (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 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 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 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 t3)) (ty3_conv g c t3 t0 H2 (THead (Flat +Cast) x t2) 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 t3 t3 (pc3_refl c t3) 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 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 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 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 t3)) (ty3_conv g c t3 t0 H2 (THead (Flat Cast) x0 x1) 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 t3 t3 (pc3_refl c t3) 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 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 +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 t3)) (ty3_conv g c3 t3 t0 (H3 i u c3 t3 +(fsubst0_fst i u c t3 c3 H6) e H7) (THead (Flat Cast) x t2) 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 t3 t3 (pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 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 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 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 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 t3)) (ty3_conv g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 +c3 H6) e H7) (THead (Flat Cast) x0 x1) 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 t3 t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 +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))))). + +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))))))))))). + +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))))))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fwd.ma new file mode 100644 index 000000000..a3ee2cbf0 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fwd.ma @@ -0,0 +1,914 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/fwd". + +include "ty3/defs.ma". + +include "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)) (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 (t0: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t2 +t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort +(next g n)) t0)))).(\lambda (H7: (eq T (THead (Bind b) u t1) (TSort n))).(let +H8 \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) H7) in +(False_ind (pc3 c0 (TSort (next g n)) (THead (Bind b) u t2)) +H8)))))))))))))))) (\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)) t2) H6))))))))))) c y x H0))) +H))))). + +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)) (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)))))) (\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 (t0: +T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t2 t0)).(\lambda (_: (((eq T t2 +(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3: +T).(pc3 (CHead c0 (Bind b) u) (lift (S n) O t3) t0)))) (\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 (t3: T).(ty3 g +e u0 t3))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: +T).(pc3 (CHead c0 (Bind b) u) (lift (S n) O u0) t0)))) (\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 (t3: T).(ty3 g +e u0 t3))))))))).(\lambda (H7: (eq T (THead (Bind b) u t1) (TLRef n))).(let +H8 \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) H7) in +(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3: +T).(pc3 c0 (lift (S n) O t3) (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 (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) (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 (t3: T).(ty3 g e u0 t3)))))) H8)))))))))))))))) (\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) +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)))))) H6))))))))))) c y x H0))) H))))). + +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 (ex4_3 T T T +(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) +x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t)))) +(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind b) u) +t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c +(Bind b) u) t2 t0))))))))))) +\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)) (ex4_3 T T T (\lambda +(t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x)))) +(\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t)))) (\lambda +(t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind b) u) t1 t2)))) +(\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c (Bind b) u) +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 (Bind b) u t1)) \to +(ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead +(Bind b) u t2) t0)))) (\lambda (_: T).(\lambda (t3: T).(\lambda (_: T).(ty3 g +c0 u t3)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 +(Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t4: T).(ty3 +g (CHead c0 (Bind b) u) t2 t4))))))))) (\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 (ex4_3 T T T (\lambda (t3: T).(\lambda (_: +T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t)))) (\lambda (_: +T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t3: +T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) +(\lambda (t3: T).(\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) +u) t3 t4)))))))).(\lambda (u0: T).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 u0 +t0)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t1)) \to (ex4_3 T T T +(\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u +t3) t0)))) (\lambda (_: T).(\lambda (t4: T).(\lambda (_: T).(ty3 g c0 u +t4)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 +(Bind b) u) t1 t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (t5: T).(ty3 +g (CHead c0 (Bind b) u) t3 t5)))))))).(\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 (ex4_3 T T T (\lambda (t4: +T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t0)))) +(\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c0 u t5)))) (\lambda +(t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 +t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c0 +(Bind b) u) t4 t6))))))) 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 (ex4_3_ind T T T +(\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u +t3) t0)))) (\lambda (_: T).(\lambda (t4: T).(\lambda (_: T).(ty3 g c0 u +t4)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 +(Bind b) u) t1 t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (t5: T).(ty3 +g (CHead c0 (Bind b) u) t3 t5)))) (ex4_3 T T T (\lambda (t3: T).(\lambda (_: +T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t2)))) (\lambda (_: +T).(\lambda (t4: T).(\lambda (_: T).(ty3 g c0 u t4)))) (\lambda (t3: +T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) +(\lambda (t3: T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) +u) t3 t5))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 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)).(\lambda (H14: (ty3 +g (CHead c0 (Bind b) u) x0 x2)).(ex4_3_intro T T T (\lambda (t3: T).(\lambda +(_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t2)))) (\lambda (_: +T).(\lambda (t4: T).(\lambda (_: T).(ty3 g c0 u t4)))) (\lambda (t3: +T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) +(\lambda (t3: T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) +u) t3 t5)))) x0 x1 x2 (pc3_t t0 c0 (THead (Bind b) u x0) H11 t2 H5) H12 H13 +H14)))))))) 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 (ex4_3 T T T (\lambda (t2: +T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (TSort (next +g m)))))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t)))) +(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) +t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c0 +(Bind b) u) t2 t0))))) 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 (ex4_3 T T T (\lambda (t2: T).(\lambda (_: +T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t)))) (\lambda (_: +T).(\lambda (t0: T).(\lambda (_: T).(ty3 g d u t0)))) (\lambda (t2: +T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 t2)))) +(\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead d (Bind b) u) +t2 t3)))))))).(\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 +(ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead +(Bind b) u t2) (lift (S n) O t))))) (\lambda (_: T).(\lambda (t0: T).(\lambda +(_: T).(ty3 g c0 u t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: +T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: +T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind b) u) t2 t3))))) 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 (ex4_3 T T T +(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) +t)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g d u t0)))) +(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) +t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead d +(Bind b) u) t2 t3)))))))).(\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 (ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: +T).(pc3 c0 (THead (Bind b) u t2) (lift (S n) O u0))))) (\lambda (_: +T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t2: +T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) +(\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind b) +u) t2 t3))))) 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 (ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 +c0 (THead (Bind b) u t2) t)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: +T).(ty3 g c0 u t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g +(CHead c0 (Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda +(t3: T).(ty3 g (CHead c0 (Bind b) u) t2 t3)))))))).(\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 (ex4_3 T T T +(\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) +(THead (Bind b) u t3) t2)))) (\lambda (_: T).(\lambda (t4: T).(\lambda (_: +T).(ty3 g (CHead c0 (Bind b0) u0) u t4)))) (\lambda (t3: T).(\lambda (_: +T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 +t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead (CHead +c0 (Bind b0) u0) (Bind b) u) t3 t5)))))))).(\lambda (t3: T).(\lambda (H5: +(ty3 g (CHead c0 (Bind b0) u0) t2 t3)).(\lambda (H6: (((eq T t2 (THead (Bind +b) u t1)) \to (ex4_3 T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: +T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t4) t3)))) (\lambda (_: +T).(\lambda (t5: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b0) u0) u t5)))) +(\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 +(Bind b0) u0) (Bind b) u) t1 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda +(t6: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t4 +t6)))))))).(\lambda (H7: (eq T (THead (Bind b0) u0 t0) (THead (Bind b) u +t1))).(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 t1) 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 _ t4 _) \Rightarrow t4])) (THead (Bind b0) u0 t0) +(THead (Bind b) u t1) 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 _ _ t4) \Rightarrow t4])) (THead (Bind b0) +u0 t0) (THead (Bind b) u t1) H7) in (\lambda (H11: (eq T u0 u)).(\lambda +(H12: (eq B b0 b)).(let H13 \def (eq_ind T t0 (\lambda (t4: T).((eq T t4 +(THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t5: T).(\lambda (_: +T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t5) t2)))) +(\lambda (_: T).(\lambda (t6: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b0) +u0) u t6)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead +(CHead c0 (Bind b0) u0) (Bind b) u) t1 t5)))) (\lambda (t5: T).(\lambda (_: +T).(\lambda (t7: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t5 +t7))))))) H4 t1 H10) in (let H14 \def (eq_ind T t0 (\lambda (t4: T).(ty3 g +(CHead c0 (Bind b0) u0) t4 t2)) H3 t1 H10) in (let H15 \def (eq_ind B b0 +(\lambda (b1: B).((eq T t2 (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda +(t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b1) u0) (THead +(Bind b) u t4) t3)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g +(CHead c0 (Bind b1) u0) u t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda +(_: T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind b) u) t1 t4)))) (\lambda +(t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead (CHead c0 (Bind b1) +u0) (Bind b) u) t4 t6))))))) H6 b H12) in (let H16 \def (eq_ind B b0 (\lambda +(b1: B).(ty3 g (CHead c0 (Bind b1) u0) t2 t3)) H5 b H12) in (let H17 \def +(eq_ind B b0 (\lambda (b1: B).((eq T t1 (THead (Bind b) u t1)) \to (ex4_3 T T +T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b1) +u0) (THead (Bind b) u t4) t2)))) (\lambda (_: T).(\lambda (t5: T).(\lambda +(_: T).(ty3 g (CHead c0 (Bind b1) u0) u t5)))) (\lambda (t4: T).(\lambda (_: +T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind b) u) t1 +t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead (CHead +c0 (Bind b1) u0) (Bind b) u) t4 t6))))))) H13 b H12) in (let H18 \def (eq_ind +B b0 (\lambda (b1: B).(ty3 g (CHead c0 (Bind b1) u0) t1 t2)) H14 b H12) in +(eq_ind_r B b (\lambda (b1: B).(ex4_3 T T T (\lambda (t4: T).(\lambda (_: +T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) (THead (Bind b1) u0 t2))))) +(\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c0 u t5)))) (\lambda +(t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 +t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c0 +(Bind b) u) t4 t6)))))) (let H19 \def (eq_ind T u0 (\lambda (t4: T).((eq T t2 +(THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t5: T).(\lambda (_: +T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t4) (THead (Bind b) u t5) t3)))) +(\lambda (_: T).(\lambda (t6: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) +t4) u t6)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead +(CHead c0 (Bind b) t4) (Bind b) u) t1 t5)))) (\lambda (t5: T).(\lambda (_: +T).(\lambda (t7: T).(ty3 g (CHead (CHead c0 (Bind b) t4) (Bind b) u) t5 +t7))))))) H15 u H11) in (let H20 \def (eq_ind T u0 (\lambda (t4: T).(ty3 g +(CHead c0 (Bind b) t4) t2 t3)) H16 u H11) in (let H21 \def (eq_ind T u0 +(\lambda (t4: T).((eq T t1 (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda +(t5: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t4) (THead +(Bind b) u t5) t2)))) (\lambda (_: T).(\lambda (t6: T).(\lambda (_: T).(ty3 g +(CHead c0 (Bind b) t4) u t6)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (_: +T).(ty3 g (CHead (CHead c0 (Bind b) t4) (Bind b) u) t1 t5)))) (\lambda (t5: +T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead (CHead c0 (Bind b) t4) +(Bind b) u) t5 t7))))))) H17 u H11) in (let H22 \def (eq_ind T u0 (\lambda +(t4: T).(ty3 g (CHead c0 (Bind b) t4) t1 t2)) H18 u H11) in (let H23 \def +(eq_ind T u0 (\lambda (t4: T).((eq T t4 (THead (Bind b) u t1)) \to (ex4_3 T T +T (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u +t5) t)))) (\lambda (_: T).(\lambda (t6: T).(\lambda (_: T).(ty3 g c0 u t6)))) +(\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) +t1 t5)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead c0 +(Bind b) u) t5 t7))))))) H2 u H11) in (let H24 \def (eq_ind T u0 (\lambda +(t4: T).(ty3 g c0 t4 t)) H1 u H11) in (eq_ind_r T u (\lambda (t4: T).(ex4_3 T +T T (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) +u t5) (THead (Bind b) t4 t2))))) (\lambda (_: T).(\lambda (t6: T).(\lambda +(_: T).(ty3 g c0 u t6)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (_: +T).(ty3 g (CHead c0 (Bind b) u) t1 t5)))) (\lambda (t5: T).(\lambda (_: +T).(\lambda (t7: T).(ty3 g (CHead c0 (Bind b) u) t5 t7)))))) (ex4_3_intro T T +T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u +t4) (THead (Bind b) u t2))))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: +T).(ty3 g c0 u t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g +(CHead c0 (Bind b) u) t1 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda +(t6: T).(ty3 g (CHead c0 (Bind b) u) t4 t6)))) t2 t t3 (pc3_refl c0 (THead +(Bind b) u t2)) H24 H22 H20) u0 H11))))))) b0 H12)))))))))) H9)) +H8)))))))))))))))) (\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 (ex4_3 +T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind +b) u t2) u0)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u +t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind +b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g +(CHead c0 (Bind b) u) t2 t0)))))))).(\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 (ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda +(_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Bind Abst) u0 t))))) (\lambda +(_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t2: +T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) +(\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind b) +u) t2 t3)))))))).(\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 (ex4_3 T T T +(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u +t2) (THead (Flat Appl) w (THead (Bind Abst) u0 t)))))) (\lambda (_: +T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t2: +T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) +(\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind b) +u) t2 t3))))) 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 (ex4_3 T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 +c0 (THead (Bind b) u t3) t2)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: +T).(ty3 g c0 u t)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(ty3 g +(CHead c0 (Bind b) u) t1 t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda +(t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))))))).(\lambda (t3: T).(\lambda +(_: (ty3 g c0 t2 t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u t1)) \to +(ex4_3 T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead +(Bind b) u t4) t3)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g +c0 u t)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 +(Bind b) u) t1 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t5: T).(ty3 +g (CHead c0 (Bind b) u) t4 t5)))))))).(\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 (ex4_3 T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: +T).(pc3 c0 (THead (Bind b) u t4) t2)))) (\lambda (_: T).(\lambda (t: +T).(\lambda (_: T).(ty3 g c0 u t)))) (\lambda (t4: T).(\lambda (_: +T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))) (\lambda (t4: +T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t4 t5))))) +H6))))))))))) c y x H0))) H))))))). + +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)) (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)))) (\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 (t0: +T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t2 t0)).(\lambda (_: (((eq T t2 +(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t3: +T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Appl) w (THead (Bind Abst) u0 t3)) +t0))) (\lambda (u0: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind b) u) v (THead +(Bind Abst) u0 t3)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind +b) u) w u0))))))).(\lambda (H7: (eq T (THead (Bind b) u t1) (THead (Flat +Appl) w v))).(let H8 \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) H7) in (False_ind (ex3_2 T T +(\lambda (u0: T).(\lambda (t3: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind +Abst) u0 t3)) (THead (Bind b) u t2)))) (\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)))) H8)))))))))))))))) (\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)) 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)))))). + +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 (land (pc3 c t2 x) (ty3 g c +t1 t2))))))) +\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)) (land (pc3 c t2 x) +(ty3 g c 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 (Flat Cast) +t2 t1)) \to (land (pc3 c0 t2 t0) (ty3 g c0 t1 t2)))))) (\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 (land (pc3 c0 t2 t) (ty3 g c0 +t1 t2))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u +t3)).(\lambda (H4: (((eq T u (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 +t3) (ty3 g c0 t1 t2))))).(\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 (land (pc3 c0 t2 t3) (ty3 g c0 t1 +t2)))) 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 (and_ind (pc3 c0 t2 t3) (ty3 +g c0 t1 t2) (land (pc3 c0 t2 t0) (ty3 g c0 t1 t2)) (\lambda (H11: (pc3 c0 t2 +t3)).(\lambda (H12: (ty3 g c0 t1 t2)).(conj (pc3 c0 t2 t0) (ty3 g c0 t1 t2) +(pc3_t t3 c0 t2 H11 t0 H5) H12))) 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 (land (pc3 c0 t2 (TSort (next g m))) (ty3 g c0 t1 +t2)) 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 +(land (pc3 d t2 t) (ty3 g d t1 t2))))).(\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 (land (pc3 c0 t2 (lift (S n) O t)) (ty3 +g c0 t1 t2)) 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 (land (pc3 d t2 t) (ty3 g d t1 t2))))).(\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 (land (pc3 c0 t2 (lift (S n) O +u)) (ty3 g c0 t1 t2)) 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 (land (pc3 c0 t2 t) (ty3 g c0 t1 t2))))).(\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 (land (pc3 (CHead +c0 (Bind b) u) t2 t3) (ty3 g (CHead c0 (Bind b) u) t1 t2))))).(\lambda (t4: +T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (_: (((eq T t3 +(THead (Flat Cast) t2 t1)) \to (land (pc3 (CHead c0 (Bind b) u) t2 t4) (ty3 g +(CHead c0 (Bind b) u) t1 t2))))).(\lambda (H7: (eq T (THead (Bind b) u t0) +(THead (Flat Cast) t2 t1))).(let H8 \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) H7) in (False_ind +(land (pc3 c0 t2 (THead (Bind b) u t3)) (ty3 g c0 t1 t2)) H8)))))))))))))))) +(\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 (land (pc3 c0 t2 u) +(ty3 g c0 t1 t2))))).(\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 (land (pc3 c0 t2 (THead (Bind Abst) u t)) (ty3 g c0 t1 t2))))).(\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 (land +(pc3 c0 t2 (THead (Flat Appl) w (THead (Bind Abst) u t))) (ty3 g c0 t1 t2)) +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 (land (pc3 c0 t2 t3) (ty3 g c0 t1 t2))))).(\lambda (t4: T).(\lambda (H3: +(ty3 g c0 t3 t4)).(\lambda (H4: (((eq T t3 (THead (Flat Cast) t2 t1)) \to +(land (pc3 c0 t2 t4) (ty3 g c0 t1 t2))))).(\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 (land (pc3 c0 t2 t4) (ty3 g c0 t1 t2)))) 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 +(land (pc3 c0 t2 t) (ty3 g c0 t1 t2)))) 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).(land (pc3 c0 t2 t) (ty3 g c0 t1 t2))) (let H13 \def (eq_ind T t0 (\lambda +(t: T).((eq T t (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 t2) (ty3 g c0 +t1 t2)))) H11 t1 H7) in (let H14 \def (eq_ind T t0 (\lambda (t: T).(ty3 g c0 +t t2)) H12 t1 H7) in (conj (pc3 c0 t2 t2) (ty3 g c0 t1 t2) (pc3_refl c0 t2) +H14))) t3 H8))))))) H6))))))))))) c y x H0))) H)))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3.ma new file mode 100644 index 000000000..0c14d34f3 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3.ma @@ -0,0 +1,732 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3". + +include "csub3/ty3.ma". + +include "ty3/subst1.ma". + +include "ty3/fsubst0.ma". + +include "pc3/pc1.ma". + +include "pc3/wcpr0.ma". + +include "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 (t4: +T).(\lambda (H4: (ty3 g (CHead c (Bind b) u) t3 t4)).(\lambda (H5: ((\forall +(c2: C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t5: T).((pr0 t3 t5) +\to (ty3 g c2 t5 t4))))))).(\lambda (c2: C).(\lambda (H6: (wcpr0 c +c2)).(\lambda (t5: T).(\lambda (H7: (pr0 (THead (Bind b) u t2) t5)).(let H8 +\def (match H7 in pr0 return (\lambda (t6: T).(\lambda (t7: T).(\lambda (_: +(pr0 t6 t7)).((eq T t6 (THead (Bind b) u t2)) \to ((eq T t7 t5) \to (ty3 g c2 +t5 (THead (Bind b) u t3))))))) with [(pr0_refl t6) \Rightarrow (\lambda (H8: +(eq T t6 (THead (Bind b) u t2))).(\lambda (H9: (eq T t6 t5)).(eq_ind T (THead +(Bind b) u t2) (\lambda (t7: T).((eq T t7 t5) \to (ty3 g c2 t5 (THead (Bind +b) u t3)))) (\lambda (H10: (eq T (THead (Bind b) u t2) t5)).(eq_ind T (THead +(Bind b) u t2) (\lambda (t7: T).(ty3 g c2 t7 (THead (Bind b) u t3))) +(ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) b t2 t3 (H3 (CHead c2 (Bind b) +u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t2 (pr0_refl t2)) t4 (H5 +(CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 +(pr0_refl t3))) t5 H10)) t6 (sym_eq T t6 (THead (Bind b) u t2) H8) H9))) | +(pr0_comp u1 u2 H8 t6 t7 H9 k) \Rightarrow (\lambda (H10: (eq T (THead k u1 +t6) (THead (Bind b) u t2))).(\lambda (H11: (eq T (THead k u2 t7) t5)).((let +H12 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t6 | (TLRef _) \Rightarrow t6 | (THead _ _ t8) +\Rightarrow t8])) (THead k u1 t6) (THead (Bind b) u t2) H10) 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 _ t8 _) +\Rightarrow t8])) (THead k u1 t6) (THead (Bind b) u t2) H10) in ((let H14 +\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 t6) (THead (Bind b) u t2) H10) in (eq_ind K +(Bind b) (\lambda (k0: K).((eq T u1 u) \to ((eq T t6 t2) \to ((eq T (THead k0 +u2 t7) t5) \to ((pr0 u1 u2) \to ((pr0 t6 t7) \to (ty3 g c2 t5 (THead (Bind b) +u t3)))))))) (\lambda (H15: (eq T u1 u)).(eq_ind T u (\lambda (t8: T).((eq T +t6 t2) \to ((eq T (THead (Bind b) u2 t7) t5) \to ((pr0 t8 u2) \to ((pr0 t6 +t7) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) (\lambda (H16: (eq T t6 +t2)).(eq_ind T t2 (\lambda (t8: T).((eq T (THead (Bind b) u2 t7) t5) \to +((pr0 u u2) \to ((pr0 t8 t7) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))) +(\lambda (H17: (eq T (THead (Bind b) u2 t7) t5)).(eq_ind T (THead (Bind b) u2 +t7) (\lambda (t8: T).((pr0 u u2) \to ((pr0 t2 t7) \to (ty3 g c2 t8 (THead +(Bind b) u t3))))) (\lambda (H18: (pr0 u u2)).(\lambda (H19: (pr0 t2 +t7)).(ex_ind T (\lambda (t8: T).(ty3 g (CHead c2 (Bind b) u) t4 t8)) (ty3 g +c2 (THead (Bind b) u2 t7) (THead (Bind b) u t3)) (\lambda (x: T).(\lambda +(H20: (ty3 g (CHead c2 (Bind b) u) t4 x)).(ex_ind T (\lambda (t8: T).(ty3 g +(CHead c2 (Bind b) u2) t3 t8)) (ty3 g c2 (THead (Bind b) u2 t7) (THead (Bind +b) u t3)) (\lambda (x0: T).(\lambda (H21: (ty3 g (CHead c2 (Bind b) u2) t3 +x0)).(ty3_conv g c2 (THead (Bind b) u t3) (THead (Bind b) u t4) (ty3_bind g +c2 u t0 (H1 c2 H6 u (pr0_refl u)) b t3 t4 (H5 (CHead c2 (Bind b) u) +(wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3)) x H20) +(THead (Bind b) u2 t7) (THead (Bind b) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H6 +u2 H18) b t7 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H6 u u2 H18 (Bind +b)) t7 H19) x0 H21) (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 H18) (Bind b) t3))))) (ty3_correct +g (CHead c2 (Bind b) u2) t7 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H6 +u u2 H18 (Bind b)) t7 H19))))) (ty3_correct g (CHead c2 (Bind b) u) t3 t4 (H5 +(CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 +(pr0_refl t3)))))) t5 H17)) t6 (sym_eq T t6 t2 H16))) u1 (sym_eq T u1 u +H15))) k (sym_eq K k (Bind b) H14))) H13)) H12)) H11 H8 H9))) | (pr0_beta u0 +v1 v2 H8 t6 t7 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 +(THead (Bind Abst) u0 t6)) (THead (Bind b) u t2))).(\lambda (H11: (eq T +(THead (Bind Abbr) v2 t7) t5)).((let H12 \def (eq_ind T (THead (Flat Appl) v1 +(THead (Bind Abst) u0 t6)) (\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 Abbr) v2 t7) t5) \to ((pr0 v1 +v2) \to ((pr0 t6 t7) \to (ty3 g c2 t5 (THead (Bind b) u t3))))) H12)) H11 H8 +H9))) | (pr0_upsilon b0 H8 v1 v2 H9 u1 u2 H10 t6 t7 H11) \Rightarrow (\lambda +(H12: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 t6)) (THead (Bind b) u +t2))).(\lambda (H13: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) +O v2) t7)) t5)).((let H14 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind +b0) u1 t6)) (\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) +H12) in (False_ind ((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) +O v2) t7)) t5) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) +\to ((pr0 t6 t7) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) H14)) H13 H8 H9 +H10 H11))) | (pr0_delta u1 u2 H8 t6 t7 H9 w H10) \Rightarrow (\lambda (H11: +(eq T (THead (Bind Abbr) u1 t6) (THead (Bind b) u t2))).(\lambda (H12: (eq T +(THead (Bind Abbr) u2 w) t5)).((let H13 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t6 | +(TLRef _) \Rightarrow t6 | (THead _ _ t8) \Rightarrow t8])) (THead (Bind +Abbr) u1 t6) (THead (Bind b) u t2) H11) 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 _ t8 _) \Rightarrow t8])) +(THead (Bind Abbr) u1 t6) (THead (Bind b) u t2) H11) in ((let H15 \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 t6) +(THead (Bind b) u t2) H11) in (eq_ind B Abbr (\lambda (b0: B).((eq T u1 u) +\to ((eq T t6 t2) \to ((eq T (THead (Bind Abbr) u2 w) t5) \to ((pr0 u1 u2) +\to ((pr0 t6 t7) \to ((subst0 O u2 t7 w) \to (ty3 g c2 t5 (THead (Bind b0) u +t3))))))))) (\lambda (H16: (eq T u1 u)).(eq_ind T u (\lambda (t8: T).((eq T +t6 t2) \to ((eq T (THead (Bind Abbr) u2 w) t5) \to ((pr0 t8 u2) \to ((pr0 t6 +t7) \to ((subst0 O u2 t7 w) \to (ty3 g c2 t5 (THead (Bind Abbr) u t3)))))))) +(\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t8: T).((eq T (THead +(Bind Abbr) u2 w) t5) \to ((pr0 u u2) \to ((pr0 t8 t7) \to ((subst0 O u2 t7 +w) \to (ty3 g c2 t5 (THead (Bind Abbr) u t3))))))) (\lambda (H18: (eq T +(THead (Bind Abbr) u2 w) t5)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda +(t8: T).((pr0 u u2) \to ((pr0 t2 t7) \to ((subst0 O u2 t7 w) \to (ty3 g c2 t8 +(THead (Bind Abbr) u t3)))))) (\lambda (H19: (pr0 u u2)).(\lambda (H20: (pr0 +t2 t7)).(\lambda (H21: (subst0 O u2 t7 w)).(let H22 \def (eq_ind_r B b +(\lambda (b0: B).(\forall (c3: C).((wcpr0 (CHead c (Bind b0) u) c3) \to +(\forall (t8: T).((pr0 t3 t8) \to (ty3 g c3 t8 t4)))))) H5 Abbr H15) in (let +H23 \def (eq_ind_r B b (\lambda (b0: B).(ty3 g (CHead c (Bind b0) u) t3 t4)) +H4 Abbr H15) in (let H24 \def (eq_ind_r B b (\lambda (b0: B).(\forall (c3: +C).((wcpr0 (CHead c (Bind b0) u) c3) \to (\forall (t8: T).((pr0 t2 t8) \to +(ty3 g c3 t8 t3)))))) H3 Abbr H15) in (let H25 \def (eq_ind_r B b (\lambda +(b0: B).(ty3 g (CHead c (Bind b0) u) t2 t3)) H2 Abbr H15) in (ex_ind T +(\lambda (t8: T).(ty3 g (CHead c2 (Bind Abbr) u) t4 t8)) (ty3 g c2 (THead +(Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x: T).(\lambda (H26: +(ty3 g (CHead c2 (Bind Abbr) u) t4 x)).(ex_ind T (\lambda (t8: T).(ty3 g +(CHead c2 (Bind Abbr) u2) t3 t8)) (ty3 g c2 (THead (Bind Abbr) u2 w) (THead +(Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (H27: (ty3 g (CHead c2 (Bind +Abbr) u2) t3 x0)).(ty3_conv g c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr) +u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) Abbr t3 t4 (H22 (CHead c2 +(Bind Abbr) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind Abbr)) t3 (pr0_refl +t3)) x H26) (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u2 t3) (ty3_bind g c2 +u2 t0 (H1 c2 H6 u2 H19) Abbr w t3 (ty3_subst0 g (CHead c2 (Bind Abbr) u2) t7 +t3 (H24 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H6 u u2 H19 (Bind Abbr)) +t7 H20) c2 u2 O (getl_refl Abbr c2 u2) w H21) x0 H27) (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 H19) (Bind Abbr) t3))))) (ty3_correct g (CHead c2 (Bind Abbr) u2) t7 t3 +(H24 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H6 u u2 H19 (Bind Abbr)) t7 +H20))))) (ty3_correct g (CHead c2 (Bind Abbr) u) t3 t4 (H22 (CHead c2 (Bind +Abbr) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind Abbr)) t3 (pr0_refl +t3))))))))))) t5 H18)) t6 (sym_eq T t6 t2 H17))) u1 (sym_eq T u1 u H16))) b +H15)) H14)) H13)) H12 H8 H9 H10))) | (pr0_zeta b0 H8 t6 t7 H9 u0) \Rightarrow +(\lambda (H10: (eq T (THead (Bind b0) u0 (lift (S O) O t6)) (THead (Bind b) u +t2))).(\lambda (H11: (eq T t7 t5)).((let H12 \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) (t8: T) on t8: T \def (match t8 +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 +t9) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t9))]) in +lref_map) (\lambda (x: nat).(plus x (S O))) O t6) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t8: T) on t8: T \def (match +t8 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 t9) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) +t9))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t6) | (THead _ _ t8) +\Rightarrow t8])) (THead (Bind b0) u0 (lift (S O) O t6)) (THead (Bind b) u +t2) H10) 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 _ t8 _) \Rightarrow t8])) (THead (Bind b0) u0 (lift (S O) O t6)) +(THead (Bind b) u t2) H10) in ((let H14 \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 t6)) (THead (Bind b) u t2) H10) in +(eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq T (lift (S O) O t6) t2) +\to ((eq T t7 t5) \to ((not (eq B b1 Abst)) \to ((pr0 t6 t7) \to (ty3 g c2 t5 +(THead (Bind b) u t3)))))))) (\lambda (H15: (eq T u0 u)).(eq_ind T u (\lambda +(_: T).((eq T (lift (S O) O t6) t2) \to ((eq T t7 t5) \to ((not (eq B b +Abst)) \to ((pr0 t6 t7) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) (\lambda +(H16: (eq T (lift (S O) O t6) t2)).(eq_ind T (lift (S O) O t6) (\lambda (_: +T).((eq T t7 t5) \to ((not (eq B b Abst)) \to ((pr0 t6 t7) \to (ty3 g c2 t5 +(THead (Bind b) u t3)))))) (\lambda (H17: (eq T t7 t5)).(eq_ind T t5 (\lambda +(t8: T).((not (eq B b Abst)) \to ((pr0 t6 t8) \to (ty3 g c2 t5 (THead (Bind +b) u t3))))) (\lambda (H18: (not (eq B b Abst))).(\lambda (H19: (pr0 t6 +t5)).(let H20 \def (eq_ind_r T t2 (\lambda (t8: T).(\forall (c3: C).((wcpr0 +(CHead c (Bind b) u) c3) \to (\forall (t9: T).((pr0 t8 t9) \to (ty3 g c3 t9 +t3)))))) H3 (lift (S O) O t6) H16) in (let H21 \def (eq_ind_r T t2 (\lambda +(t8: T).(ty3 g (CHead c (Bind b) u) t8 t3)) H2 (lift (S O) O t6) H16) in +(ex_ind T (\lambda (t8: T).(ty3 g (CHead c2 (Bind b) u) t4 t8)) (ty3 g c2 t5 +(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H22: (ty3 g (CHead c2 (Bind +b) u) t4 x)).(B_ind (\lambda (b1: B).((not (eq B b1 Abst)) \to ((ty3 g (CHead +c2 (Bind b1) u) t3 t4) \to ((ty3 g (CHead c2 (Bind b1) u) t4 x) \to ((ty3 g +(CHead c2 (Bind b1) u) (lift (S O) O t5) t3) \to (ty3 g c2 t5 (THead (Bind +b1) u t3))))))) (\lambda (H23: (not (eq B Abbr Abst))).(\lambda (H24: (ty3 g +(CHead c2 (Bind Abbr) u) t3 t4)).(\lambda (H25: (ty3 g (CHead c2 (Bind Abbr) +u) t4 x)).(\lambda (H26: (ty3 g (CHead c2 (Bind Abbr) u) (lift (S O) O t5) +t3)).(let H27 \def (ty3_gen_cabbr g (CHead c2 (Bind Abbr) u) (lift (S O) O +t5) t3 H26 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 t5) (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 t5 (THead (Bind Abbr) u t3)) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H28: (subst1 O u (lift (S O) O t5) (lift (S O) +O x0))).(\lambda (H29: (subst1 O u t3 (lift (S O) O x1))).(\lambda (H30: (ty3 +g c2 x0 x1)).(let H31 \def (eq_ind T x0 (\lambda (t8: T).(ty3 g c2 t8 x1)) +H30 t5 (lift_inj x0 t5 (S O) O (subst1_gen_lift_eq t5 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_comm O (S O))) H28))) in (ty3_conv +g c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr) u t4) (ty3_bind g c2 u t0 +(H1 c2 H6 u (pr0_refl u)) Abbr t3 t4 H24 x H25) t5 x1 H31 (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) H29))) 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 H23 x1 x1 (pr0_refl x1) u)))))))))))) H27)))))) (\lambda (H23: (not (eq +B Abst Abst))).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) t3 t4)).(\lambda +(_: (ty3 g (CHead c2 (Bind Abst) u) t4 x)).(\lambda (_: (ty3 g (CHead c2 +(Bind Abst) u) (lift (S O) O t5) t3)).(let H27 \def (match (H23 (refl_equal B +Abst)) in False return (\lambda (_: False).(ty3 g c2 t5 (THead (Bind Abst) u +t3))) with []) in H27))))) (\lambda (H23: (not (eq B Void Abst))).(\lambda +(H24: (ty3 g (CHead c2 (Bind Void) u) t3 t4)).(\lambda (H25: (ty3 g (CHead c2 +(Bind Void) u) t4 x)).(\lambda (H26: (ty3 g (CHead c2 (Bind Void) u) (lift (S +O) O t5) t3)).(let H27 \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u) (lift +(S O) O t5) t3 H26 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 t5) (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 t5 (THead (Bind Void) u t3)) (\lambda (x0: T).(\lambda +(x1: T).(\lambda (H28: (eq T (lift (S O) O t5) (lift (S O) O x0))).(\lambda +(H29: (eq T t3 (lift (S O) O x1))).(\lambda (H30: (ty3 g c2 x0 x1)).(let H31 +\def (eq_ind T t3 (\lambda (t8: T).(ty3 g (CHead c2 (Bind Void) u) t8 t4)) +H24 (lift (S O) O x1) H29) in (eq_ind_r T (lift (S O) O x1) (\lambda (t8: +T).(ty3 g c2 t5 (THead (Bind Void) u t8))) (let H32 \def (eq_ind_r T x0 +(\lambda (t8: T).(ty3 g c2 t8 x1)) H30 t5 (lift_inj t5 x0 (S O) O H28)) in +(ty3_conv g c2 (THead (Bind Void) u (lift (S O) O x1)) (THead (Bind Void) u +t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) Void (lift (S O) O x1) t4 +H31 x H25) t5 x1 H32 (pc3_pr2_x c2 x1 (THead (Bind Void) u (lift (S O) O x1)) +(pr2_free c2 (THead (Bind Void) u (lift (S O) O x1)) x1 (pr0_zeta Void H23 x1 +x1 (pr0_refl x1) u))))) t3 H29))))))) H27)))))) b H18 (H5 (CHead c2 (Bind b) +u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3)) H22 (H20 +(CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) (lift (S +O) O t5) (pr0_lift t6 t5 H19 (S O) O))))) (ty3_correct g (CHead c2 (Bind b) +u) t3 t4 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind +b)) t3 (pr0_refl t3)))))))) t7 (sym_eq T t7 t5 H17))) t2 H16)) u0 (sym_eq T +u0 u H15))) b0 (sym_eq B b0 b H14))) H13)) H12)) H11 H8 H9))) | (pr0_epsilon +t6 t7 H8 u0) \Rightarrow (\lambda (H9: (eq T (THead (Flat Cast) u0 t6) (THead +(Bind b) u t2))).(\lambda (H10: (eq T t7 t5)).((let H11 \def (eq_ind T (THead +(Flat Cast) u0 t6) (\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) +H9) in (False_ind ((eq T t7 t5) \to ((pr0 t6 t7) \to (ty3 g c2 t5 (THead +(Bind b) u t3)))) H11)) H10 H8)))]) in (H8 (refl_equal T (THead (Bind b) u +t2)) (refl_equal T t5)))))))))))))))))))) (\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)).(ex4_3_ind T T T (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 +(THead (Bind Abst) u t5) x)))) (\lambda (_: T).(\lambda (t6: T).(\lambda (_: +T).(ty3 g c2 u t6)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(ty3 g +(CHead c2 (Bind Abst) u) t0 t5)))) (\lambda (t5: T).(\lambda (_: T).(\lambda +(t7: T).(ty3 g (CHead c2 (Bind Abst) u) t5 t7)))) (ty3 g c2 (THead (Flat +Appl) u2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (x2: 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)).(\lambda (H22: (ty3 g (CHead c2 (Bind Abst) u) x0 +x2)).(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 x2 H22)) (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)).(ex4_3_ind T T T (\lambda (t5: T).(\lambda (_: T).(\lambda (_: +T).(pc3 c2 (THead (Bind Abst) u t5) x)))) (\lambda (_: T).(\lambda (t6: +T).(\lambda (_: T).(ty3 g c2 u t6)))) (\lambda (t5: T).(\lambda (_: +T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5)))) (\lambda (t5: +T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead c2 (Bind Abst) u) t5 t7)))) +(ty3 g c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u +t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 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)).(\lambda (H22: (ty3 g (CHead c2 (Bind +Abst) u) x0 x2)).(ex4_3_ind T T T (\lambda (t5: T).(\lambda (_: T).(\lambda +(_: T).(pc3 c2 (THead (Bind Abst) u0 t5) (THead (Bind Abst) u t0))))) +(\lambda (_: T).(\lambda (t6: T).(\lambda (_: T).(ty3 g c2 u0 t6)))) (\lambda +(t5: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u0) t4 +t5)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead c2 +(Bind Abst) u0) t5 t7)))) (ty3 g c2 (THead (Bind Abbr) v2 t4) (THead (Flat +Appl) w (THead (Bind Abst) u t0))) (\lambda (x3: T).(\lambda (x4: T).(\lambda +(x5: T).(\lambda (H23: (pc3 c2 (THead (Bind Abst) u0 x3) (THead (Bind Abst) u +t0))).(\lambda (H24: (ty3 g c2 u0 x4)).(\lambda (H25: (ty3 g (CHead c2 (Bind +Abst) u0) t4 x3)).(\lambda (H26: (ty3 g (CHead c2 (Bind Abst) u0) x3 +x5)).(and_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u1: T).(pc3 (CHead c2 +(Bind b) u1) x3 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) +w (THead (Bind Abst) u t0))) (\lambda (H27: (pc3 c2 u0 u)).(\lambda (H28: +((\forall (b: B).(\forall (u1: T).(pc3 (CHead c2 (Bind b) u1) x3 +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 x2 H22)) (THead (Bind Abbr) v2 t4) (THead (Bind Abbr) v2 x3) (ty3_bind g +c2 v2 u (H1 c2 H4 v2 H14) Abbr t4 x3 (csub3_ty3_ld g c2 v2 u0 (ty3_conv g c2 +u0 x4 H24 v2 u (H1 c2 H4 v2 H14) (pc3_s c2 u u0 H27)) t4 x3 H25) x5 +(csub3_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x4 H24 v2 u (H1 c2 H4 v2 H14) +(pc3_s c2 u u0 H27)) x3 x5 H26)) (pc3_t (THead (Bind Abbr) v2 t0) c2 (THead +(Bind Abbr) v2 x3) (pc3_head_2 c2 v2 x3 t0 (Bind Abbr) (H28 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 x3 t0 H23))))))))) +(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 (ex4_3_ind T T T (\lambda (t5: T).(\lambda (_: T).(\lambda +(_: T).(pc3 c2 (THead (Bind Abst) u t5) x)))) (\lambda (_: T).(\lambda (t6: +T).(\lambda (_: T).(ty3 g c2 u t6)))) (\lambda (t5: T).(\lambda (_: +T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5)))) (\lambda (t5: +T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead c2 (Bind Abst) u) t5 t7)))) +(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 (x2: 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)).(\lambda (H27: (ty3 g (CHead c2 (Bind Abst) u) x0 +x2)).(ex4_3_ind T T T (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(pc3 +c2 (THead (Bind b) u2 t5) (THead (Bind Abst) u t0))))) (\lambda (_: +T).(\lambda (t6: T).(\lambda (_: T).(ty3 g c2 u2 t6)))) (\lambda (t5: +T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) u2) t4 t5)))) +(\lambda (t5: T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead c2 (Bind b) +u2) t5 t7)))) (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 (x3: +T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H28: (pc3 c2 (THead (Bind b) +u2 x3) (THead (Bind Abst) u t0))).(\lambda (H29: (ty3 g c2 u2 x4)).(\lambda +(H30: (ty3 g (CHead c2 (Bind b) u2) t4 x3)).(\lambda (_: (ty3 g (CHead c2 +(Bind b) u2) x3 x5)).(let H32 \def (eq_ind T (lift (S O) O (THead (Bind Abst) +u t0)) (\lambda (t5: T).(pc3 (CHead c2 (Bind b) u2) x3 t5)) (pc3_gen_not_abst +b H16 c2 x3 t0 u2 u H28) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S +O) t0)) (lift_bind Abst u t0 (S O) O)) in (let H33 \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 (ex4_3_ind T T T (\lambda (t5: T).(\lambda (_: 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).(\lambda (_: T).(ty3 g (CHead +c2 (Bind b) u2) (lift (S O) O u) t6)))) (\lambda (t5: T).(\lambda (_: +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)))) (\lambda (t5: T).(\lambda (_: +T).(\lambda (t7: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S +O) O u)) t5 t7)))) (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 (x6: +T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (_: (pc3 (CHead c2 (Bind b) u2) +(THead (Bind Abst) (lift (S O) O u) x6) (lift (S O) O x))).(\lambda (H35: +(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) x7)).(\lambda (H36: (ty3 g +(CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O) +t0) x6)).(\lambda (H37: (ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) +(lift (S O) O u)) x6 x8)).(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 x2 H27)) (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 x4 +H29 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) x6) (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O +u) x7 H35 Abst (lift (S O) (S O) t0) x6 H36 x8 H37) t4 x3 H30 H32)) (THead +(Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S O) O u) x6)) +(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)) (THead (Bind Abst) (lift (S +O) O u) (lift (S O) (S O) t0)) x6 (ty3_bind g (CHead c2 (Bind b) u2) (lift (S +O) O u) x7 H35 Abst (lift (S O) (S O) t0) x6 H36 x8 H37))) (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_s +(THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u +t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head w v2 +(pc1_pr0_r w v2 H17) (THead (Bind Abst) u t0) (THead (Bind b) u2 (lift (S O) +O (THead (Bind Abst) u t0))) (pc1_pr0_x (THead (Bind Abst) u t0) (THead (Bind +b) u2 (lift (S O) O (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) H33))))))))))) +(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_epsilon 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 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 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 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 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 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 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 t3)))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 +t6)).(ty3_conv g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) (THead (Flat Cast) u2 +t6) 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_pr2_x c2 u2 t3 (pr2_free c2 t3 u2 H16))))) 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 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 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 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 t3)))) H10)) H9 H6 +H7))) | (pr0_epsilon 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 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 t3)))) +(\lambda (H12: (eq T t6 t4)).(eq_ind T t4 (\lambda (t7: T).((pr0 t2 t7) \to +(ty3 g c2 t4 t3))) (\lambda (H13: (pr0 t2 t4)).(H1 c2 H4 t4 H13)) 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))))). + +theorem ty3_sred_pr1: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr1 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: (pr1 t1 +t2)).(pr1_ind (\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: (pr0 t4 t3)).(\lambda (t5: T).(\lambda (_: +(pr1 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_wcpr0_pr0 g c t4 t H3 c (wcpr0_refl c) t3 +H0))))))))))) t1 t2 H)))). + +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)))). + +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)))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3_props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3_props.ma new file mode 100644 index 000000000..f429f69d5 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3_props.ma @@ -0,0 +1,501 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/pr3_props". + +include "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))))). + +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))))). + +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)) (\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)))).(and_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))))).(and_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_plus_r x1 h n 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)))).(and_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))))).(and_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_plus_r x1 h n 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 (t0: T).(\lambda (H5: (ty3 g (CHead c0 +(Bind b) u) t3 t0)).(\lambda (H6: ((\forall (x0: T).(\forall (x1: nat).((eq T +t3 (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) t0)) +(\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: +nat).(\lambda (H7: (eq T (THead (Bind b) u t2) (lift h x1 x0))).(\lambda (e: +C).(\lambda (H8: (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 +(H9: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (H10: (eq T u (lift h x1 +x2))).(\lambda (H11: (eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind +b) x2 x3) (\lambda (t4: T).(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x1 t5) +(THead (Bind b) u t3))) (\lambda (t5: T).(ty3 g e t4 t5)))) (let H12 \def +(eq_ind T t2 (\lambda (t4: T).(\forall (x4: T).(\forall (x5: nat).((eq T t4 +(lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b) u) e0) +\to (ex2 T (\lambda (t5: T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t5) t3)) +(\lambda (t5: T).(ty3 g e0 x4 t5))))))))) H4 (lift h (S x1) x3) H11) in (let +H13 \def (eq_ind T t2 (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t4 t3)) +H3 (lift h (S x1) x3) H11) in (let H14 \def (eq_ind T u (\lambda (t4: T).(ty3 +g (CHead c0 (Bind b) t4) (lift h (S x1) x3) t3)) H13 (lift h x1 x2) H10) in +(let H15 \def (eq_ind T u (\lambda (t4: 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) t4) e0) \to (ex2 T (\lambda (t5: T).(pc3 (CHead c0 +(Bind b) t4) (lift h x5 t5) t3)) (\lambda (t5: T).(ty3 g e0 x4 t5))))))))) +H12 (lift h x1 x2) H10) in (let H16 \def (eq_ind T u (\lambda (t4: +T).(\forall (x4: T).(\forall (x5: nat).((eq T t3 (lift h x5 x4)) \to (\forall +(e0: C).((drop h x5 (CHead c0 (Bind b) t4) e0) \to (ex2 T (\lambda (t5: +T).(pc3 (CHead c0 (Bind b) t4) (lift h x5 t5) t0)) (\lambda (t5: T).(ty3 g e0 +x4 t5))))))))) H6 (lift h x1 x2) H10) in (let H17 \def (eq_ind T u (\lambda +(t4: T).(ty3 g (CHead c0 (Bind b) t4) t3 t0)) H5 (lift h x1 x2) H10) in (let +H18 \def (eq_ind T u (\lambda (t4: T).(\forall (x4: T).(\forall (x5: +nat).((eq T t4 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to +(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x5 t5) t)) (\lambda (t5: T).(ty3 g e0 +x4 t5))))))))) H2 (lift h x1 x2) H10) in (let H19 \def (eq_ind T u (\lambda +(t4: T).(ty3 g c0 t4 t)) H1 (lift h x1 x2) H10) in (eq_ind_r T (lift h x1 x2) +(\lambda (t4: T).(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x1 t5) (THead (Bind +b) t4 t3))) (\lambda (t5: T).(ty3 g e (THead (Bind b) x2 x3) t5)))) (let H20 +\def (H18 x2 x1 (refl_equal T (lift h x1 x2)) e H8) 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 (H22: (ty3 g e x2 +x4)).(let H23 \def (H15 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e +(Bind b) x2) (drop_skip_bind h x1 c0 e H8 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 (H24: (pc3 (CHead c0 +(Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H25: (ty3 g (CHead +e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t4: T).(ty3 g (CHead e (Bind b) +x2) x5 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 (x6: T).(\lambda (H26: (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 (t4: T).(pc3 c0 t4 (THead (Bind b) (lift h x1 x2) t3))) +(pc3_head_2 c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H24) (lift h x1 +(THead (Bind b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H22 b +x3 x5 H25 x6 H26)))) (ty3_correct g (CHead e (Bind b) x2) x3 x5 H25))))) +H23))))) H20)) u H10))))))))) x0 H9)))))) (lift_gen_bind b u t2 x0 h x1 +H7)))))))))))))))))))) (\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)).(ex4_3_ind T T +T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 e (THead (Bind Abst) +x6 t2) x8)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g e x6 +t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead e (Bind +Abst) x6) x7 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g +(CHead e (Bind Abst) x6) t2 t3)))) (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 (x11: 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)).(\lambda (H27: (ty3 g (CHead e (Bind Abst) x6) x9 +x11)).(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 x11 H27) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7) +H20))))))))))) (ty3_gen_bind g Abst e x6 x7 x8 (ty3_sred_pr3 e x5 (THead +(Bind Abst) x6 x7) H20 g x8 H23))))) (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) 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) 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) 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) (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) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e (THead +(Flat Cast) x2 x3) t4))) (\lambda (x5: T).(\lambda (_: (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) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) +x2 x3) t4)) x2 (pc3_refl c0 (lift h x1 x2)) (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))))))). + +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)).(ty3_conv g c t2 x (ty3_sred_pr3 c t1 t2 H0 g x H1) u t1 H +(pc3_pr3_r c t1 t2 H0)))) (ty3_correct g c u t1 H)))))))). + +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)).(ty3_unique g c x +t1 (ty3_sred_pr3 c u1 x H3 g t1 H) t2 (ty3_sred_pr3 c u2 x H4 g t2 H0))))) +H2)))))))))). + +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)))))))))). + +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)))))))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/props.ma new file mode 100644 index 000000000..d7615f70f --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/props.ma @@ -0,0 +1,422 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/props". + +include "ty3/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_comm 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_comm 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_comm 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_comm 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 (t4: T).(\lambda (_: (ty3 g +(CHead c (Bind b) u) t3 t4)).(\lambda (H5: ((\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 t3) (lift h d t4)))))))).(\lambda (c0: C).(\lambda (d: +nat).(\lambda (h: nat).(\lambda (H6: (drop h d c0 c)).(eq_ind_r T (THead +(Bind b) (lift h d u) (lift h (s (Bind b) d) t0)) (\lambda (t5: T).(ty3 g c0 +t5 (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 (t5: T).(ty3 g c0 (THead (Bind b) +(lift h d u) (lift h (s (Bind b) d) t0)) t5)) (ty3_bind g c0 (lift h d u) +(lift h d t) (H1 c0 d h H6) 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 H6 b u)) (lift h +(S d) t4) (H5 (CHead c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0 +c H6 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 t3))) (ty3_cast g +c0 (lift h (s (Flat Cast) d) t0) (lift h d t3) (H1 c0 d h H4) (lift h d t4) +(H3 c0 d h H4)) (lift h d (THead (Flat Cast) t3 t0)) (lift_head (Flat Cast) +t3 t0 h d)))))))))))))) e t1 t2 H))))). + +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 (_: (ex T +(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))).(\lambda (t4: +T).(\lambda (H4: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H5: (ex T +(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t4 t5)))).(let H6 \def H5 in +(ex_ind T (\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t4 t5)) (ex T +(\lambda (t5: T).(ty3 g c0 (THead (Bind b) u t3) t5))) (\lambda (x: +T).(\lambda (H7: (ty3 g (CHead c0 (Bind b) u) t4 x)).(ex_intro T (\lambda +(t5: T).(ty3 g c0 (THead (Bind b) u t3) t5)) (THead (Bind b) u t4) (ty3_bind +g c0 u t H0 b t3 t4 H4 x H7)))) H6))))))))))))))) (\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)).(ex4_3_ind T T T (\lambda (t3: T).(\lambda +(_: T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u t3) x0)))) (\lambda (_: +T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t3: +T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u) t t3)))) +(\lambda (t3: T).(\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind +Abst) u) t3 t4)))) (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 +(x3: 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)).(\lambda +(H11: (ty3 g (CHead c0 (Bind Abst) u) x1 x3)).(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 x3 H11)))))))))) (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 (H1: (ex T +(\lambda (t: T).(ty3 g c0 t3 t)))).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 +t4)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g c0 t4 t)))).H1)))))))) c t1 t2 +H))))). + +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 (_: (ty3 g (CHead c0 (Bind b) u0) t2 +t3)).(\lambda (_: ((\forall (t4: T).((ty3 g (CHead c0 (Bind b) u0) t2 t4) \to +(pc3 (CHead c0 (Bind b) u0) t3 t4))))).(\lambda (t4: T).(\lambda (H6: (ty3 g +c0 (THead (Bind b) u0 t0) t4)).(ex4_3_ind T T T (\lambda (t5: T).(\lambda (_: +T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u0 t5) t4)))) (\lambda (_: +T).(\lambda (t6: T).(\lambda (_: T).(ty3 g c0 u0 t6)))) (\lambda (t5: +T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u0) t0 t5)))) +(\lambda (t5: T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead c0 (Bind b) +u0) t5 t7)))) (pc3 c0 (THead (Bind b) u0 t2) t4) (\lambda (x0: T).(\lambda +(x1: T).(\lambda (x2: T).(\lambda (H7: (pc3 c0 (THead (Bind b) u0 x0) +t4)).(\lambda (_: (ty3 g c0 u0 x1)).(\lambda (H9: (ty3 g (CHead c0 (Bind b) +u0) t0 x0)).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) x0 x2)).(pc3_t (THead +(Bind b) u0 x0) c0 (THead (Bind b) u0 t2) (pc3_head_2 c0 u0 t2 x0 (Bind b) +(H3 x0 H9)) t4 H7)))))))) (ty3_gen_bind g b c0 u0 t0 t4 H6))))))))))))))))) +(\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 (_: ((\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)).(and_ind (pc3 c0 t2 t4) (ty3 g c0 t0 t2) (pc3 c0 t2 t4) (\lambda (H5: +(pc3 c0 t2 t4)).(\lambda (_: (ty3 g c0 t0 t2)).H5)) (ty3_gen_cast g c0 t0 t2 +t4 H4)))))))))))) c u t1 H))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1.ma new file mode 100644 index 000000000..eb09bb829 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1.ma @@ -0,0 +1,1149 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1". + +include "ty3/props.ma". + +include "pc3/subst1.ma". + +include "pc3/fwd.ma". + +include "csubst1/getl.ma". + +include "csubst1/fwd.ma". + +include "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_comm 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_comm (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_comm 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_comm (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 (t0: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) +t4 t0)).(\lambda (H5: ((\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 t4 +(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t0 (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 +(H6: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H7: +(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H8: (drop (S O) d a0 a)).(let +H9 \def (H1 e u0 d H6 a0 H7 a H8) 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 (H10: +(subst1 d u0 u (lift (S O) d x0))).(\lambda (_: (subst1 d u0 t (lift (S O) d +x1))).(\lambda (H12: (ty3 g a x0 x1)).(let H13 \def (H5 e u0 (S d) (getl_head +(Bind b) d c0 (CHead e (Bind Abbr) u0) H6 u) (CHead a0 (Bind b) (lift (S O) d +x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H10 c0 a0 H7) (CHead a (Bind b) +x0) (drop_skip_bind (S O) d a0 a H8 b x0)) in (ex3_2_ind T T (\lambda (y1: +T).(\lambda (_: T).(subst1 (S d) u0 t4 (lift (S O) (S d) y1)))) (\lambda (_: +T).(\lambda (y2: T).(subst1 (S d) u0 t0 (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 (H14: (subst1 (S d) u0 t4 (lift (S +O) (S d) x2))).(\lambda (_: (subst1 (S d) u0 t0 (lift (S O) (S d) +x3))).(\lambda (H16: (ty3 g (CHead a (Bind b) x0) x2 x3)).(let H17 \def (H3 e +u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind Abbr) u0) H6 u) (CHead a0 +(Bind b) (lift (S O) d x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H10 c0 +a0 H7) (CHead a (Bind b) x0) (drop_skip_bind (S O) d a0 a H8 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 (x4: T).(\lambda (x5: T).(\lambda (H18: +(subst1 (S d) u0 t3 (lift (S O) (S d) x4))).(\lambda (H19: (subst1 (S d) u0 +t4 (lift (S O) (S d) x5))).(\lambda (H20: (ty3 g (CHead a (Bind b) x0) x4 +x5)).(let H21 \def (eq_ind T x5 (\lambda (t5: T).(ty3 g (CHead a (Bind b) x0) +x4 t5)) H20 x2 (subst1_confluence_lift t4 x5 u0 (S d) H19 x2 H14)) in +(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 x4) (THead (Bind b) x0 x2) (eq_ind_r +T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x4)) (\lambda (t5: +T).(subst1 d u0 (THead (Bind b) u t3) t5)) (subst1_head u0 u (lift (S O) d +x0) d H10 (Bind b) t3 (lift (S O) (S d) x4) H18) (lift (S O) d (THead (Bind +b) x0 x4)) (lift_bind b x0 x4 (S O) d)) (eq_ind_r T (THead (Bind b) (lift (S +O) d x0) (lift (S O) (S d) x2)) (\lambda (t5: T).(subst1 d u0 (THead (Bind b) +u t4) t5)) (subst1_head u0 u (lift (S O) d x0) d H10 (Bind b) t4 (lift (S O) +(S d) x2) H14) (lift (S O) d (THead (Bind b) x0 x2)) (lift_bind b x0 x2 (S O) +d)) (ty3_bind g a x0 x1 H12 b x4 x2 H21 x3 H16)))))))) H17))))))) H13))))))) +H9))))))))))))))))))))))) (\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)).(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 (Bind Abst) 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 (H15: (eq T (lift (S O) d x1) (THead (Bind Abst) x4 +x5))).(\lambda (H16: (subst1 d u0 u x4)).(\lambda (H17: (subst1 (s (Bind +Abst) d) u0 t x5)).(let H18 \def (sym_equal T (lift (S O) d x1) (THead (Bind +Abst) x4 x5) H15) 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 (H19: (eq T x1 (THead (Bind Abst) x6 x7))).(\lambda (H20: (eq T +x4 (lift (S O) d x6))).(\lambda (H21: (eq T x5 (lift (S O) (S d) x7))).(let +H22 \def (eq_ind T x5 (\lambda (t0: T).(subst1 (s (Bind Abst) d) u0 t t0)) +H17 (lift (S O) (S d) x7) H21) in (let H23 \def (eq_ind T x4 (\lambda (t0: +T).(subst1 d u0 u t0)) H16 (lift (S O) d x6) H20) in (let H24 \def (eq_ind T +x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead (Bind Abst) x6 x7) H19) in +(let H25 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g a x0 (THead (Bind Abst) t0 +x7))) H24 x3 (subst1_confluence_lift u x6 u0 d H23 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) H22) (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 H25))))))))))) (lift_gen_bind Abst x4 x5 x1 (S O) d H18)))))))) +(subst1_gen_head (Bind Abst) u0 u t (lift (S O) d x1) d H9))))))) 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 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 (_: (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 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 t4 (lift (S O) d y2)))) +(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2) +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)) H8 (ty3_cast g a +x2 x0 H15 x1 H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))). + +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_comm 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_comm (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_comm 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_comm (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 (t0: +T).(\lambda (H4: (ty3 g (CHead c0 (Bind b) u) t4 t0)).(\lambda (H5: ((\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 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 (u0: T).(\lambda (d: nat).(\lambda (H6: (getl d c0 (CHead e (Bind +Void) u0))).(\lambda (a: C).(\lambda (H7: (drop (S O) d c0 a)).(let H8 \def +(H1 e u0 d H6 a H7) 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 (H9: (eq T u (lift (S O) d x0))).(\lambda +(H10: (eq T t (lift (S O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 +\def (eq_ind T t (\lambda (t5: T).(ty3 g c0 u t5)) H0 (lift (S O) d x1) H10) +in (let H13 \def (eq_ind T u (\lambda (t5: T).(ty3 g c0 t5 (lift (S O) d +x1))) H12 (lift (S O) d x0) H9) in (let H14 \def (eq_ind T u (\lambda (t5: +T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 (CHead c0 +(Bind b) t5) (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 +(CHead c0 (Bind b) t5) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(eq T t4 (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))))))))))) H5 (lift (S O) d x0) H9) in (let H15 \def (eq_ind T u (\lambda +(t5: T).(ty3 g (CHead c0 (Bind b) t5) t4 t0)) H4 (lift (S O) d x0) H9) in +(let H16 \def (eq_ind T u (\lambda (t5: T).(\forall (e0: C).(\forall (u1: +T).(\forall (d0: nat).((getl d0 (CHead c0 (Bind b) t5) (CHead e0 (Bind Void) +u1)) \to (\forall (a0: C).((drop (S O) d0 (CHead c0 (Bind b) t5) 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) H9) in +(let H17 \def (eq_ind T u (\lambda (t5: T).(ty3 g (CHead c0 (Bind b) t5) t3 +t4)) H2 (lift (S O) d x0) H9) in (eq_ind_r T (lift (S O) d x0) (\lambda (t5: +T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) t5 t3) +(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) +t5 t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 +y2))))) (let H18 \def (H16 e u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind +Void) u0) H6 (lift (S O) d x0)) (CHead a (Bind b) x0) (drop_skip_bind (S O) d +c0 a H7 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 (H19: (eq T t3 (lift (S O) (S d) x2))).(\lambda (H20: (eq T t4 +(lift (S O) (S d) x3))).(\lambda (H21: (ty3 g (CHead a (Bind b) x0) x2 +x3)).(let H22 \def (eq_ind T t4 (\lambda (t5: T).(\forall (e0: C).(\forall +(u1: T).(\forall (d0: nat).((getl d0 (CHead c0 (Bind b) (lift (S O) d x0)) +(CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 (CHead c0 +(Bind b) (lift (S O) d x0)) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(eq T t5 (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))))))))))) H14 (lift (S O) (S d) x3) H20) in (eq_ind_r T (lift (S O) (S d) +x3) (\lambda (t5: 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) t5) (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 (t5: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T +(THead (Bind b) (lift (S O) d x0) t5) (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))))) (let H23 \def (H22 e u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind +Void) u0) H6 (lift (S O) d x0)) (CHead a (Bind b) x0) (drop_skip_bind (S O) d +c0 a H7 b x0)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift +(S O) (S d) x3) (lift (S O) (S d) y1)))) (\lambda (_: T).(\lambda (y2: T).(eq +T t0 (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) (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)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H24: (eq T +(lift (S O) (S d) x3) (lift (S O) (S d) x4))).(\lambda (_: (eq T t0 (lift (S +O) (S d) x5))).(\lambda (H26: (ty3 g (CHead a (Bind b) x0) x4 x5)).(let H27 +\def (eq_ind_r T x4 (\lambda (t5: T).(ty3 g (CHead a (Bind b) x0) t5 x5)) H26 +x3 (lift_inj x3 x4 (S O) (S d) H24)) in (eq_ind T (lift (S O) d (THead (Bind +b) x0 x2)) (\lambda (t5: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T +t5 (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))))) (eq_ind T (lift (S O) d (THead +(Bind b) x0 x3)) (\lambda (t5: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: +T).(eq T (lift (S O) d (THead (Bind b) x0 x2)) (lift (S O) d y1)))) (\lambda +(_: T).(\lambda (y2: T).(eq T t5 (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 (Bind b) x0 x2)) (lift (S O) d +y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Bind b) +x0 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) (refl_equal T (lift (S O) +d (THead (Bind b) x0 x2))) (refl_equal T (lift (S O) d (THead (Bind b) x0 +x3))) (ty3_bind g a x0 x1 H11 b x2 x3 H21 x5 H27)) (THead (Bind b) (lift (S +O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3 (S O) d)) (THead (Bind b) +(lift (S O) d x0) (lift (S O) (S d) x2)) (lift_bind b x0 x2 (S O) d)))))))) +H23)) t3 H19) t4 H20))))))) H18)) u H9)))))))))))) H8))))))))))))))))))))) +(\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 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 (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 +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 (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 (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 (lift (S O) d x0) +(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 x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: +T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2) x0 (refl_equal T (lift (S O) d +(THead (Flat Cast) x0 x2))) (refl_equal T (lift (S O) d x0)) (ty3_cast g a x2 +x0 H19 x1 H9)) (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)))))))))) +H6)))))))))))))))) c t1 t2 H))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/tau0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/tau0.ma new file mode 100644 index 000000000..ff9b2da36 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/tau0.ma @@ -0,0 +1,634 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/tau0". + +include "ty3/pr3_props.ma". + +include "tau0/defs.ma". + +theorem ty3_tau0: + \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u +t1) \to (\forall (t2: T).((tau0 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).((tau0 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).((tau0 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).((tau0 g c0 u0 t4) \to (ty3 g c0 u0 t4))))).(\lambda (_: +(pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (tau0 g c0 u0 t0)).(H3 t0 +H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda +(H0: (tau0 g c0 (TSort m) t2)).(let H1 \def (match H0 in tau0 return (\lambda +(c1: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (tau0 ? c1 t t0)).((eq +C c1 c0) \to ((eq T t (TSort m)) \to ((eq T t0 t2) \to (ty3 g c0 (TSort m) +t2)))))))) with [(tau0_sort c1 n) \Rightarrow (\lambda (H1: (eq C c1 +c0)).(\lambda (H2: (eq T (TSort n) (TSort m))).(\lambda (H3: (eq T (TSort +(next g n)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) (TSort m)) \to +((eq T (TSort (next g n)) t2) \to (ty3 g c0 (TSort m) t2)))) (\lambda (H4: +(eq T (TSort n) (TSort m))).(let H5 \def (f_equal T nat (\lambda (e: +T).(match e in T return (\lambda (_: T).nat) with [(TSort n0) \Rightarrow n0 +| (TLRef _) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TSort n) (TSort +m) H4) in (eq_ind nat m (\lambda (n0: nat).((eq T (TSort (next g n0)) t2) \to +(ty3 g c0 (TSort m) t2))) (\lambda (H6: (eq T (TSort (next g m)) t2)).(eq_ind +T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t)) (ty3_sort g c0 +m) t2 H6)) n (sym_eq nat n m H5)))) c1 (sym_eq C c1 c0 H1) H2 H3)))) | +(tau0_abbr c1 d v i H1 w H2) \Rightarrow (\lambda (H3: (eq C c1 c0)).(\lambda +(H4: (eq T (TLRef i) (TSort m))).(\lambda (H5: (eq T (lift (S i) O w) +t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) (TSort m)) \to ((eq T +(lift (S i) O w) t2) \to ((getl i c2 (CHead d (Bind Abbr) v)) \to ((tau0 g d +v w) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H6: (eq T (TLRef i) (TSort +m))).(let H7 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort m) H6) in +(False_ind ((eq T (lift (S i) O w) t2) \to ((getl i c0 (CHead d (Bind Abbr) +v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))) H7))) c1 (sym_eq C c1 +c0 H3) H4 H5 H1 H2)))) | (tau0_abst c1 d v i H1 w H2) \Rightarrow (\lambda +(H3: (eq C c1 c0)).(\lambda (H4: (eq T (TLRef i) (TSort m))).(\lambda (H5: +(eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) +(TSort m)) \to ((eq T (lift (S i) O v) t2) \to ((getl i c2 (CHead d (Bind +Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H6: +(eq T (TLRef i) (TSort m))).(let H7 \def (eq_ind T (TLRef i) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I +(TSort m) H6) in (False_ind ((eq T (lift (S i) O v) t2) \to ((getl i c0 +(CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))) +H7))) c1 (sym_eq C c1 c0 H3) H4 H5 H1 H2)))) | (tau0_bind b c1 v t0 t3 H1) +\Rightarrow (\lambda (H2: (eq C c1 c0)).(\lambda (H3: (eq T (THead (Bind b) v +t0) (TSort m))).(\lambda (H4: (eq T (THead (Bind b) v t3) t2)).(eq_ind C c0 +(\lambda (c2: C).((eq T (THead (Bind b) v t0) (TSort m)) \to ((eq T (THead +(Bind b) v t3) t2) \to ((tau0 g (CHead c2 (Bind b) v) t0 t3) \to (ty3 g c0 +(TSort m) t2))))) (\lambda (H5: (eq T (THead (Bind b) v t0) (TSort m))).(let +H6 \def (eq_ind T (THead (Bind b) v t0) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort m) H5) in +(False_ind ((eq T (THead (Bind b) v t3) t2) \to ((tau0 g (CHead c0 (Bind b) +v) t0 t3) \to (ty3 g c0 (TSort m) t2))) H6))) c1 (sym_eq C c1 c0 H2) H3 H4 +H1)))) | (tau0_appl c1 v t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c1 +c0)).(\lambda (H3: (eq T (THead (Flat Appl) v t0) (TSort m))).(\lambda (H4: +(eq T (THead (Flat Appl) v t3) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T +(THead (Flat Appl) v t0) (TSort m)) \to ((eq T (THead (Flat Appl) v t3) t2) +\to ((tau0 g c2 t0 t3) \to (ty3 g c0 (TSort m) t2))))) (\lambda (H5: (eq T +(THead (Flat Appl) v t0) (TSort m))).(let H6 \def (eq_ind T (THead (Flat +Appl) v t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) +\Rightarrow True])) I (TSort m) H5) in (False_ind ((eq T (THead (Flat Appl) v +t3) t2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (TSort m) t2))) H6))) c1 (sym_eq +C c1 c0 H2) H3 H4 H1)))) | (tau0_cast c1 v1 v2 H1 t0 t3 H2) \Rightarrow +(\lambda (H3: (eq C c1 c0)).(\lambda (H4: (eq T (THead (Flat Cast) v1 t0) +(TSort m))).(\lambda (H5: (eq T (THead (Flat Cast) v2 t3) t2)).(eq_ind C c0 +(\lambda (c2: C).((eq T (THead (Flat Cast) v1 t0) (TSort m)) \to ((eq T +(THead (Flat Cast) v2 t3) t2) \to ((tau0 g c2 v1 v2) \to ((tau0 g c2 t0 t3) +\to (ty3 g c0 (TSort m) t2)))))) (\lambda (H6: (eq T (THead (Flat Cast) v1 +t0) (TSort m))).(let H7 \def (eq_ind T (THead (Flat Cast) v1 t0) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I +(TSort m) H6) in (False_ind ((eq T (THead (Flat Cast) v2 t3) t2) \to ((tau0 g +c0 v1 v2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (TSort m) t2)))) H7))) c1 +(sym_eq C c1 c0 H3) H4 H5 H1 H2))))]) in (H1 (refl_equal C c0) (refl_equal T +(TSort m)) (refl_equal T t2))))))) (\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).((tau0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: +T).(\lambda (H3: (tau0 g c0 (TLRef n) t2)).(let H4 \def (match H3 in tau0 +return (\lambda (c1: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (tau0 +? c1 t0 t3)).((eq C c1 c0) \to ((eq T t0 (TLRef n)) \to ((eq T t3 t2) \to +(ty3 g c0 (TLRef n) t2)))))))) with [(tau0_sort c1 n0) \Rightarrow (\lambda +(H4: (eq C c1 c0)).(\lambda (H5: (eq T (TSort n0) (TLRef n))).(\lambda (H6: +(eq T (TSort (next g n0)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n0) +(TLRef n)) \to ((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2)))) +(\lambda (H7: (eq T (TSort n0) (TLRef n))).(let H8 \def (eq_ind T (TSort n0) +(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow +False])) I (TLRef n) H7) in (False_ind ((eq T (TSort (next g n0)) t2) \to +(ty3 g c0 (TLRef n) t2)) H8))) c1 (sym_eq C c1 c0 H4) H5 H6)))) | (tau0_abbr +c1 d0 v i H4 w H5) \Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq +T (TLRef i) (TLRef n))).(\lambda (H8: (eq T (lift (S i) O w) t2)).(eq_ind C +c0 (\lambda (c2: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O w) +t2) \to ((getl i c2 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g +c0 (TLRef n) t2)))))) (\lambda (H9: (eq T (TLRef i) (TLRef n))).(let H10 \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) H9) in (eq_ind nat n (\lambda (n0: +nat).((eq T (lift (S n0) O w) t2) \to ((getl n0 c0 (CHead d0 (Bind Abbr) v)) +\to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H11: (eq T +(lift (S n) O w) t2)).(eq_ind T (lift (S n) O w) (\lambda (t0: T).((getl n c0 +(CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t0)))) +(\lambda (H12: (getl n c0 (CHead d0 (Bind Abbr) v))).(\lambda (H13: (tau0 g +d0 v w)).(let H14 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c2: +C).(getl n c0 c2)) H0 (CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind +Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H12)) in (let H15 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow d | (CHead c2 _ _) \Rightarrow c2])) (CHead d (Bind Abbr) u0) +(CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead +d0 (Bind Abbr) v) H12)) in ((let H16 \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 d0 (Bind Abbr) v) +(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H12)) in +(\lambda (H17: (eq C d d0)).(let H18 \def (eq_ind_r T v (\lambda (t0: +T).(getl n c0 (CHead d0 (Bind Abbr) t0))) H14 u0 H16) in (let H19 \def +(eq_ind_r T v (\lambda (t0: T).(tau0 g d0 t0 w)) H13 u0 H16) in (let H20 \def +(eq_ind_r C d0 (\lambda (c2: C).(getl n c0 (CHead c2 (Bind Abbr) u0))) H18 d +H17) in (let H21 \def (eq_ind_r C d0 (\lambda (c2: C).(tau0 g c2 u0 w)) H19 d +H17) in (ty3_abbr g n c0 d u0 H20 w (H2 w H21)))))))) H15))))) t2 H11)) i +(sym_eq nat i n H10)))) c1 (sym_eq C c1 c0 H6) H7 H8 H4 H5)))) | (tau0_abst +c1 d0 v i H4 w H5) \Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq +T (TLRef i) (TLRef n))).(\lambda (H8: (eq T (lift (S i) O v) t2)).(eq_ind C +c0 (\lambda (c2: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O v) +t2) \to ((getl i c2 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g +c0 (TLRef n) t2)))))) (\lambda (H9: (eq T (TLRef i) (TLRef n))).(let H10 \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) H9) in (eq_ind nat n (\lambda (n0: +nat).((eq T (lift (S n0) O v) t2) \to ((getl n0 c0 (CHead d0 (Bind Abst) v)) +\to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H11: (eq T +(lift (S n) O v) t2)).(eq_ind T (lift (S n) O v) (\lambda (t0: T).((getl n c0 +(CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t0)))) +(\lambda (H12: (getl n c0 (CHead d0 (Bind Abst) v))).(\lambda (_: (tau0 g d0 +v w)).(let H14 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c2: C).(getl +n c0 c2)) H0 (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abbr) u0) +n H0 (CHead d0 (Bind Abst) v) H12)) in (let H15 \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 d0 (Bind +Abst) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abst) v) +H12)) in (False_ind (ty3 g c0 (TLRef n) (lift (S n) O v)) H15))))) t2 H11)) i +(sym_eq nat i n H10)))) c1 (sym_eq C c1 c0 H6) H7 H8 H4 H5)))) | (tau0_bind b +c1 v t0 t3 H4) \Rightarrow (\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T +(THead (Bind b) v t0) (TLRef n))).(\lambda (H7: (eq T (THead (Bind b) v t3) +t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Bind b) v t0) (TLRef n)) +\to ((eq T (THead (Bind b) v t3) t2) \to ((tau0 g (CHead c2 (Bind b) v) t0 +t3) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H8: (eq T (THead (Bind b) v t0) +(TLRef n))).(let H9 \def (eq_ind T (THead (Bind b) v t0) (\lambda (e: +T).(match e 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) v t3) t2) \to ((tau0 g +(CHead c0 (Bind b) v) t0 t3) \to (ty3 g c0 (TLRef n) t2))) H9))) c1 (sym_eq C +c1 c0 H5) H6 H7 H4)))) | (tau0_appl c1 v t0 t3 H4) \Rightarrow (\lambda (H5: +(eq C c1 c0)).(\lambda (H6: (eq T (THead (Flat Appl) v t0) (TLRef +n))).(\lambda (H7: (eq T (THead (Flat Appl) v t3) t2)).(eq_ind C c0 (\lambda +(c2: C).((eq T (THead (Flat Appl) v t0) (TLRef n)) \to ((eq T (THead (Flat +Appl) v t3) t2) \to ((tau0 g c2 t0 t3) \to (ty3 g c0 (TLRef n) t2))))) +(\lambda (H8: (eq T (THead (Flat Appl) v t0) (TLRef n))).(let H9 \def (eq_ind +T (THead (Flat Appl) v t0) (\lambda (e: T).(match e 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 +(Flat Appl) v t3) t2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (TLRef n) t2))) +H9))) c1 (sym_eq C c1 c0 H5) H6 H7 H4)))) | (tau0_cast c1 v1 v2 H4 t0 t3 H5) +\Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq T (THead (Flat +Cast) v1 t0) (TLRef n))).(\lambda (H8: (eq T (THead (Flat Cast) v2 t3) +t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Cast) v1 t0) (TLRef +n)) \to ((eq T (THead (Flat Cast) v2 t3) t2) \to ((tau0 g c2 v1 v2) \to +((tau0 g c2 t0 t3) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H9: (eq T +(THead (Flat Cast) v1 t0) (TLRef n))).(let H10 \def (eq_ind T (THead (Flat +Cast) v1 t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) +\Rightarrow True])) I (TLRef n) H9) in (False_ind ((eq T (THead (Flat Cast) +v2 t3) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (TLRef +n) t2)))) H10))) c1 (sym_eq C c1 c0 H6) H7 H8 H4 H5))))]) in (H4 (refl_equal +C c0) (refl_equal T (TLRef n)) (refl_equal T t2))))))))))))) (\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).((tau0 g d u0 t2) \to (ty3 g d u0 +t2))))).(\lambda (t2: T).(\lambda (H3: (tau0 g c0 (TLRef n) t2)).(let H4 \def +(match H3 in tau0 return (\lambda (c1: C).(\lambda (t0: T).(\lambda (t3: +T).(\lambda (_: (tau0 ? c1 t0 t3)).((eq C c1 c0) \to ((eq T t0 (TLRef n)) \to +((eq T t3 t2) \to (ty3 g c0 (TLRef n) t2)))))))) with [(tau0_sort c1 n0) +\Rightarrow (\lambda (H4: (eq C c1 c0)).(\lambda (H5: (eq T (TSort n0) (TLRef +n))).(\lambda (H6: (eq T (TSort (next g n0)) t2)).(eq_ind C c0 (\lambda (_: +C).((eq T (TSort n0) (TLRef n)) \to ((eq T (TSort (next g n0)) t2) \to (ty3 g +c0 (TLRef n) t2)))) (\lambda (H7: (eq T (TSort n0) (TLRef n))).(let H8 \def +(eq_ind T (TSort n0) (\lambda (e: T).(match e in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | +(THead _ _ _) \Rightarrow False])) I (TLRef n) H7) in (False_ind ((eq T +(TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2)) H8))) c1 (sym_eq C c1 c0 +H4) H5 H6)))) | (tau0_abbr c1 d0 v i H4 w H5) \Rightarrow (\lambda (H6: (eq C +c1 c0)).(\lambda (H7: (eq T (TLRef i) (TLRef n))).(\lambda (H8: (eq T (lift +(S i) O w) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) (TLRef n)) \to +((eq T (lift (S i) O w) t2) \to ((getl i c2 (CHead d0 (Bind Abbr) v)) \to +((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H9: (eq T (TLRef +i) (TLRef n))).(let H10 \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) H9) in +(eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O w) t2) \to ((getl n0 +c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) +t2))))) (\lambda (H11: (eq T (lift (S n) O w) t2)).(eq_ind T (lift (S n) O w) +(\lambda (t0: T).((getl n c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) +\to (ty3 g c0 (TLRef n) t0)))) (\lambda (H12: (getl n c0 (CHead d0 (Bind +Abbr) v))).(\lambda (_: (tau0 g d0 v w)).(let H14 \def (eq_ind C (CHead d +(Bind Abst) u0) (\lambda (c2: C).(getl n c0 c2)) H0 (CHead d0 (Bind Abbr) v) +(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abbr) v) H12)) in +(let H15 \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 d0 (Bind Abbr) v) (getl_mono c0 (CHead d +(Bind Abst) u0) n H0 (CHead d0 (Bind Abbr) v) H12)) in (False_ind (ty3 g c0 +(TLRef n) (lift (S n) O w)) H15))))) t2 H11)) i (sym_eq nat i n H10)))) c1 +(sym_eq C c1 c0 H6) H7 H8 H4 H5)))) | (tau0_abst c1 d0 v i H4 w H5) +\Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq T (TLRef i) (TLRef +n))).(\lambda (H8: (eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c2: +C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O v) t2) \to ((getl i +c2 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) +t2)))))) (\lambda (H9: (eq T (TLRef i) (TLRef n))).(let H10 \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) H9) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S +n0) O v) t2) \to ((getl n0 c0 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) +\to (ty3 g c0 (TLRef n) t2))))) (\lambda (H11: (eq T (lift (S n) O v) +t2)).(eq_ind T (lift (S n) O v) (\lambda (t0: T).((getl n c0 (CHead d0 (Bind +Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t0)))) (\lambda (H12: +(getl n c0 (CHead d0 (Bind Abst) v))).(\lambda (H13: (tau0 g d0 v w)).(let +H14 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c2: C).(getl n c0 c2)) +H0 (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 +(CHead d0 (Bind Abst) v) H12)) in (let H15 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | +(CHead c2 _ _) \Rightarrow c2])) (CHead d (Bind Abst) u0) (CHead d0 (Bind +Abst) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abst) v) +H12)) in ((let H16 \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 d0 (Bind Abst) v) +(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abst) v) H12)) in +(\lambda (H17: (eq C d d0)).(let H18 \def (eq_ind_r T v (\lambda (t0: +T).(getl n c0 (CHead d0 (Bind Abst) t0))) H14 u0 H16) in (let H19 \def +(eq_ind_r T v (\lambda (t0: T).(tau0 g d0 t0 w)) H13 u0 H16) in (eq_ind T u0 +(\lambda (t0: T).(ty3 g c0 (TLRef n) (lift (S n) O t0))) (let H20 \def +(eq_ind_r C d0 (\lambda (c2: C).(getl n c0 (CHead c2 (Bind Abst) u0))) H18 d +H17) in (let H21 \def (eq_ind_r C d0 (\lambda (c2: C).(tau0 g c2 u0 w)) H19 d +H17) in (ty3_abst g n c0 d u0 H20 t H1))) v H16))))) H15))))) t2 H11)) i +(sym_eq nat i n H10)))) c1 (sym_eq C c1 c0 H6) H7 H8 H4 H5)))) | (tau0_bind b +c1 v t0 t3 H4) \Rightarrow (\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T +(THead (Bind b) v t0) (TLRef n))).(\lambda (H7: (eq T (THead (Bind b) v t3) +t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Bind b) v t0) (TLRef n)) +\to ((eq T (THead (Bind b) v t3) t2) \to ((tau0 g (CHead c2 (Bind b) v) t0 +t3) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H8: (eq T (THead (Bind b) v t0) +(TLRef n))).(let H9 \def (eq_ind T (THead (Bind b) v t0) (\lambda (e: +T).(match e 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) v t3) t2) \to ((tau0 g +(CHead c0 (Bind b) v) t0 t3) \to (ty3 g c0 (TLRef n) t2))) H9))) c1 (sym_eq C +c1 c0 H5) H6 H7 H4)))) | (tau0_appl c1 v t0 t3 H4) \Rightarrow (\lambda (H5: +(eq C c1 c0)).(\lambda (H6: (eq T (THead (Flat Appl) v t0) (TLRef +n))).(\lambda (H7: (eq T (THead (Flat Appl) v t3) t2)).(eq_ind C c0 (\lambda +(c2: C).((eq T (THead (Flat Appl) v t0) (TLRef n)) \to ((eq T (THead (Flat +Appl) v t3) t2) \to ((tau0 g c2 t0 t3) \to (ty3 g c0 (TLRef n) t2))))) +(\lambda (H8: (eq T (THead (Flat Appl) v t0) (TLRef n))).(let H9 \def (eq_ind +T (THead (Flat Appl) v t0) (\lambda (e: T).(match e 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 +(Flat Appl) v t3) t2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (TLRef n) t2))) +H9))) c1 (sym_eq C c1 c0 H5) H6 H7 H4)))) | (tau0_cast c1 v1 v2 H4 t0 t3 H5) +\Rightarrow (\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq T (THead (Flat +Cast) v1 t0) (TLRef n))).(\lambda (H8: (eq T (THead (Flat Cast) v2 t3) +t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Cast) v1 t0) (TLRef +n)) \to ((eq T (THead (Flat Cast) v2 t3) t2) \to ((tau0 g c2 v1 v2) \to +((tau0 g c2 t0 t3) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H9: (eq T +(THead (Flat Cast) v1 t0) (TLRef n))).(let H10 \def (eq_ind T (THead (Flat +Cast) v1 t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) +\Rightarrow True])) I (TLRef n) H9) in (False_ind ((eq T (THead (Flat Cast) +v2 t3) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (TLRef +n) t2)))) H10))) c1 (sym_eq C c1 c0 H6) H7 H8 H4 H5))))]) in (H4 (refl_equal +C c0) (refl_equal T (TLRef n)) (refl_equal T t2))))))))))))) (\lambda (c0: +C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda +(_: ((\forall (t2: T).((tau0 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).((tau0 g (CHead c0 (Bind b) +u0) t2 t4) \to (ty3 g (CHead c0 (Bind b) u0) t2 t4))))).(\lambda (t0: +T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t3 t0)).(\lambda (_: ((\forall +(t4: T).((tau0 g (CHead c0 (Bind b) u0) t3 t4) \to (ty3 g (CHead c0 (Bind b) +u0) t3 t4))))).(\lambda (t4: T).(\lambda (H6: (tau0 g c0 (THead (Bind b) u0 +t2) t4)).(let H7 \def (match H6 in tau0 return (\lambda (c1: C).(\lambda (t5: +T).(\lambda (t6: T).(\lambda (_: (tau0 ? c1 t5 t6)).((eq C c1 c0) \to ((eq T +t5 (THead (Bind b) u0 t2)) \to ((eq T t6 t4) \to (ty3 g c0 (THead (Bind b) u0 +t2) t4)))))))) with [(tau0_sort c1 n) \Rightarrow (\lambda (H7: (eq C c1 +c0)).(\lambda (H8: (eq T (TSort n) (THead (Bind b) u0 t2))).(\lambda (H9: (eq +T (TSort (next g n)) t4)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) +(THead (Bind b) u0 t2)) \to ((eq T (TSort (next g n)) t4) \to (ty3 g c0 +(THead (Bind b) u0 t2) t4)))) (\lambda (H10: (eq T (TSort n) (THead (Bind b) +u0 t2))).(let H11 \def (eq_ind T (TSort n) (\lambda (e: T).(match e in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u0 +t2) H10) in (False_ind ((eq T (TSort (next g n)) t4) \to (ty3 g c0 (THead +(Bind b) u0 t2) t4)) H11))) c1 (sym_eq C c1 c0 H7) H8 H9)))) | (tau0_abbr c1 +d v i H7 w H8) \Rightarrow (\lambda (H9: (eq C c1 c0)).(\lambda (H10: (eq T +(TLRef i) (THead (Bind b) u0 t2))).(\lambda (H11: (eq T (lift (S i) O w) +t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) (THead (Bind b) u0 t2)) +\to ((eq T (lift (S i) O w) t4) \to ((getl i c2 (CHead d (Bind Abbr) v)) \to +((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda (H12: +(eq T (TLRef i) (THead (Bind b) u0 t2))).(let H13 \def (eq_ind T (TLRef i) +(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow +False])) I (THead (Bind b) u0 t2) H12) in (False_ind ((eq T (lift (S i) O w) +t4) \to ((getl i c0 (CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g +c0 (THead (Bind b) u0 t2) t4)))) H13))) c1 (sym_eq C c1 c0 H9) H10 H11 H7 +H8)))) | (tau0_abst c1 d v i H7 w H8) \Rightarrow (\lambda (H9: (eq C c1 +c0)).(\lambda (H10: (eq T (TLRef i) (THead (Bind b) u0 t2))).(\lambda (H11: +(eq T (lift (S i) O v) t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) +(THead (Bind b) u0 t2)) \to ((eq T (lift (S i) O v) t4) \to ((getl i c2 +(CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 +t2) t4)))))) (\lambda (H12: (eq T (TLRef i) (THead (Bind b) u0 t2))).(let H13 +\def (eq_ind T (TLRef i) (\lambda (e: T).(match e in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | +(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u0 t2) H12) in +(False_ind ((eq T (lift (S i) O v) t4) \to ((getl i c0 (CHead d (Bind Abst) +v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) H13))) c1 +(sym_eq C c1 c0 H9) H10 H11 H7 H8)))) | (tau0_bind b0 c1 v t5 t6 H7) +\Rightarrow (\lambda (H8: (eq C c1 c0)).(\lambda (H9: (eq T (THead (Bind b0) +v t5) (THead (Bind b) u0 t2))).(\lambda (H10: (eq T (THead (Bind b0) v t6) +t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Bind b0) v t5) (THead (Bind +b) u0 t2)) \to ((eq T (THead (Bind b0) v t6) t4) \to ((tau0 g (CHead c2 (Bind +b0) v) t5 t6) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))) (\lambda (H11: (eq +T (THead (Bind b0) v t5) (THead (Bind b) u0 t2))).(let H12 \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 b0) v t5) (THead (Bind b) u0 t2) H11) in ((let H13 \def (f_equal +T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t7 _) \Rightarrow t7])) +(THead (Bind b0) v t5) (THead (Bind b) u0 t2) H11) in ((let H14 \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 t5) (THead (Bind b) u0 t2) H11) in +(eq_ind B b (\lambda (b1: B).((eq T v u0) \to ((eq T t5 t2) \to ((eq T (THead +(Bind b1) v t6) t4) \to ((tau0 g (CHead c0 (Bind b1) v) t5 t6) \to (ty3 g c0 +(THead (Bind b) u0 t2) t4)))))) (\lambda (H15: (eq T v u0)).(eq_ind T u0 +(\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Bind b) t7 t6) t4) \to +((tau0 g (CHead c0 (Bind b) t7) t5 t6) \to (ty3 g c0 (THead (Bind b) u0 t2) +t4))))) (\lambda (H16: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T +(THead (Bind b) u0 t6) t4) \to ((tau0 g (CHead c0 (Bind b) u0) t7 t6) \to +(ty3 g c0 (THead (Bind b) u0 t2) t4)))) (\lambda (H17: (eq T (THead (Bind b) +u0 t6) t4)).(eq_ind T (THead (Bind b) u0 t6) (\lambda (t7: T).((tau0 g (CHead +c0 (Bind b) u0) t2 t6) \to (ty3 g c0 (THead (Bind b) u0 t2) t7))) (\lambda +(H18: (tau0 g (CHead c0 (Bind b) u0) t2 t6)).(let H_y \def (H3 t6 H18) in +(ex_ind T (\lambda (t7: T).(ty3 g (CHead c0 (Bind b) u0) t6 t7)) (ty3 g c0 +(THead (Bind b) u0 t2) (THead (Bind b) u0 t6)) (\lambda (x: T).(\lambda (H19: +(ty3 g (CHead c0 (Bind b) u0) t6 x)).(ty3_bind g c0 u0 t H0 b t2 t6 H_y x +H19))) (ty3_correct g (CHead c0 (Bind b) u0) t2 t6 H_y)))) t4 H17)) t5 +(sym_eq T t5 t2 H16))) v (sym_eq T v u0 H15))) b0 (sym_eq B b0 b H14))) H13)) +H12))) c1 (sym_eq C c1 c0 H8) H9 H10 H7)))) | (tau0_appl c1 v t5 t6 H7) +\Rightarrow (\lambda (H8: (eq C c1 c0)).(\lambda (H9: (eq T (THead (Flat +Appl) v t5) (THead (Bind b) u0 t2))).(\lambda (H10: (eq T (THead (Flat Appl) +v t6) t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Appl) v t5) +(THead (Bind b) u0 t2)) \to ((eq T (THead (Flat Appl) v t6) t4) \to ((tau0 g +c2 t5 t6) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))) (\lambda (H11: (eq T +(THead (Flat Appl) v t5) (THead (Bind b) u0 t2))).(let H12 \def (eq_ind T +(THead (Flat Appl) v 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) u0 t2) H11) in (False_ind ((eq T (THead (Flat Appl) v t6) t4) \to ((tau0 g +c0 t5 t6) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))) H12))) c1 (sym_eq C c1 +c0 H8) H9 H10 H7)))) | (tau0_cast c1 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda +(H9: (eq C c1 c0)).(\lambda (H10: (eq T (THead (Flat Cast) v1 t5) (THead +(Bind b) u0 t2))).(\lambda (H11: (eq T (THead (Flat Cast) v2 t6) t4)).(eq_ind +C c0 (\lambda (c2: C).((eq T (THead (Flat Cast) v1 t5) (THead (Bind b) u0 +t2)) \to ((eq T (THead (Flat Cast) v2 t6) t4) \to ((tau0 g c2 v1 v2) \to +((tau0 g c2 t5 t6) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda +(H12: (eq T (THead (Flat Cast) v1 t5) (THead (Bind b) u0 t2))).(let H13 \def +(eq_ind T (THead (Flat Cast) v1 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) u0 t2) H12) in (False_ind ((eq T (THead (Flat +Cast) v2 t6) t4) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t5 t6) \to (ty3 g c0 +(THead (Bind b) u0 t2) t4)))) H13))) c1 (sym_eq C c1 c0 H9) H10 H11 H7 +H8))))]) in (H7 (refl_equal C c0) (refl_equal T (THead (Bind b) u0 t2)) +(refl_equal T t4)))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda +(u0: T).(\lambda (H0: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((tau0 +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).((tau0 g c0 v t2) \to (ty3 g c0 v t2))))).(\lambda (t2: T).(\lambda (H4: +(tau0 g c0 (THead (Flat Appl) w v) t2)).(let H5 \def (match H4 in tau0 return +(\lambda (c1: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (tau0 ? c1 t0 +t3)).((eq C c1 c0) \to ((eq T t0 (THead (Flat Appl) w v)) \to ((eq T t3 t2) +\to (ty3 g c0 (THead (Flat Appl) w v) t2)))))))) with [(tau0_sort c1 n) +\Rightarrow (\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T (TSort n) (THead +(Flat Appl) w v))).(\lambda (H7: (eq T (TSort (next g n)) t2)).(eq_ind C c0 +(\lambda (_: C).((eq T (TSort n) (THead (Flat Appl) w v)) \to ((eq T (TSort +(next g n)) t2) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) (\lambda (H8: +(eq T (TSort n) (THead (Flat Appl) w v))).(let H9 \def (eq_ind T (TSort n) +(\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) w v) H8) in (False_ind ((eq T (TSort (next g +n)) t2) \to (ty3 g c0 (THead (Flat Appl) w v) t2)) H9))) c1 (sym_eq C c1 c0 +H5) H6 H7)))) | (tau0_abbr c1 d v0 i H5 w0 H6) \Rightarrow (\lambda (H7: (eq +C c1 c0)).(\lambda (H8: (eq T (TLRef i) (THead (Flat Appl) w v))).(\lambda +(H9: (eq T (lift (S i) O w0) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef +i) (THead (Flat Appl) w v)) \to ((eq T (lift (S i) O w0) t2) \to ((getl i c2 +(CHead d (Bind Abbr) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat +Appl) w v) t2)))))) (\lambda (H10: (eq T (TLRef i) (THead (Flat Appl) w +v))).(let H11 \def (eq_ind T (TLRef i) (\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) w +v) H10) in (False_ind ((eq T (lift (S i) O w0) t2) \to ((getl i c0 (CHead d +(Bind Abbr) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) +t2)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6)))) | (tau0_abst c1 d v0 i +H5 w0 H6) \Rightarrow (\lambda (H7: (eq C c1 c0)).(\lambda (H8: (eq T (TLRef +i) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (lift (S i) O v0) +t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef i) (THead (Flat Appl) w v)) +\to ((eq T (lift (S i) O v0) t2) \to ((getl i c2 (CHead d (Bind Abst) v0)) +\to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))))) (\lambda +(H10: (eq T (TLRef i) (THead (Flat Appl) w v))).(let H11 \def (eq_ind T +(TLRef i) (\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) w v) H10) in (False_ind ((eq T +(lift (S i) O v0) t2) \to ((getl i c0 (CHead d (Bind Abst) v0)) \to ((tau0 g +d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) H11))) c1 (sym_eq C c1 +c0 H7) H8 H9 H5 H6)))) | (tau0_bind b c1 v0 t0 t3 H5) \Rightarrow (\lambda +(H6: (eq C c1 c0)).(\lambda (H7: (eq T (THead (Bind b) v0 t0) (THead (Flat +Appl) w v))).(\lambda (H8: (eq T (THead (Bind b) v0 t3) t2)).(eq_ind C c0 +(\lambda (c2: C).((eq T (THead (Bind b) v0 t0) (THead (Flat Appl) w v)) \to +((eq T (THead (Bind b) v0 t3) t2) \to ((tau0 g (CHead c2 (Bind b) v0) t0 t3) +\to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H9: (eq T (THead +(Bind b) v0 t0) (THead (Flat Appl) w v))).(let H10 \def (eq_ind T (THead +(Bind b) v0 t0) (\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 b) v0 t3) t2) \to ((tau0 g (CHead c0 +(Bind b) v0) t0 t3) \to (ty3 g c0 (THead (Flat Appl) w v) t2))) H10))) c1 +(sym_eq C c1 c0 H6) H7 H8 H5)))) | (tau0_appl c1 v0 t0 t3 H5) \Rightarrow +(\lambda (H6: (eq C c1 c0)).(\lambda (H7: (eq T (THead (Flat Appl) v0 t0) +(THead (Flat Appl) w v))).(\lambda (H8: (eq T (THead (Flat Appl) v0 t3) +t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Appl) v0 t0) (THead +(Flat Appl) w v)) \to ((eq T (THead (Flat Appl) v0 t3) t2) \to ((tau0 g c2 t0 +t3) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H9: (eq T (THead +(Flat Appl) v0 t0) (THead (Flat Appl) w v))).(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 _ _ t4) \Rightarrow t4])) +(THead (Flat Appl) v0 t0) (THead (Flat Appl) w v) H9) in ((let H11 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t4 _) +\Rightarrow t4])) (THead (Flat Appl) v0 t0) (THead (Flat Appl) w v) H9) in +(eq_ind T w (\lambda (t4: T).((eq T t0 v) \to ((eq T (THead (Flat Appl) t4 +t3) t2) \to ((tau0 g c0 t0 t3) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) +(\lambda (H12: (eq T t0 v)).(eq_ind T v (\lambda (t4: T).((eq T (THead (Flat +Appl) w t3) t2) \to ((tau0 g c0 t4 t3) \to (ty3 g c0 (THead (Flat Appl) w v) +t2)))) (\lambda (H13: (eq T (THead (Flat Appl) w t3) t2)).(eq_ind T (THead +(Flat Appl) w t3) (\lambda (t4: T).((tau0 g c0 v t3) \to (ty3 g c0 (THead +(Flat Appl) w v) t4))) (\lambda (H14: (tau0 g c0 v t3)).(let H_y \def (H3 t3 +H14) in (let H15 \def (ty3_unique g c0 v t3 H_y (THead (Bind Abst) u0 t) H2) +in (ex_ind T (\lambda (t4: T).(ty3 g c0 t3 t4)) (ty3 g c0 (THead (Flat Appl) +w v) (THead (Flat Appl) w t3)) (\lambda (x: T).(\lambda (H16: (ty3 g c0 t3 +x)).(ex_ind T (\lambda (t4: T).(ty3 g c0 u0 t4)) (ty3 g c0 (THead (Flat Appl) +w v) (THead (Flat Appl) w t3)) (\lambda (x0: T).(\lambda (_: (ty3 g c0 u0 +x0)).(ex_ind T (\lambda (t4: T).(ty3 g c0 (THead (Bind Abst) u0 t) t4)) (ty3 +g c0 (THead (Flat Appl) w v) (THead (Flat Appl) w t3)) (\lambda (x1: +T).(\lambda (H18: (ty3 g c0 (THead (Bind Abst) u0 t) x1)).(ex4_3_ind T T T +(\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) +u0 t4) x1)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c0 u0 +t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 +(Bind Abst) u0) t t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: +T).(ty3 g (CHead c0 (Bind Abst) u0) t4 t6)))) (ty3 g c0 (THead (Flat Appl) w +v) (THead (Flat Appl) w t3)) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: +T).(\lambda (_: (pc3 c0 (THead (Bind Abst) u0 x2) x1)).(\lambda (H20: (ty3 g +c0 u0 x3)).(\lambda (H21: (ty3 g (CHead c0 (Bind Abst) u0) t x2)).(\lambda +(H22: (ty3 g (CHead c0 (Bind Abst) u0) x2 x4)).(ty3_conv g c0 (THead (Flat +Appl) w t3) (THead (Flat Appl) w (THead (Bind Abst) u0 x2)) (ty3_appl g c0 w +u0 H0 t3 x2 (ty3_sconv g c0 t3 x H16 (THead (Bind Abst) u0 t) (THead (Bind +Abst) u0 x2) (ty3_bind g c0 u0 x3 H20 Abst t x2 H21 x4 H22) H15)) (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_s c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t)) +(THead (Flat Appl) w t3) (pc3_thin_dx c0 t3 (THead (Bind Abst) u0 t) H15 w +Appl)))))))))) (ty3_gen_bind g Abst c0 u0 t x1 H18)))) (ty3_correct g c0 v +(THead (Bind Abst) u0 t) H2)))) (ty3_correct g c0 w u0 H0)))) (ty3_correct g +c0 v t3 H_y))))) t2 H13)) t0 (sym_eq T t0 v H12))) v0 (sym_eq T v0 w H11))) +H10))) c1 (sym_eq C c1 c0 H6) H7 H8 H5)))) | (tau0_cast c1 v1 v2 H5 t0 t3 H6) +\Rightarrow (\lambda (H7: (eq C c1 c0)).(\lambda (H8: (eq T (THead (Flat +Cast) v1 t0) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead (Flat Cast) +v2 t3) t2)).(eq_ind C c0 (\lambda (c2: C).((eq T (THead (Flat Cast) v1 t0) +(THead (Flat Appl) w v)) \to ((eq T (THead (Flat Cast) v2 t3) t2) \to ((tau0 +g c2 v1 v2) \to ((tau0 g c2 t0 t3) \to (ty3 g c0 (THead (Flat Appl) w v) +t2)))))) (\lambda (H10: (eq T (THead (Flat Cast) v1 t0) (THead (Flat Appl) w +v))).(let H11 \def (eq_ind T (THead (Flat Cast) v1 t0) (\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) H10) in (False_ind +((eq T (THead (Flat Cast) v2 t3) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 +t0 t3) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) H11))) c1 (sym_eq C c1 c0 +H7) H8 H9 H5 H6))))]) in (H5 (refl_equal C c0) (refl_equal T (THead (Flat +Appl) w v)) (refl_equal T t2)))))))))))))) (\lambda (c0: C).(\lambda (t2: +T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2 t3)).(\lambda (H1: ((\forall +(t4: T).((tau0 g c0 t2 t4) \to (ty3 g c0 t2 t4))))).(\lambda (t0: T).(\lambda +(_: (ty3 g c0 t3 t0)).(\lambda (H3: ((\forall (t4: T).((tau0 g c0 t3 t4) \to +(ty3 g c0 t3 t4))))).(\lambda (t4: T).(\lambda (H4: (tau0 g c0 (THead (Flat +Cast) t3 t2) t4)).(let H5 \def (match H4 in tau0 return (\lambda (c1: +C).(\lambda (t: T).(\lambda (t5: T).(\lambda (_: (tau0 ? c1 t t5)).((eq C c1 +c0) \to ((eq T t (THead (Flat Cast) t3 t2)) \to ((eq T t5 t4) \to (ty3 g c0 +(THead (Flat Cast) t3 t2) t4)))))))) with [(tau0_sort c1 n) \Rightarrow +(\lambda (H5: (eq C c1 c0)).(\lambda (H6: (eq T (TSort n) (THead (Flat Cast) +t3 t2))).(\lambda (H7: (eq T (TSort (next g n)) t4)).(eq_ind C c0 (\lambda +(_: C).((eq T (TSort n) (THead (Flat Cast) t3 t2)) \to ((eq T (TSort (next g +n)) t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))) (\lambda (H8: (eq T +(TSort n) (THead (Flat Cast) t3 t2))).(let H9 \def (eq_ind T (TSort n) +(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow +False])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T (TSort (next g +n)) t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)) H9))) c1 (sym_eq C c1 c0 +H5) H6 H7)))) | (tau0_abbr c1 d v i H5 w H6) \Rightarrow (\lambda (H7: (eq C +c1 c0)).(\lambda (H8: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(\lambda +(H9: (eq T (lift (S i) O w) t4)).(eq_ind C c0 (\lambda (c2: C).((eq T (TLRef +i) (THead (Flat Cast) t3 t2)) \to ((eq T (lift (S i) O w) t4) \to ((getl i c2 +(CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) +t3 t2) t4)))))) (\lambda (H10: (eq T (TLRef i) (THead (Flat Cast) t3 +t2))).(let H11 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t3 +t2) H10) in (False_ind ((eq T (lift (S i) O w) t4) \to ((getl i c0 (CHead d +(Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) +t4)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6)))) | (tau0_abst c1 d v i H5 +w H6) \Rightarrow (\lambda (H7: (eq C c1 c0)).(\lambda (H8: (eq T (TLRef i) +(THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T (lift (S i) O v) t4)).(eq_ind +C c0 (\lambda (c2: C).((eq T (TLRef i) (THead (Flat Cast) t3 t2)) \to ((eq T +(lift (S i) O v) t4) \to ((getl i c2 (CHead d (Bind Abst) v)) \to ((tau0 g d +v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H10: (eq T +(TLRef i) (THead (Flat Cast) t3 t2))).(let H11 \def (eq_ind T (TLRef i) +(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow +False])) I (THead (Flat Cast) t3 t2) H10) in (False_ind ((eq T (lift (S i) O +v) t4) \to ((getl i c0 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 +g c0 (THead (Flat Cast) t3 t2) t4)))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 +H6)))) | (tau0_bind b c1 v t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1 +c0)).(\lambda (H7: (eq T (THead (Bind b) v t5) (THead (Flat Cast) t3 +t2))).(\lambda (H8: (eq T (THead (Bind b) v t6) t4)).(eq_ind C c0 (\lambda +(c2: C).((eq T (THead (Bind b) v t5) (THead (Flat Cast) t3 t2)) \to ((eq T +(THead (Bind b) v t6) t4) \to ((tau0 g (CHead c2 (Bind b) v) t5 t6) \to (ty3 +g c0 (THead (Flat Cast) t3 t2) t4))))) (\lambda (H9: (eq T (THead (Bind b) v +t5) (THead (Flat Cast) t3 t2))).(let H10 \def (eq_ind T (THead (Bind b) v 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 b) v t6) t4) \to ((tau0 g (CHead c0 (Bind b) v) t5 t6) +\to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))) H10))) c1 (sym_eq C c1 c0 H6) +H7 H8 H5)))) | (tau0_appl c1 v t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1 +c0)).(\lambda (H7: (eq T (THead (Flat Appl) v t5) (THead (Flat Cast) t3 +t2))).(\lambda (H8: (eq T (THead (Flat Appl) v t6) t4)).(eq_ind C c0 (\lambda +(c2: C).((eq T (THead (Flat Appl) v t5) (THead (Flat Cast) t3 t2)) \to ((eq T +(THead (Flat Appl) v t6) t4) \to ((tau0 g c2 t5 t6) \to (ty3 g c0 (THead +(Flat Cast) t3 t2) t4))))) (\lambda (H9: (eq T (THead (Flat Appl) v t5) +(THead (Flat Cast) t3 t2))).(let H10 \def (eq_ind T (THead (Flat Appl) v 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) +H9) in (False_ind ((eq T (THead (Flat Appl) v t6) t4) \to ((tau0 g c0 t5 t6) +\to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))) H10))) c1 (sym_eq C c1 c0 H6) +H7 H8 H5)))) | (tau0_cast c1 v1 v2 H5 t5 t6 H6) \Rightarrow (\lambda (H7: (eq +C c1 c0)).(\lambda (H8: (eq T (THead (Flat Cast) v1 t5) (THead (Flat Cast) t3 +t2))).(\lambda (H9: (eq T (THead (Flat Cast) v2 t6) t4)).(eq_ind C c0 +(\lambda (c2: C).((eq T (THead (Flat Cast) v1 t5) (THead (Flat Cast) t3 t2)) +\to ((eq T (THead (Flat Cast) v2 t6) t4) \to ((tau0 g c2 v1 v2) \to ((tau0 g +c2 t5 t6) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H10: (eq +T (THead (Flat Cast) v1 t5) (THead (Flat Cast) t3 t2))).(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 _ _ t) +\Rightarrow t])) (THead (Flat Cast) v1 t5) (THead (Flat Cast) t3 t2) H10) in +((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 _ t +_) \Rightarrow t])) (THead (Flat Cast) v1 t5) (THead (Flat Cast) t3 t2) H10) +in (eq_ind T t3 (\lambda (t: T).((eq T t5 t2) \to ((eq T (THead (Flat Cast) +v2 t6) t4) \to ((tau0 g c0 t v2) \to ((tau0 g c0 t5 t6) \to (ty3 g c0 (THead +(Flat Cast) t3 t2) t4)))))) (\lambda (H13: (eq T t5 t2)).(eq_ind T t2 +(\lambda (t: T).((eq T (THead (Flat Cast) v2 t6) t4) \to ((tau0 g c0 t3 v2) +\to ((tau0 g c0 t t6) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))))) +(\lambda (H14: (eq T (THead (Flat Cast) v2 t6) t4)).(eq_ind T (THead (Flat +Cast) v2 t6) (\lambda (t: T).((tau0 g c0 t3 v2) \to ((tau0 g c0 t2 t6) \to +(ty3 g c0 (THead (Flat Cast) t3 t2) t)))) (\lambda (H15: (tau0 g c0 t3 +v2)).(\lambda (H16: (tau0 g c0 t2 t6)).(let H_y \def (H1 t6 H16) in (let H_y0 +\def (H3 v2 H15) in (let H17 \def (ty3_unique g c0 t2 t6 H_y t3 H0) in +(ex_ind T (\lambda (t: T).(ty3 g c0 v2 t)) (ty3 g c0 (THead (Flat Cast) t3 +t2) (THead (Flat Cast) v2 t6)) (\lambda (x: T).(\lambda (H18: (ty3 g c0 v2 +x)).(ex_ind T (\lambda (t: T).(ty3 g c0 t6 t)) (ty3 g c0 (THead (Flat Cast) +t3 t2) (THead (Flat Cast) v2 t6)) (\lambda (x0: T).(\lambda (H19: (ty3 g c0 +t6 x0)).(ty3_conv g c0 (THead (Flat Cast) v2 t6) v2 (ty3_cast g c0 t6 v2 +(ty3_sconv g c0 t6 x0 H19 t3 v2 H_y0 H17) x H18) (THead (Flat Cast) t3 t2) t3 +(ty3_cast g c0 t2 t3 H0 v2 H_y0) (pc3_s c0 t3 (THead (Flat Cast) v2 t6) +(pc3_pr2_u c0 t6 (THead (Flat Cast) v2 t6) (pr2_free c0 (THead (Flat Cast) v2 +t6) t6 (pr0_epsilon t6 t6 (pr0_refl t6) v2)) t3 H17))))) (ty3_correct g c0 t2 +t6 H_y)))) (ty3_correct g c0 t3 v2 H_y0))))))) t4 H14)) t5 (sym_eq T t5 t2 +H13))) v1 (sym_eq T v1 t3 H12))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 +H6))))]) in (H5 (refl_equal C c0) (refl_equal T (THead (Flat Cast) t3 t2)) +(refl_equal T t4))))))))))))) c u t1 H))))). + -- 2.39.2