From: Ferruccio Guidi Date: Wed, 6 Sep 2006 16:49:52 +0000 (+0000) Subject: some theorems about getl X-Git-Tag: 0.4.95@7852~1070 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=9a81424c15d8b80a50f62ffe5f5b3ced32dfa463;p=helm.git some theorems about getl --- diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma b/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma index 4e37576d2..98d1c163b 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma @@ -136,6 +136,7 @@ alias id "lt_n_Sn" = "cic:/Coq/Arith/Lt/lt_n_Sn.con". alias id "lt_le_trans" = "cic:/Coq/Arith/Lt/lt_le_trans.con". alias id "lt_wf_ind" = "cic:/Coq/Arith/Wf_nat/lt_wf_ind.con". alias id "bool_ind" = "cic:/Coq/Init/Datatypes/bool_ind.con". +alias id "ex_ind" = "cic:/Coq/Init/Logic/ex_ind.con". theorem f_equal: \forall A,B:Type. \forall f:A \to B. \forall x,y:A. x = y \to f x = f y. diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/cimp/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/cimp/defs.ma new file mode 100644 index 000000000..84e8f45bb --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/cimp/defs.ma @@ -0,0 +1,27 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/cimp/defs". + +include "getl/defs.ma". + +definition cimp: + C \to (C \to Prop) +\def + \lambda (c1: C).(\lambda (c2: C).(\forall (b: B).(\forall (d1: C).(\forall +(w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to (ex C +(\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w)))))))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/defs.ma new file mode 100644 index 000000000..3853cbad5 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/defs.ma @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/clear/defs". + +include "C/defs.ma". + +inductive clear: C \to (C \to Prop) \def +| clear_bind: \forall (b: B).(\forall (e: C).(\forall (u: T).(clear (CHead e +(Bind b) u) (CHead e (Bind b) u)))) +| clear_flat: \forall (e: C).(\forall (c: C).((clear e c) \to (\forall (f: +F).(\forall (u: T).(clear (CHead e (Flat f) u) c))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/drop.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/drop.ma new file mode 100644 index 000000000..d30e4dbde --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/drop.ma @@ -0,0 +1,175 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / 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/clear/drop". + +include "clear/fwd.ma". + +include "drop/fwd.ma". + +theorem drop_clear: + \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to +(ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c1 (CHead +e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e +c2)))))))) +\def + \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (i: +nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: +C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda +(e: C).(\lambda (_: T).(drop i O e c2))))))))) (\lambda (n: nat).(\lambda +(c2: C).(\lambda (i: nat).(\lambda (H: (drop (S i) O (CSort n) c2)).(and3_ind +(eq C c2 (CSort n)) (eq nat (S i) O) (eq nat O O) (ex2_3 B C T (\lambda (b: +B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) +(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda +(_: (eq C c2 (CSort n))).(\lambda (H1: (eq nat (S i) O)).(\lambda (_: (eq nat +O O)).(let H3 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat +return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow +True])) I O H1) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (e: +C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) (\lambda (_: +B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) H3))))) (drop_gen_sort +n (S i) O c2 H)))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall +(i: nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: +C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda +(e: C).(\lambda (_: T).(drop i O e c2)))))))))).(\lambda (k: K).(\lambda (t: +T).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O (CHead c k +t) c2)).((match k in K return (\lambda (k0: K).((drop (r k0 i) O c c2) \to +(ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c +k0 t) (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: +T).(drop i O e c2))))))) with [(Bind b) \Rightarrow (\lambda (H1: (drop (r +(Bind b) i) O c c2)).(ex2_3_intro B C T (\lambda (b0: B).(\lambda (e: +C).(\lambda (v: T).(clear (CHead c (Bind b) t) (CHead e (Bind b0) v))))) +(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) b c t +(clear_bind b c t) H1)) | (Flat f) \Rightarrow (\lambda (H1: (drop (r (Flat +f) i) O c c2)).(let H2 \def (H c2 i H1) in (ex2_3_ind B C T (\lambda (b: +B).(\lambda (e: C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda +(_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) (ex2_3 B C T +(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) +(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: +T).(drop i O e c2))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: +T).(\lambda (H3: (clear c (CHead x1 (Bind x0) x2))).(\lambda (H4: (drop i O +x1 c2)).(ex2_3_intro B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: +T).(clear (CHead c (Flat f) t) (CHead e (Bind b) v))))) (\lambda (_: +B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) x0 x1 x2 (clear_flat c +(CHead x1 (Bind x0) x2) H3 f t) H4)))))) H2)))]) (drop_gen_drop k c c2 t i +H0))))))))) c1). + +theorem drop_clear_O: + \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (u: T).((clear c +(CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1 +e2) \to (drop (S i) O c e2)))))))) +\def + \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: +C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: +C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 e2)))))))) +(\lambda (n: nat).(\lambda (e1: C).(\lambda (u: T).(\lambda (H: (clear (CSort +n) (CHead e1 (Bind b) u))).(\lambda (e2: C).(\lambda (i: nat).(\lambda (_: +(drop i O e1 e2)).(clear_gen_sort (CHead e1 (Bind b) u) n H (drop (S i) O +(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: +C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: +C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 +e2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (u: +T).(\lambda (H0: (clear (CHead c0 k t) (CHead e1 (Bind b) u))).(\lambda (e2: +C).(\lambda (i: nat).(\lambda (H1: (drop i O e1 e2)).((match k in K return +(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) u)) \to (drop (S +i) O (CHead c0 k0 t) e2))) with [(Bind b0) \Rightarrow (\lambda (H2: (clear +(CHead c0 (Bind b0) t) (CHead e1 (Bind b) u))).(let H3 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 (Bind b) u) (CHead +c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in ((let +H4 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) +with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k in K +return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow +b])])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 +(CHead e1 (Bind b) u) t H2)) in ((let H5 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t) \Rightarrow t])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) +(clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in (\lambda (H6: (eq B b +b0)).(\lambda (H7: (eq C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c: +C).(drop i O c e2)) H1 c0 H7) in (eq_ind B b (\lambda (b1: B).(drop (S i) O +(CHead c0 (Bind b1) t) e2)) (drop_drop (Bind b) i c0 e2 H8 t) b0 H6))))) H4)) +H3))) | (Flat f) \Rightarrow (\lambda (H2: (clear (CHead c0 (Flat f) t) +(CHead e1 (Bind b) u))).(drop_drop (Flat f) i c0 e2 (H e1 u (clear_gen_flat f +c0 (CHead e1 (Bind b) u) t H2) e2 i H1) t))]) H0))))))))))) c)). + +theorem drop_clear_S: + \forall (x2: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop +h (S d) x1 x2) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear +x2 (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 +(Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))))))))))) +\def + \lambda (x2: C).(C_ind (\lambda (c: C).(\forall (x1: C).(\forall (h: +nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: +C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: +C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 +c2)))))))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (h: nat).(\lambda +(d: nat).(\lambda (_: (drop h (S d) x1 (CSort n))).(\lambda (b: B).(\lambda +(c2: C).(\lambda (u: T).(\lambda (H0: (clear (CSort n) (CHead c2 (Bind b) +u))).(clear_gen_sort (CHead c2 (Bind b) u) n H0 (ex2 C (\lambda (c1: +C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 +c2))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (x1: C).(\forall (h: +nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: +C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: +C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 +c2))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H0: (drop h (S d) x1 (CHead c k +t))).(\lambda (b: B).(\lambda (c2: C).(\lambda (u: T).(\lambda (H1: (clear +(CHead c k t) (CHead c2 (Bind b) u))).(ex2_ind C (\lambda (e: C).(eq C x1 +(CHead e k (lift h (r k d) t)))) (\lambda (e: C).(drop h (r k d) e c)) (ex2 C +(\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: +C).(drop h d c1 c2))) (\lambda (x: C).(\lambda (H2: (eq C x1 (CHead x k (lift +h (r k d) t)))).(\lambda (H3: (drop h (r k d) x c)).(eq_ind_r C (CHead x k +(lift h (r k d) t)) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear c0 (CHead +c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))) ((match k in +K return (\lambda (k0: K).((clear (CHead c k0 t) (CHead c2 (Bind b) u)) \to +((drop h (r k0 d) x c) \to (ex2 C (\lambda (c1: C).(clear (CHead x k0 (lift h +(r k0 d) t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 +c2)))))) with [(Bind b0) \Rightarrow (\lambda (H4: (clear (CHead c (Bind b0) +t) (CHead c2 (Bind b) u))).(\lambda (H5: (drop h (r (Bind b0) d) x c)).(let +H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) +with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 +(Bind b) u) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) +t H4)) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return +(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow +(match k in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat +_) \Rightarrow b])])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t) +(clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in ((let H8 \def (f_equal C +T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) +\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u) (CHead c +(Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in (\lambda +(H9: (eq B b b0)).(\lambda (H10: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: +T).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) +t)) (CHead c1 (Bind b) (lift h d t0)))) (\lambda (c1: C).(drop h d c1 c2)))) +(eq_ind_r C c (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind +b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b) (lift h d t)))) (\lambda +(c1: C).(drop h d c1 c0)))) (eq_ind_r B b0 (\lambda (b1: B).(ex2 C (\lambda +(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind +b1) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)))) (ex_intro2 C (\lambda +(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind +b0) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)) x (clear_bind b0 x +(lift h d t)) H5) b H9) c2 H10) u H8)))) H7)) H6)))) | (Flat f) \Rightarrow +(\lambda (H4: (clear (CHead c (Flat f) t) (CHead c2 (Bind b) u))).(\lambda +(H5: (drop h (r (Flat f) d) x c)).(let H6 \def (H x h d H5 b c2 u +(clear_gen_flat f c (CHead c2 (Bind b) u) t H4)) in (ex2_ind C (\lambda (c1: +C).(clear x (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 +c2)) (ex2 C (\lambda (c1: C).(clear (CHead x (Flat f) (lift h (r (Flat f) d) +t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))) +(\lambda (x0: C).(\lambda (H7: (clear x (CHead x0 (Bind b) (lift h d +u)))).(\lambda (H8: (drop h d x0 c2)).(ex_intro2 C (\lambda (c1: C).(clear +(CHead x (Flat f) (lift h (r (Flat f) d) t)) (CHead c1 (Bind b) (lift h d +u)))) (\lambda (c1: C).(drop h d c1 c2)) x0 (clear_flat x (CHead x0 (Bind b) +(lift h d u)) H7 f (lift h (r (Flat f) d) t)) H8)))) H6))))]) H1 H3) x1 +H2)))) (drop_gen_skip_r c x1 t h d k H0)))))))))))))) x2). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/fwd.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/fwd.ma new file mode 100644 index 000000000..2c8a2c80a --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/fwd.ma @@ -0,0 +1,159 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/clear/fwd". + +include "clear/defs.ma". + +theorem clear_gen_sort: + \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P: +Prop).P))) +\def + \lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda +(P: Prop).(let H0 \def (match H in clear return (\lambda (c: C).(\lambda (c0: +C).(\lambda (_: (clear c c0)).((eq C c (CSort n)) \to ((eq C c0 x) \to P))))) +with [(clear_bind b e u) \Rightarrow (\lambda (H0: (eq C (CHead e (Bind b) u) +(CSort n))).(\lambda (H1: (eq C (CHead e (Bind b) u) x)).((let H2 \def +(eq_ind C (CHead e (Bind b) u) (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) +\Rightarrow True])) I (CSort n) H0) in (False_ind ((eq C (CHead e (Bind b) u) +x) \to P) H2)) H1))) | (clear_flat e c H0 f u) \Rightarrow (\lambda (H1: (eq +C (CHead e (Flat f) u) (CSort n))).(\lambda (H2: (eq C c x)).((let H3 \def +(eq_ind C (CHead e (Flat f) u) (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) +\Rightarrow True])) I (CSort n) H1) in (False_ind ((eq C c x) \to ((clear e +c) \to P)) H3)) H2 H0)))]) in (H0 (refl_equal C (CSort n)) (refl_equal C +x)))))). + +theorem clear_gen_bind: + \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear +(CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u)))))) +\def + \lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: +(clear (CHead e (Bind b) u) x)).(let H0 \def (match H in clear return +(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (clear c c0)).((eq C c (CHead e +(Bind b) u)) \to ((eq C c0 x) \to (eq C x (CHead e (Bind b) u))))))) with +[(clear_bind b0 e0 u0) \Rightarrow (\lambda (H0: (eq C (CHead e0 (Bind b0) +u0) (CHead e (Bind b) u))).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) +x)).((let H2 \def (f_equal C T (\lambda (e1: C).(match e1 in C return +(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow +t])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H0) in ((let H3 \def +(f_equal C B (\lambda (e1: C).(match e1 in C return (\lambda (_: C).B) with +[(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return +(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow +b0])])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H0) in ((let H4 \def +(f_equal C C (\lambda (e1: C).(match e1 in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow c])) (CHead e0 (Bind +b0) u0) (CHead e (Bind b) u) H0) in (eq_ind C e (\lambda (c: C).((eq B b0 b) +\to ((eq T u0 u) \to ((eq C (CHead c (Bind b0) u0) x) \to (eq C x (CHead e +(Bind b) u)))))) (\lambda (H5: (eq B b0 b)).(eq_ind B b (\lambda (b1: B).((eq +T u0 u) \to ((eq C (CHead e (Bind b1) u0) x) \to (eq C x (CHead e (Bind b) +u))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead e +(Bind b) t) x) \to (eq C x (CHead e (Bind b) u)))) (\lambda (H7: (eq C (CHead +e (Bind b) u) x)).(eq_ind C (CHead e (Bind b) u) (\lambda (c: C).(eq C c +(CHead e (Bind b) u))) (refl_equal C (CHead e (Bind b) u)) x H7)) u0 (sym_eq +T u0 u H6))) b0 (sym_eq B b0 b H5))) e0 (sym_eq C e0 e H4))) H3)) H2)) H1))) +| (clear_flat e0 c H0 f u0) \Rightarrow (\lambda (H1: (eq C (CHead e0 (Flat +f) u0) (CHead e (Bind b) u))).(\lambda (H2: (eq C c x)).((let H3 \def (eq_ind +C (CHead e0 (Flat f) u0) (\lambda (e1: C).(match e1 in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match +k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat +_) \Rightarrow True])])) I (CHead e (Bind b) u) H1) in (False_ind ((eq C c x) +\to ((clear e0 c) \to (eq C x (CHead e (Bind b) u)))) H3)) H2 H0)))]) in (H0 +(refl_equal C (CHead e (Bind b) u)) (refl_equal C x))))))). + +theorem clear_gen_flat: + \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear +(CHead e (Flat f) u) x) \to (clear e x))))) +\def + \lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: +(clear (CHead e (Flat f) u) x)).(let H0 \def (match H in clear return +(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (clear c c0)).((eq C c (CHead e +(Flat f) u)) \to ((eq C c0 x) \to (clear e x)))))) with [(clear_bind b e0 u0) +\Rightarrow (\lambda (H0: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f) +u))).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) x)).((let H2 \def (eq_ind C +(CHead e0 (Bind b) u0) (\lambda (e1: C).(match e1 in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match +k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat +_) \Rightarrow False])])) I (CHead e (Flat f) u) H0) in (False_ind ((eq C +(CHead e0 (Bind b) u0) x) \to (clear e x)) H2)) H1))) | (clear_flat e0 c H0 +f0 u0) \Rightarrow (\lambda (H1: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat +f) u))).(\lambda (H2: (eq C c x)).((let H3 \def (f_equal C T (\lambda (e1: +C).(match e1 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | +(CHead _ _ t) \Rightarrow t])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) +H1) in ((let H4 \def (f_equal C F (\lambda (e1: C).(match e1 in C return +(\lambda (_: C).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _) \Rightarrow +(match k in K return (\lambda (_: K).F) with [(Bind _) \Rightarrow f0 | (Flat +f) \Rightarrow f])])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) H1) in +((let H5 \def (f_equal C C (\lambda (e1: C).(match e1 in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow c])) +(CHead e0 (Flat f0) u0) (CHead e (Flat f) u) H1) in (eq_ind C e (\lambda (c0: +C).((eq F f0 f) \to ((eq T u0 u) \to ((eq C c x) \to ((clear c0 c) \to (clear +e x)))))) (\lambda (H6: (eq F f0 f)).(eq_ind F f (\lambda (_: F).((eq T u0 u) +\to ((eq C c x) \to ((clear e c) \to (clear e x))))) (\lambda (H7: (eq T u0 +u)).(eq_ind T u (\lambda (_: T).((eq C c x) \to ((clear e c) \to (clear e +x)))) (\lambda (H8: (eq C c x)).(eq_ind C x (\lambda (c0: C).((clear e c0) +\to (clear e x))) (\lambda (H9: (clear e x)).H9) c (sym_eq C c x H8))) u0 +(sym_eq T u0 u H7))) f0 (sym_eq F f0 f H6))) e0 (sym_eq C e0 e H5))) H4)) +H3)) H2 H0)))]) in (H0 (refl_equal C (CHead e (Flat f) u)) (refl_equal C +x))))))). + +theorem clear_gen_flat_r: + \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x +(CHead e (Flat f) u)) \to (\forall (P: Prop).P))))) +\def + \lambda (f: F).(\lambda (x: C).(\lambda (e: C).(\lambda (u: T).(\lambda (H: +(clear x (CHead e (Flat f) u))).(\lambda (P: Prop).(insert_eq C (CHead e +(Flat f) u) (\lambda (c: C).(clear x c)) P (\lambda (y: C).(\lambda (H0: +(clear x y)).(clear_ind (\lambda (_: C).(\lambda (c0: C).((eq C c0 (CHead e +(Flat f) u)) \to P))) (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: +T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f) u))).(let H2 +\def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat f) u) H1) +in (False_ind P H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda (H1: +(clear e0 c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u)) \to P))).(\lambda +(_: F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat f) u))).(let H4 +\def (eq_ind C c (\lambda (c: C).((eq C c (CHead e (Flat f) u)) \to P)) H2 +(CHead e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda (c: C).(clear +e0 c)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C (CHead e (Flat f) +u)))))))))))) x y H0))) H)))))). + +theorem clear_gen_all: + \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b: +B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u)))))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (clear c1 c2)).(clear_ind +(\lambda (_: C).(\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda (e: +C).(\lambda (u: T).(eq C c0 (CHead e (Bind b) u)))))))) (\lambda (b: +B).(\lambda (e: C).(\lambda (u: T).(ex_3_intro B C T (\lambda (b0: +B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead e (Bind b) u) (CHead e0 +(Bind b0) u0))))) b e u (refl_equal C (CHead e (Bind b) u)))))) (\lambda (e: +C).(\lambda (c: C).(\lambda (H0: (clear e c)).(\lambda (H1: (ex_3 B C T +(\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c (CHead e (Bind b) +u))))))).(\lambda (_: F).(\lambda (_: T).(let H2 \def H1 in (ex_3_ind B C T +(\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c (CHead e0 (Bind b) +u0))))) (ex_3 B C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c +(CHead e0 (Bind b) u0)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: +T).(\lambda (H3: (eq C c (CHead x1 (Bind x0) x2))).(let H4 \def (eq_ind C c +(\lambda (c: C).(clear e c)) H0 (CHead x1 (Bind x0) x2) H3) in (eq_ind_r C +(CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda +(e0: C).(\lambda (u0: T).(eq C c0 (CHead e0 (Bind b) u0))))))) (ex_3_intro B +C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead x1 (Bind +x0) x2) (CHead e0 (Bind b) u0))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) +x2))) c H3)))))) H2)))))))) c1 c2 H))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/props.ma new file mode 100644 index 000000000..d6c135094 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clear/props.ma @@ -0,0 +1,144 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / 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/clear/props". + +include "clear/fwd.ma". + +theorem clear_clear: + \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2))) +\def + \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to +(clear c2 c2)))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear +(CSort n) c2)).(clear_gen_sort c2 n H (clear c2 c2))))) (\lambda (c: +C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (clear c2 +c2))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear +(CHead c k t) c2)).((match k in K return (\lambda (k0: K).((clear (CHead c k0 +t) c2) \to (clear c2 c2))) with [(Bind b) \Rightarrow (\lambda (H1: (clear +(CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: +C).(clear c0 c0)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H1))) | +(Flat f) \Rightarrow (\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 +(clear_gen_flat f c c2 t H1)))]) H0))))))) c1). + +theorem clear_mono: + \forall (c: C).(\forall (c1: C).((clear c c1) \to (\forall (c2: C).((clear c +c2) \to (eq C c1 c2))))) +\def + \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).((clear c0 c1) \to +(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2)))))) (\lambda (n: +nat).(\lambda (c1: C).(\lambda (_: (clear (CSort n) c1)).(\lambda (c2: +C).(\lambda (H0: (clear (CSort n) c2)).(clear_gen_sort c2 n H0 (eq C c1 +c2))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to +(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k: +K).(\lambda (t: T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t) +c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).((match k in K +return (\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 +t) c2) \to (eq C c1 c2)))) with [(Bind b) \Rightarrow (\lambda (H2: (clear +(CHead c0 (Bind b) t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) +c2)).(eq_ind_r C (CHead c0 (Bind b) t) (\lambda (c3: C).(eq C c1 c3)) +(eq_ind_r C (CHead c0 (Bind b) t) (\lambda (c3: C).(eq C c3 (CHead c0 (Bind +b) t))) (refl_equal C (CHead c0 (Bind b) t)) c1 (clear_gen_bind b c0 c1 t +H2)) c2 (clear_gen_bind b c0 c2 t H3)))) | (Flat f) \Rightarrow (\lambda (H2: +(clear (CHead c0 (Flat f) t) c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) +c2)).(H c1 (clear_gen_flat f c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t +H3))))]) H0 H1))))))))) c). + +theorem clear_trans: + \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (c2: C).((clear c +c2) \to (clear c1 c2))))) +\def + \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c0: C).((clear c c0) \to +(\forall (c2: C).((clear c0 c2) \to (clear c c2)))))) (\lambda (n: +nat).(\lambda (c: C).(\lambda (H: (clear (CSort n) c)).(\lambda (c2: +C).(\lambda (_: (clear c c2)).(clear_gen_sort c n H (clear (CSort n) +c2))))))) (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0) \to +(\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k: +K).(\lambda (t: T).(\lambda (c0: C).(\lambda (H0: (clear (CHead c k t) +c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).((match k in K return +(\lambda (k0: K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) +with [(Bind b) \Rightarrow (\lambda (H2: (clear (CHead c (Bind b) t) +c0)).(let H3 \def (eq_ind C c0 (\lambda (c: C).(clear c c2)) H1 (CHead c +(Bind b) t) (clear_gen_bind b c c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) +(\lambda (c3: C).(clear (CHead c (Bind b) t) c3)) (clear_bind b c t) c2 +(clear_gen_bind b c c2 t H3)))) | (Flat f) \Rightarrow (\lambda (H2: (clear +(CHead c (Flat f) t) c0)).(clear_flat c c2 (H c0 (clear_gen_flat f c c0 t H2) +c2 H1) f t))]) H0))))))))) c1). + +theorem clear_ctail: + \forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1 +(CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k +u1 c1) (CHead (CTail k u1 c2) (Bind b) u2)))))))) +\def + \lambda (b: B).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: +C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: +K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) +u2)))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H: +(clear (CSort n) (CHead c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1: +T).(match k in K return (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) +(CHead (CTail k0 u1 c2) (Bind b) u2))) with [(Bind b0) \Rightarrow +(clear_gen_sort (CHead c2 (Bind b) u2) n H (clear (CHead (CSort n) (Bind b0) +u1) (CHead (CTail (Bind b0) u1 c2) (Bind b) u2))) | (Flat f) \Rightarrow +(clear_gen_sort (CHead c2 (Bind b) u2) n H (clear (CHead (CSort n) (Flat f) +u1) (CHead (CTail (Flat f) u1 c2) (Bind b) u2)))]))))))) (\lambda (c: +C).(\lambda (H: ((\forall (c2: C).(\forall (u2: T).((clear c (CHead c2 (Bind +b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k u1 c) (CHead +(CTail k u1 c2) (Bind b) u2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda +(c2: C).(\lambda (u2: T).(\lambda (H0: (clear (CHead c k t) (CHead c2 (Bind +b) u2))).(\lambda (k0: K).(\lambda (u1: T).((match k in K return (\lambda +(k1: K).((clear (CHead c k1 t) (CHead c2 (Bind b) u2)) \to (clear (CHead +(CTail k0 u1 c) k1 t) (CHead (CTail k0 u1 c2) (Bind b) u2)))) with [(Bind b0) +\Rightarrow (\lambda (H1: (clear (CHead c (Bind b0) t) (CHead c2 (Bind b) +u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) +(CHead c2 (Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 +(Bind b) u2) t H1)) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in +C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead c +(Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H4 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 +(Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) +u2) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C c2 c)).(eq_ind_r +T t (\lambda (t0: T).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail +k0 u1 c2) (Bind b) t0))) (eq_ind_r C c (\lambda (c0: C).(clear (CHead (CTail +k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c0) (Bind b) t))) (eq_ind B b +(\lambda (b1: B).(clear (CHead (CTail k0 u1 c) (Bind b1) t) (CHead (CTail k0 +u1 c) (Bind b) t))) (clear_bind b (CTail k0 u1 c) t) b0 H5) c2 H6) u2 H4)))) +H3)) H2))) | (Flat f) \Rightarrow (\lambda (H1: (clear (CHead c (Flat f) t) +(CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1 c) (CHead (CTail k0 u1 c2) +(Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead c2 (Bind b) u2) t H1) k0 u1) +f t))]) H0)))))))))) c1)). + +theorem clear_cle: + \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1))) +\def + \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to +(le (cweight c2) (cweight c))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda +(H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (le (cweight c2) O))))) +(\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le (cweight +c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: +C).(\lambda (H0: (clear (CHead c k t) c2)).((match k in K return (\lambda +(k0: K).((clear (CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) +(tweight t))))) with [(Bind b) \Rightarrow (\lambda (H1: (clear (CHead c +(Bind b) t) c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(le +(cweight c0) (plus (cweight c) (tweight t)))) (le_n (plus (cweight c) +(tweight t))) c2 (clear_gen_bind b c c2 t H1))) | (Flat f) \Rightarrow +(\lambda (H1: (clear (CHead c (Flat f) t) c2)).(le_S_n (cweight c2) (plus +(cweight c) (tweight t)) (le_n_S (cweight c2) (plus (cweight c) (tweight t)) +(le_plus_trans (cweight c2) (cweight c) (tweight t) (H c2 (clear_gen_flat f c +c2 t H1))))))]) H0))))))) c1). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clen/getl.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clen/getl.ma new file mode 100644 index 000000000..2e027fe83 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/clen/getl.ma @@ -0,0 +1,326 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / 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/clen/getl". + +include "clen/defs.ma". + +include "getl/props.ma". + +theorem getl_ctail_clen: + \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n: +nat).(getl (clen c) (CTail (Bind b) t c) (CHead (CSort n) (Bind b) t)))))) +\def + \lambda (b: B).(\lambda (t: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(ex +nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) +(Bind b) t))))) (\lambda (n: nat).(ex_intro nat (\lambda (n0: nat).(getl O +(CHead (CSort n) (Bind b) t) (CHead (CSort n0) (Bind b) t))) n (getl_refl b +(CSort n) t))) (\lambda (c0: C).(\lambda (H: (ex nat (\lambda (n: nat).(getl +(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))).(\lambda (k: +K).(\lambda (t0: T).(let H0 \def H in (ex_ind nat (\lambda (n: nat).(getl +(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))) (ex nat +(\lambda (n: nat).(getl (s k (clen c0)) (CHead (CTail (Bind b) t c0) k t0) +(CHead (CSort n) (Bind b) t)))) (\lambda (x: nat).(\lambda (H1: (getl (clen +c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t))).(match k in K return +(\lambda (k0: K).(ex nat (\lambda (n: nat).(getl (s k0 (clen c0)) (CHead +(CTail (Bind b) t c0) k0 t0) (CHead (CSort n) (Bind b) t))))) with [(Bind b0) +\Rightarrow (ex_intro nat (\lambda (n: nat).(getl (S (clen c0)) (CHead (CTail +(Bind b) t c0) (Bind b0) t0) (CHead (CSort n) (Bind b) t))) x (getl_head +(Bind b0) (clen c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t) H1 +t0)) | (Flat f) \Rightarrow (ex_intro nat (\lambda (n: nat).(getl (clen c0) +(CHead (CTail (Bind b) t c0) (Flat f) t0) (CHead (CSort n) (Bind b) t))) x +(getl_flat (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t) (clen c0) H1 f +t0))]))) H0)))))) c))). + +theorem getl_gen_tail: + \forall (k: K).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).(\forall +(c2: C).(\forall (c1: C).(\forall (i: nat).((getl i (CTail k u1 c1) (CHead c2 +(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) +(\lambda (e: C).(getl i c1 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: +nat).(eq nat i (clen c1))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: +nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))))))))) +\def + \lambda (k: K).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: T).(\lambda +(c2: C).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (i: nat).((getl i +(CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C +c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 +nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind +b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort +n)))))))) (\lambda (n: nat).(\lambda (i: nat).(match i in nat return (\lambda +(n0: nat).((getl n0 (CTail k u1 (CSort n)) (CHead c2 (Bind b) u2)) \to (or +(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 +(CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 +(clen (CSort n)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq +T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort n1))))))) with [O \Rightarrow +(\lambda (H: (getl O (CHead (CSort n) k u1) (CHead c2 (Bind b) u2))).((match +k in K return (\lambda (k0: K).((clear (CHead (CSort n) k0 u1) (CHead c2 +(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k0 u1 e))) +(\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda +(_: nat).(eq nat O O)) (\lambda (_: nat).(eq K k0 (Bind b))) (\lambda (_: +nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))))) with [(Bind +b0) \Rightarrow (\lambda (H0: (clear (CHead (CSort n) (Bind b0) u1) (CHead c2 +(Bind b) u2))).(let H1 \def (f_equal C C (\lambda (e: C).(match e in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow +c])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 +(CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H2 \def (f_equal C B +(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) +\Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: +K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c2 +(Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) +(CHead c2 (Bind b) u2) u1 H0)) in ((let H3 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | +(CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind +b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in +(\lambda (H4: (eq B b b0)).(\lambda (H5: (eq C c2 (CSort n))).(eq_ind_r C +(CSort n) (\lambda (c: C).(or (ex2 C (\lambda (e: C).(eq C c (CTail (Bind b0) +u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat +(\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) +(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c (CSort n0)))))) +(eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (e: C).(eq C (CSort n) +(CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) +t)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind +b0) (Bind b))) (\lambda (_: nat).(eq T u1 t)) (\lambda (n0: nat).(eq C (CSort +n) (CSort n0)))))) (eq_ind_r B b0 (\lambda (b1: B).(or (ex2 C (\lambda (e: +C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) +(CHead e (Bind b1) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda +(_: nat).(eq K (Bind b0) (Bind b1))) (\lambda (_: nat).(eq T u1 u1)) (\lambda +(n0: nat).(eq C (CSort n) (CSort n0)))))) (or_intror (ex2 C (\lambda (e: +C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) +(CHead e (Bind b0) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda +(_: nat).(eq K (Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda +(n0: nat).(eq C (CSort n) (CSort n0)))) (ex4_intro nat (\lambda (_: nat).(eq +nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b0))) (\lambda (_: nat).(eq +T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0))) n (refl_equal nat +O) (refl_equal K (Bind b0)) (refl_equal T u1) (refl_equal C (CSort n)))) b +H4) u2 H3) c2 H5)))) H2)) H1))) | (Flat f) \Rightarrow (\lambda (H0: (clear +(CHead (CSort n) (Flat f) u1) (CHead c2 (Bind b) u2))).(clear_gen_sort (CHead +c2 (Bind b) u2) n (clear_gen_flat f (CSort n) (CHead c2 (Bind b) u2) u1 H0) +(or (ex2 C (\lambda (e: C).(eq C c2 (CTail (Flat f) u1 e))) (\lambda (e: +C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq +nat O O)) (\lambda (_: nat).(eq K (Flat f) (Bind b))) (\lambda (_: nat).(eq T +u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))))))]) (getl_gen_O (CHead +(CSort n) k u1) (CHead c2 (Bind b) u2) H))) | (S n0) \Rightarrow (\lambda (H: +(getl (S n0) (CHead (CSort n) k u1) (CHead c2 (Bind b) u2))).(getl_gen_sort n +(r k n0) (CHead c2 (Bind b) u2) (getl_gen_S k (CSort n) (CHead c2 (Bind b) +u2) u1 n0 H) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda +(e: C).(getl (S n0) (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: +nat).(eq nat (S n0) O)) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: +nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort n1)))))))]))) (\lambda +(c: C).(\lambda (H: ((\forall (i: nat).((getl i (CTail k u1 c) (CHead c2 +(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) +(\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: +nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: +nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))))))).(\lambda (k0: +K).(\lambda (t: T).(\lambda (i: nat).(match i in nat return (\lambda (n: +nat).((getl n (CTail k u1 (CHead c k0 t)) (CHead c2 (Bind b) u2)) \to (or +(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n +(CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n +(clen (CHead c k0 t)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: +nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))))) with [O +\Rightarrow (\lambda (H0: (getl O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind +b) u2))).((match k0 in K return (\lambda (k1: K).((clear (CHead (CTail k u1 +c) k1 t) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 +(CTail k u1 e))) (\lambda (e: C).(getl O (CHead c k1 t) (CHead e (Bind b) +u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s k1 (clen c)))) (\lambda (_: +nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq +C c2 (CSort n))))))) with [(Bind b0) \Rightarrow (\lambda (H1: (clear (CHead +(CTail k u1 c) (Bind b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C +C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead +(CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind +b) u2) t H1)) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in C +return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead +(CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind +b) u2) t H1)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e in C +return (\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) +\Rightarrow t])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) +(clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in (\lambda +(H5: (eq B b b0)).(\lambda (H6: (eq C c2 (CTail k u1 c))).(eq_ind T u2 +(\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) +(\lambda (e: C).(getl O (CHead c (Bind b0) t0) (CHead e (Bind b) u2)))) (ex4 +nat (\lambda (_: nat).(eq nat O (s (Bind b0) (clen c)))) (\lambda (_: +nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq +C c2 (CSort n)))))) (eq_ind B b (\lambda (b1: B).(or (ex2 C (\lambda (e: +C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b1) u2) +(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b1) +(clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 +u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (let H7 \def (eq_ind C c2 +(\lambda (c0: C).(\forall (i: nat).((getl i (CTail k u1 c) (CHead c0 (Bind b) +u2)) \to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: +C).(getl i c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat i +(clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 +u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))))) H (CTail k u1 c) H6) in +(eq_ind_r C (CTail k u1 c) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C +c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e +(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) (clen c)))) +(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda +(n: nat).(eq C c0 (CSort n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C +(CTail k u1 c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) +(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) +(clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 +u2)) (\lambda (n: nat).(eq C (CTail k u1 c) (CSort n)))) (ex_intro2 C +(\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1 e))) (\lambda (e: C).(getl O +(CHead c (Bind b) u2) (CHead e (Bind b) u2))) c (refl_equal C (CTail k u1 c)) +(getl_refl b c u2))) c2 H6)) b0 H5) t H4)))) H3)) H2))) | (Flat f) +\Rightarrow (\lambda (H1: (clear (CHead (CTail k u1 c) (Flat f) t) (CHead c2 +(Bind b) u2))).(let H2 \def (H O (getl_intro O (CTail k u1 c) (CHead c2 (Bind +b) u2) (CTail k u1 c) (drop_refl (CTail k u1 c)) (clear_gen_flat f (CTail k +u1 c) (CHead c2 (Bind b) u2) t H1))) in (or_ind (ex2 C (\lambda (e: C).(eq C +c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2)))) (ex4 +nat (\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind +b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))) +(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O +(CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq +nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda +(_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (H3: +(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c +(CHead e (Bind b) u2))))).(ex2_ind C (\lambda (e: C).(eq C c2 (CTail k u1 +e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))) (or (ex2 C (\lambda +(e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) +(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) +(clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 +u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (x: C).(\lambda (H4: +(eq C c2 (CTail k u1 x))).(\lambda (H5: (getl O c (CHead x (Bind b) +u2))).(eq_ind_r C (CTail k u1 x) (\lambda (c0: C).(or (ex2 C (\lambda (e: +C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) +(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) +(clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 +u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (or_introl (ex2 C (\lambda (e: +C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c +(Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s +(Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: +nat).(eq T u1 u2)) (\lambda (n: nat).(eq C (CTail k u1 x) (CSort n)))) +(ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda +(e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2))) x (refl_equal C +(CTail k u1 x)) (getl_flat c (CHead x (Bind b) u2) O H5 f t))) c2 H4)))) H3)) +(\lambda (H3: (ex4 nat (\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: +nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq +C c2 (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq nat O (clen c))) +(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda +(n: nat).(eq C c2 (CSort n))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 +e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) +(ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: +nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq +C c2 (CSort n))))) (\lambda (x0: nat).(\lambda (H4: (eq nat O (clen +c))).(\lambda (H5: (eq K k (Bind b))).(\lambda (H6: (eq T u1 u2)).(\lambda +(H7: (eq C c2 (CSort x0))).(eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C +(\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c +(Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s +(Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: +nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (eq_ind T u1 +(\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) +(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) t0)))) (ex4 +nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq +K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) (\lambda (n: nat).(eq C (CSort +x0) (CSort n)))))) (eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda +(e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: C).(getl O (CHead c +(Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s +(Flat f) (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: +nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) +(or_intror (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) +(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4 +nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq +K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C +(CSort x0) (CSort n)))) (ex4_intro nat (\lambda (_: nat).(eq nat O (s (Flat +f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: +nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort n))) x0 H4 +(refl_equal K (Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) k H5) +u2 H6) c2 H7)))))) H3)) H2)))]) (getl_gen_O (CHead (CTail k u1 c) k0 t) +(CHead c2 (Bind b) u2) H0))) | (S n) \Rightarrow (\lambda (H0: (getl (S n) +(CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2))).(let H_x \def (H (r k0 +n) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H0)) in (let H1 +\def H_x in (or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda +(e: C).(getl (r k0 n) c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: +nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b))) +(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))) (or +(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) +(CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S +n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: +nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (H2: +(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 +n) c (CHead e (Bind b) u2))))).(ex2_ind C (\lambda (e: C).(eq C c2 (CTail k +u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2))) (or (ex2 C +(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead +c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s +k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T +u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x: C).(\lambda +(H3: (eq C c2 (CTail k u1 x))).(\lambda (H4: (getl (r k0 n) c (CHead x (Bind +b) u2))).(let H5 \def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k +u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind +b) u2) t n H0) (CTail k u1 x) H3) in (eq_ind_r C (CTail k u1 x) (\lambda (c0: +C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl +(S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq +nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: +nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (or_introl +(ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: +C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: +nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) +(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C (CTail k u1 x) +(CSort n0)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 +e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2))) x +(refl_equal C (CTail k u1 x)) (getl_head k0 n c (CHead x (Bind b) u2) H4 t))) +c2 H3))))) H2)) (\lambda (H2: (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) +(clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 +u2)) (\lambda (n: nat).(eq C c2 (CSort n))))).(ex4_ind nat (\lambda (_: +nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b))) +(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))) (or +(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) +(CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S +n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: +nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x0: +nat).(\lambda (H3: (eq nat (r k0 n) (clen c))).(\lambda (H4: (eq K k (Bind +b))).(\lambda (H5: (eq T u1 u2)).(\lambda (H6: (eq C c2 (CSort x0))).(let H7 +\def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 +(Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H0) +(CSort x0) H6) in (eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C (\lambda +(e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) +(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen +c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) +(\lambda (n0: nat).(eq C c0 (CSort n0)))))) (let H8 \def (eq_ind_r T u2 +(\lambda (t: T).(getl (r k0 n) (CTail k u1 c) (CHead (CSort x0) (Bind b) t))) +H7 u1 H5) in (eq_ind T u1 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C +(CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead +e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) +(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) (\lambda +(n0: nat).(eq C (CSort x0) (CSort n0)))))) (let H9 \def (eq_ind K k (\lambda +(k: K).(getl (r k0 n) (CTail k u1 c) (CHead (CSort x0) (Bind b) u1))) H8 +(Bind b) H4) in (eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda (e: +C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 +t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 +(clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 +u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))))) (eq_ind nat (r k0 n) +(\lambda (n0: nat).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind +b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) +u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 n0))) (\lambda (_: +nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n1: +nat).(eq C (CSort x0) (CSort n1)))))) (eq_ind_r nat (S n) (\lambda (n0: +nat).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) +(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat +(\lambda (_: nat).(eq nat (S n) n0)) (\lambda (_: nat).(eq K (Bind b) (Bind +b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) +(CSort n1)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail +(Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) +u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq +K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq +C (CSort x0) (CSort n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat (S n) (S +n))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 +u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0))) x0 (refl_equal nat (S +n)) (refl_equal K (Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) (s +k0 (r k0 n)) (s_r k0 n)) (clen c) H3) k H4)) u2 H5)) c2 H6))))))) H2)) +H1))))])))))) c1)))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/clear.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/clear.ma new file mode 100644 index 000000000..2329dd98e --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/clear.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/getl/clear". + +include "getl/props.ma". + +include "clear/drop.ma". + +theorem clear_getl_trans: + \forall (i: nat).(\forall (c2: C).(\forall (c3: C).((getl i c2 c3) \to +(\forall (c1: C).((clear c1 c2) \to (getl i c1 c3)))))) +\def + \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c2: C).(\forall (c3: +C).((getl n c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl n c1 +c3))))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (H: (getl O c2 +c3)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(getl_intro O c1 c3 c1 +(drop_refl c1) (clear_trans c1 c2 H0 c3 (getl_gen_O c2 c3 H)))))))) (\lambda +(n: nat).(\lambda (_: ((\forall (c2: C).(\forall (c3: C).((getl n c2 c3) \to +(\forall (c1: C).((clear c1 c2) \to (getl n c1 c3)))))))).(\lambda (c2: +C).(C_ind (\lambda (c: C).(\forall (c3: C).((getl (S n) c c3) \to (\forall +(c1: C).((clear c1 c) \to (getl (S n) c1 c3)))))) (\lambda (n0: nat).(\lambda +(c3: C).(\lambda (H0: (getl (S n) (CSort n0) c3)).(\lambda (c1: C).(\lambda +(_: (clear c1 (CSort n0))).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c1 +c3))))))) (\lambda (c: C).(\lambda (_: ((\forall (c3: C).((getl (S n) c c3) +\to (\forall (c1: C).((clear c1 c) \to (getl (S n) c1 c3))))))).(\lambda (k: +K).(\lambda (t: T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) +c3)).(\lambda (c1: C).(\lambda (H2: (clear c1 (CHead c k t))).((match k in K +return (\lambda (k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear c1 (CHead +c k0 t)) \to (getl (S n) c1 c3)))) with [(Bind b) \Rightarrow (\lambda (H3: +(getl (S n) (CHead c (Bind b) t) c3)).(\lambda (H4: (clear c1 (CHead c (Bind +b) t))).(let H5 \def (getl_gen_all c c3 (r (Bind b) n) (getl_gen_S (Bind b) c +c3 t n H3)) in (ex2_ind C (\lambda (e: C).(drop n O c e)) (\lambda (e: +C).(clear e c3)) (getl (S n) c1 c3) (\lambda (x: C).(\lambda (H6: (drop n O c +x)).(\lambda (H7: (clear x c3)).(getl_intro (S n) c1 c3 x (drop_clear_O b c1 +c t H4 x n H6) H7)))) H5)))) | (Flat f) \Rightarrow (\lambda (_: (getl (S n) +(CHead c (Flat f) t) c3)).(\lambda (H4: (clear c1 (CHead c (Flat f) +t))).(clear_gen_flat_r f c1 c t H4 (getl (S n) c1 c3))))]) H1 H2))))))))) +c2)))) i). + +theorem getl_clear_trans: + \forall (i: nat).(\forall (c1: C).(\forall (c2: C).((getl i c1 c2) \to +(\forall (c3: C).((clear c2 c3) \to (getl i c1 c3)))))) +\def + \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (getl i c1 +c2)).(\lambda (c3: C).(\lambda (H0: (clear c2 c3)).(let H1 \def (getl_gen_all +c1 c2 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: +C).(clear e c2)) (getl i c1 c3) (\lambda (x: C).(\lambda (H2: (drop i O c1 +x)).(\lambda (H3: (clear x c2)).(let H4 \def (clear_gen_all x c2 H3) in +(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c2 +(CHead e (Bind b) u))))) (getl i c1 c3) (\lambda (x0: B).(\lambda (x1: +C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(let H6 +\def (eq_ind C c2 (\lambda (c: C).(clear x c)) H3 (CHead x1 (Bind x0) x2) H5) +in (let H7 \def (eq_ind C c2 (\lambda (c: C).(clear c c3)) H0 (CHead x1 (Bind +x0) x2) H5) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: C).(getl i c1 +c)) (getl_intro i c1 (CHead x1 (Bind x0) x2) x H2 H6) c3 (clear_gen_bind x0 +x1 c3 x2 H7)))))))) H4))))) H1))))))). + +theorem getl_clear_bind: + \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (v: T).((clear c +(CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2) +\to (getl (S n) c e2)))))))) +\def + \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: +C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: +C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2)))))))) (\lambda +(n: nat).(\lambda (e1: C).(\lambda (v: T).(\lambda (H: (clear (CSort n) +(CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n0: nat).(\lambda (_: +(getl n0 e1 e2)).(clear_gen_sort (CHead e1 (Bind b) v) n H (getl (S n0) +(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: +C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: +C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2))))))))).(\lambda +(k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (v: T).(\lambda (H0: (clear +(CHead c0 k t) (CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n: +nat).(\lambda (H1: (getl n e1 e2)).((match k in K return (\lambda (k0: +K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) v)) \to (getl (S n) (CHead c0 +k0 t) e2))) with [(Bind b0) \Rightarrow (\lambda (H2: (clear (CHead c0 (Bind +b0) t) (CHead e1 (Bind b) v))).(let H3 \def (f_equal C C (\lambda (e: +C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e1 | +(CHead c _ _) \Rightarrow c])) (CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) +(clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in ((let H4 \def (f_equal +C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) +\Rightarrow b | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: +K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead e1 +(Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) +v) t H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return +(\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow +t])) (CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 +(CHead e1 (Bind b) v) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq +C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c: C).(getl n c e2)) H1 c0 H7) +in (eq_ind B b (\lambda (b1: B).(getl (S n) (CHead c0 (Bind b1) t) e2)) +(getl_head (Bind b) n c0 e2 H8 t) b0 H6))))) H4)) H3))) | (Flat f) +\Rightarrow (\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 (Bind b) +v))).(getl_flat c0 e2 (S n) (H e1 v (clear_gen_flat f c0 (CHead e1 (Bind b) +v) t H2) e2 n H1) f t))]) H0))))))))))) c)). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/dec.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/dec.ma new file mode 100644 index 000000000..59c3775bb --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/dec.ma @@ -0,0 +1,97 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/getl/dec". + +include "getl/props.ma". + +theorem getl_dec: + \forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda +(b: B).(\lambda (v: T).(getl i c (CHead e (Bind b) v)))))) (\forall (d: +C).((getl i c d) \to (\forall (P: Prop).P))))) +\def + \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (i: nat).(or (ex_3 C B T +(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b) +v)))))) (\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P)))))) +(\lambda (n: nat).(\lambda (i: nat).(or_intror (ex_3 C B T (\lambda (e: +C).(\lambda (b: B).(\lambda (v: T).(getl i (CSort n) (CHead e (Bind b) +v)))))) (\forall (d: C).((getl i (CSort n) d) \to (\forall (P: Prop).P))) +(\lambda (d: C).(\lambda (H: (getl i (CSort n) d)).(\lambda (P: +Prop).(getl_gen_sort n i d H P))))))) (\lambda (c0: C).(\lambda (H: ((\forall +(i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: +T).(getl i c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c0 d) \to +(\forall (P: Prop).P))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (i: +nat).(match i in nat return (\lambda (n: nat).(or (ex_3 C B T (\lambda (e: +C).(\lambda (b: B).(\lambda (v: T).(getl n (CHead c0 k t) (CHead e (Bind b) +v)))))) (\forall (d: C).((getl n (CHead c0 k t) d) \to (\forall (P: +Prop).P))))) with [O \Rightarrow (match k in K return (\lambda (k0: K).(or +(ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 +k0 t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 k0 t) d) +\to (\forall (P: Prop).P))))) with [(Bind b) \Rightarrow (or_introl (ex_3 C B +T (\lambda (e: C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) +t) (CHead e (Bind b0) v)))))) (\forall (d: C).((getl O (CHead c0 (Bind b) t) +d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e: C).(\lambda +(b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) t) (CHead e (Bind b0) +v))))) c0 b t (getl_refl b c0 t))) | (Flat f) \Rightarrow (let H_x \def (H O) +in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: +B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v)))))) (\forall (d: +C).((getl O c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T (\lambda (e: +C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e +(Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to +(\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda +(b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v))))))).(ex_3_ind C B T +(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) +v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl +O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O +(CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: +C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl O c0 (CHead x0 (Bind +x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: +T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: +C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) (ex_3_intro +C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat +f) t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_flat c0 (CHead x0 (Bind x1) x2) +O H2 f t))))))) H1)) (\lambda (H1: ((\forall (d: C).((getl O c0 d) \to +(\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b: +B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) +(\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) +(\lambda (d: C).(\lambda (H2: (getl O (CHead c0 (Flat f) t) d)).(\lambda (P: +Prop).(H1 d (getl_intro O c0 d c0 (drop_refl c0) (clear_gen_flat f c0 d t +(getl_gen_O (CHead c0 (Flat f) t) d H2))) P)))))) H0)))]) | (S n) \Rightarrow +(let H_x \def (H (r k 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 (r k n) c0 (CHead e (Bind b) +v)))))) (\forall (d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))) (or +(ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) +(CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead +c0 k t) d) \to (\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda +(e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) +v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: +T).(getl (r k n) c0 (CHead e (Bind b) v))))) (or (ex_3 C B T (\lambda (e: +C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind +b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: +Prop).P)))) (\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: +(getl (r k n) c0 (CHead x0 (Bind x1) x2))).(or_introl (ex_3 C B T (\lambda +(e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e +(Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall +(P: Prop).P))) (ex_3_intro C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: +T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_head k +n c0 (CHead x0 (Bind x1) x2) H2 t))))))) H1)) (\lambda (H1: ((\forall (d: +C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))))).(or_intror (ex_3 C B T +(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) +(CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to +(\forall (P: Prop).P))) (\lambda (d: C).(\lambda (H2: (getl (S n) (CHead c0 k +t) d)).(\lambda (P: Prop).(H1 d (getl_gen_S k c0 d t n H2) P)))))) +H0)))])))))) c). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/defs.ma new file mode 100644 index 000000000..40845209b --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/defs.ma @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/getl/defs". + +include "drop/defs.ma". + +include "clear/defs.ma". + +inductive getl (h:nat) (c1:C) (c2:C): Prop \def +| getl_intro: \forall (e: C).((drop h O c1 e) \to ((clear e c2) \to (getl h +c1 c2))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/drop.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/drop.ma new file mode 100644 index 000000000..593ed87bb --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/drop.ma @@ -0,0 +1,66 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/getl/drop". + +include "getl/fwd.ma". + +include "clear/drop.ma". + +include "r/props.ma". + +theorem getl_drop: + \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: +nat).((getl h c (CHead e (Bind b) u)) \to (drop (S h) O c e)))))) +\def + \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: +C).(\forall (u: T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to +(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: +T).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) (CHead e (Bind b) +u))).(getl_gen_sort n h (CHead e (Bind b) u) H (drop (S h) O (CSort n) +e))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: +T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to (drop (S h) O c0 +e))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: +T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t) +(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e))) (\lambda (H0: +(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear +(CHead c0 k0 t) (CHead e (Bind b) u)) \to (drop (S O) O (CHead c0 k0 t) e))) +(\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind +b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow +c])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead +e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0: C).(match e0 +in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u) (CHead c0 +(Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in ((let H4 +\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead e (Bind +b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t +H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e c0)).(eq_ind_r C c0 +(\lambda (c1: C).(drop (S O) O (CHead c0 (Bind b0) t) c1)) (eq_ind B b +(\lambda (b1: B).(drop (S O) O (CHead c0 (Bind b1) t) c0)) (drop_drop (Bind +b) O c0 c0 (drop_refl c0) t) b0 H5) e H6)))) H3)) H2)))) (\lambda (f: +F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b) +u))).(drop_clear_O b (CHead c0 (Flat f) t) e u (clear_flat c0 (CHead e (Bind +b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1) f t) e O (drop_refl +e)))) k (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n: +nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S +n) O (CHead c0 k t) e)))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead e +(Bind b) u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0: +nat).(drop n0 O c0 e)) (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind b) u) t +n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/flt.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/flt.ma new file mode 100644 index 000000000..589f2dd50 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/flt.ma @@ -0,0 +1,66 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/getl/flt". + +include "getl/fwd.ma". + +include "clear/props.ma". + +include "flt/props.ma". + +theorem getl_flt: + \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: +nat).((getl i c (CHead e (Bind b) u)) \to (flt e u c (TLRef i))))))) +\def + \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: +C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to +(flt e u c0 (TLRef i))))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: +T).(\lambda (i: nat).(\lambda (H: (getl i (CSort n) (CHead e (Bind b) +u))).(getl_gen_sort n i (CHead e (Bind b) u) H (flt e u (CSort n) (TLRef +i)))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: +T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to (flt e u c0 (TLRef +i)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: +T).(\lambda (i: nat).(match i in nat return (\lambda (n: nat).((getl n (CHead +c0 k t) (CHead e (Bind b) u)) \to (flt e u (CHead c0 k t) (TLRef n)))) with +[O \Rightarrow (\lambda (H0: (getl O (CHead c0 k t) (CHead e (Bind b) +u))).((match k in K return (\lambda (k0: K).((clear (CHead c0 k0 t) (CHead e +(Bind b) u)) \to (flt e u (CHead c0 k0 t) (TLRef O)))) with [(Bind b0) +\Rightarrow (\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind b) +u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C return +(\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow +c])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead +e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0: C).(match e0 +in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +\Rightarrow b | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u) (CHead c0 +(Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in ((let H4 +\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead e (Bind +b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t +H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e c0)).(eq_ind_r T t +(\lambda (t0: T).(flt e t0 (CHead c0 (Bind b0) t) (TLRef O))) (eq_ind_r C c0 +(\lambda (c1: C).(flt c1 t (CHead c0 (Bind b0) t) (TLRef O))) (eq_ind B b +(\lambda (b1: B).(flt c0 t (CHead c0 (Bind b1) t) (TLRef O))) (flt_arith0 +(Bind b) c0 t O) b0 H5) e H6) u H4)))) H3)) H2))) | (Flat f) \Rightarrow +(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b) u))).(flt_arith1 +(Bind b) e c0 u (clear_cle c0 (CHead e (Bind b) u) (clear_gen_flat f c0 +(CHead e (Bind b) u) t H1)) (Flat f) t O))]) (getl_gen_O (CHead c0 k t) +(CHead e (Bind b) u) H0))) | (S n) \Rightarrow (\lambda (H0: (getl (S n) +(CHead c0 k t) (CHead e (Bind b) u))).(let H_y \def (H e u (r k n) +(getl_gen_S k c0 (CHead e (Bind b) u) t n H0)) in (flt_arith2 e c0 u (r k n) +H_y k t (S n))))])))))))) c)). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/fwd.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/fwd.ma new file mode 100644 index 000000000..afa14d8f5 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/fwd.ma @@ -0,0 +1,107 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / 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/getl/fwd". + +include "getl/defs.ma". + +include "drop/fwd.ma". + +include "clear/fwd.ma". + +theorem getl_gen_all: + \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex2 +C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)))))) +\def + \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1 +c2)).(let H0 \def (match H in getl return (\lambda (_: (getl ? ? ?)).(ex2 C +(\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)))) with +[(getl_intro e H0 H1) \Rightarrow (ex_intro2 C (\lambda (e0: C).(drop i O c1 +e0)) (\lambda (e0: C).(clear e0 c2)) e H0 H1)]) in H0)))). + +theorem getl_gen_sort: + \forall (n: nat).(\forall (h: nat).(\forall (x: C).((getl h (CSort n) x) \to +(\forall (P: Prop).P)))) +\def + \lambda (n: nat).(\lambda (h: nat).(\lambda (x: C).(\lambda (H: (getl h +(CSort n) x)).(\lambda (P: Prop).(let H0 \def (getl_gen_all (CSort n) x h H) +in (ex2_ind C (\lambda (e: C).(drop h O (CSort n) e)) (\lambda (e: C).(clear +e x)) P (\lambda (x0: C).(\lambda (H1: (drop h O (CSort n) x0)).(\lambda (H2: +(clear x0 x)).(and3_ind (eq C x0 (CSort n)) (eq nat h O) (eq nat O O) P +(\lambda (H3: (eq C x0 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_: +(eq nat O O)).(let H6 \def (eq_ind C x0 (\lambda (c: C).(clear c x)) H2 +(CSort n) H3) in (clear_gen_sort x n H6 P))))) (drop_gen_sort n h O x0 +H1))))) H0)))))). + +theorem getl_gen_O: + \forall (e: C).(\forall (x: C).((getl O e x) \to (clear e x))) +\def + \lambda (e: C).(\lambda (x: C).(\lambda (H: (getl O e x)).(let H0 \def +(getl_gen_all e x O H) in (ex2_ind C (\lambda (e0: C).(drop O O e e0)) +(\lambda (e0: C).(clear e0 x)) (clear e x) (\lambda (x0: C).(\lambda (H1: +(drop O O e x0)).(\lambda (H2: (clear x0 x)).(let H3 \def (eq_ind_r C x0 +(\lambda (c: C).(clear c x)) H2 e (drop_gen_refl e x0 H1)) in H3)))) H0)))). + +theorem getl_gen_S: + \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: +nat).((getl (S h) (CHead c k u) x) \to (getl (r k h) c x)))))) +\def + \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: +nat).(\lambda (H: (getl (S h) (CHead c k u) x)).(let H0 \def (getl_gen_all +(CHead c k u) x (S h) H) in (ex2_ind C (\lambda (e: C).(drop (S h) O (CHead c +k u) e)) (\lambda (e: C).(clear e x)) (getl (r k h) c x) (\lambda (x0: +C).(\lambda (H1: (drop (S h) O (CHead c k u) x0)).(\lambda (H2: (clear x0 +x)).(getl_intro (r k h) c x x0 (drop_gen_drop k c x0 u h H1) H2)))) H0))))))). + +theorem getl_gen_flat: + \forall (f: F).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: +nat).((getl i (CHead e (Flat f) v) d) \to (getl i e d)))))) +\def + \lambda (f: F).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: +nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Flat f) v) d) \to (getl n +e d))) (\lambda (H: (getl O (CHead e (Flat f) v) d)).(getl_intro O e d e +(drop_refl e) (clear_gen_flat f e d v (getl_gen_O (CHead e (Flat f) v) d +H)))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead e (Flat f) v) d) \to +(getl n e d)))).(\lambda (H0: (getl (S n) (CHead e (Flat f) v) +d)).(getl_gen_S (Flat f) e d v n H0)))) i))))). + +theorem getl_gen_bind: + \forall (b: B).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: +nat).((getl i (CHead e (Bind b) v) d) \to (or (land (eq nat i O) (eq C d +(CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda +(j: nat).(getl j e d))))))))) +\def + \lambda (b: B).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: +nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Bind b) v) d) \to (or +(land (eq nat n O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: +nat).(eq nat n (S j))) (\lambda (j: nat).(getl j e d)))))) (\lambda (H: (getl +O (CHead e (Bind b) v) d)).(eq_ind_r C (CHead e (Bind b) v) (\lambda (c: +C).(or (land (eq nat O O) (eq C c (CHead e (Bind b) v))) (ex2 nat (\lambda +(j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e c))))) (or_introl +(land (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind b) v))) (ex2 nat +(\lambda (j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e (CHead e +(Bind b) v)))) (conj (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind +b) v)) (refl_equal nat O) (refl_equal C (CHead e (Bind b) v)))) d +(clear_gen_bind b e d v (getl_gen_O (CHead e (Bind b) v) d H)))) (\lambda (n: +nat).(\lambda (_: (((getl n (CHead e (Bind b) v) d) \to (or (land (eq nat n +O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat n (S +j))) (\lambda (j: nat).(getl j e d))))))).(\lambda (H0: (getl (S n) (CHead e +(Bind b) v) d)).(or_intror (land (eq nat (S n) O) (eq C d (CHead e (Bind b) +v))) (ex2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: nat).(getl +j e d))) (ex_intro2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: +nat).(getl j e d)) n (refl_equal nat (S n)) (getl_gen_S (Bind b) e d v n +H0)))))) i))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/props.ma new file mode 100644 index 000000000..eb20484df --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/getl/props.ma @@ -0,0 +1,74 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/getl/props". + +include "getl/fwd.ma". + +include "drop/props.ma". + +include "clear/props.ma". + +theorem getl_refl: + \forall (b: B).(\forall (c: C).(\forall (u: T).(getl O (CHead c (Bind b) u) +(CHead c (Bind b) u)))) +\def + \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(getl_intro O (CHead c (Bind +b) u) (CHead c (Bind b) u) (CHead c (Bind b) u) (drop_refl (CHead c (Bind b) +u)) (clear_bind b c u)))). + +theorem getl_head: + \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: C).((getl (r k +h) c e) \to (\forall (u: T).(getl (S h) (CHead c k u) e)))))) +\def + \lambda (k: K).(\lambda (h: nat).(\lambda (c: C).(\lambda (e: C).(\lambda +(H: (getl (r k h) c e)).(\lambda (u: T).(let H0 \def (getl_gen_all c e (r k +h) H) in (ex2_ind C (\lambda (e0: C).(drop (r k h) O c e0)) (\lambda (e0: +C).(clear e0 e)) (getl (S h) (CHead c k u) e) (\lambda (x: C).(\lambda (H1: +(drop (r k h) O c x)).(\lambda (H2: (clear x e)).(getl_intro (S h) (CHead c k +u) e x (drop_drop k h c x H1 u) H2)))) H0))))))). + +theorem getl_flat: + \forall (c: C).(\forall (e: C).(\forall (h: nat).((getl h c e) \to (\forall +(f: F).(\forall (u: T).(getl h (CHead c (Flat f) u) e)))))) +\def + \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (H: (getl h c +e)).(\lambda (f: F).(\lambda (u: T).(let H0 \def (getl_gen_all c e h H) in +(ex2_ind C (\lambda (e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) +(getl h (CHead c (Flat f) u) e) (\lambda (x: C).(\lambda (H1: (drop h O c +x)).(\lambda (H2: (clear x e)).((match h in nat return (\lambda (n: +nat).((drop n O c x) \to (getl n (CHead c (Flat f) u) e))) with [O +\Rightarrow (\lambda (H3: (drop O O c x)).(let H4 \def (eq_ind_r C x (\lambda +(c: C).(clear c e)) H2 c (drop_gen_refl c x H3)) in (getl_intro O (CHead c +(Flat f) u) e (CHead c (Flat f) u) (drop_refl (CHead c (Flat f) u)) +(clear_flat c e H4 f u)))) | (S n) \Rightarrow (\lambda (H3: (drop (S n) O c +x)).(getl_intro (S n) (CHead c (Flat f) u) e x (drop_drop (Flat f) n c x H3 +u) H2))]) H1)))) H0))))))). + +theorem getl_ctail: + \forall (b: B).(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: +nat).((getl i c (CHead d (Bind b) u)) \to (\forall (k: K).(\forall (v: +T).(getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u))))))))) +\def + \lambda (b: B).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: +nat).(\lambda (H: (getl i c (CHead d (Bind b) u))).(\lambda (k: K).(\lambda +(v: T).(let H0 \def (getl_gen_all c (CHead d (Bind b) u) i H) in (ex2_ind C +(\lambda (e: C).(drop i O c e)) (\lambda (e: C).(clear e (CHead d (Bind b) +u))) (getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)) (\lambda (x: +C).(\lambda (H1: (drop i O c x)).(\lambda (H2: (clear x (CHead d (Bind b) +u))).(getl_intro i (CTail k v c) (CHead (CTail k v d) (Bind b) u) (CTail k v +x) (drop_ctail c x O i H1 k v) (clear_ctail b x d u H2 k v))))) H0))))))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma index c0b188265..7f15aaf08 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma @@ -46,6 +46,8 @@ include "r/props.ma". include "clen/defs.ma". +include "clen/getl.ma". + include "flt/defs.ma". include "flt/props.ma". @@ -78,6 +80,30 @@ include "drop1/defs.ma". include "drop1/props.ma". +include "clear/defs.ma". + +include "clear/fwd.ma". + +include "clear/props.ma". + +include "clear/drop.ma". + +include "getl/defs.ma". + +include "getl/fwd.ma". + +include "getl/props.ma". + +include "getl/clear.ma". + +include "getl/drop.ma". + +include "getl/dec.ma". + +include "getl/flt.ma". + +include "cimp/defs.ma". + include "G/defs.ma". include "next_plus/defs.ma".