+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubst1/defs.ma".
-
-include "basic_1/csubst0/fwd.ma".
-
-include "basic_1/subst1/defs.ma".
-
-include "basic_1/s/fwd.ma".
-
-implied lemma csubst1_ind:
- \forall (i: nat).(\forall (v: T).(\forall (c1: C).(\forall (P: ((C \to
-Prop))).((P c1) \to (((\forall (c2: C).((csubst0 i v c1 c2) \to (P c2)))) \to
-(\forall (c: C).((csubst1 i v c1 c) \to (P c))))))))
-\def
- \lambda (i: nat).(\lambda (v: T).(\lambda (c1: C).(\lambda (P: ((C \to
-Prop))).(\lambda (f: (P c1)).(\lambda (f0: ((\forall (c2: C).((csubst0 i v c1
-c2) \to (P c2))))).(\lambda (c: C).(\lambda (c0: (csubst1 i v c1 c)).(match
-c0 with [csubst1_refl \Rightarrow f | (csubst1_sing x x0) \Rightarrow (f0 x
-x0)])))))))).
-
-lemma csubst1_gen_head:
- \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall
-(v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T
-C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2:
-T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2:
-C).(csubst1 i v c1 c2))))))))))
-\def
- \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
-(v: T).(\lambda (i: nat).(\lambda (H: (csubst1 (s k i) v (CHead c1 k u1)
-x)).(csubst1_ind (s k i) v (CHead c1 k u1) (\lambda (c: C).(ex3_2 T C
-(\lambda (u2: T).(\lambda (c2: C).(eq C c (CHead c2 k u2)))) (\lambda (u2:
-T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2:
-C).(csubst1 i v c1 c2))))) (ex3_2_intro T C (\lambda (u2: T).(\lambda (c2:
-C).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_:
-C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1
-c2))) u1 c1 (refl_equal C (CHead c1 k u1)) (subst1_refl i v u1) (csubst1_refl
-i v c1)) (\lambda (c2: C).(\lambda (H0: (csubst0 (s k i) v (CHead c1 k u1)
-c2)).(let H1 \def (csubst0_gen_head k c1 c2 u1 v (s k i) H0) in (or3_ind
-(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j))))
-(\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2:
-T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_:
-C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0
-j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
-nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda
-(_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_:
-C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3:
-C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2 T C (\lambda (u2:
-T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_:
-C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1
-c3)))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat
-(s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k
-u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))))).(ex3_2_ind T
-nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda
-(u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2:
-T).(\lambda (j: nat).(subst0 j v u1 u2))) (ex3_2 T C (\lambda (u2:
-T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_:
-C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1
-c3)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat (s k i) (s k
-x1))).(\lambda (H4: (eq C c2 (CHead c1 k x0))).(\lambda (H5: (subst0 x1 v u1
-x0)).(eq_ind_r C (CHead c1 k x0) (\lambda (c: C).(ex3_2 T C (\lambda (u2:
-T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_:
-C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1
-c3))))) (let H_y \def (s_inj k i x1 H3) in (let H6 \def (eq_ind_r nat x1
-(\lambda (n: nat).(subst0 n v u1 x0)) H5 i H_y) in (ex3_2_intro T C (\lambda
-(u2: T).(\lambda (c3: C).(eq C (CHead c1 k x0) (CHead c3 k u2)))) (\lambda
-(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
-C).(csubst1 i v c1 c3))) x0 c1 (refl_equal C (CHead c1 k x0)) (subst1_single
-i v u1 x0 H6) (csubst1_refl i v c1)))) c2 H4)))))) H2)) (\lambda (H2: (ex3_2
-C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda
-(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3:
-C).(\lambda (j: nat).(csubst0 j v c1 c3))))).(ex3_2_ind C nat (\lambda (_:
-C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_:
-nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0
-j v c1 c3))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3
-k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_:
-T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: C).(\lambda (x1:
-nat).(\lambda (H3: (eq nat (s k i) (s k x1))).(\lambda (H4: (eq C c2 (CHead
-x0 k u1))).(\lambda (H5: (csubst0 x1 v c1 x0)).(eq_ind_r C (CHead x0 k u1)
-(\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead
-c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda
-(_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H_y \def (s_inj k i x1
-H3) in (let H6 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c1 x0))
-H5 i H_y) in (ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead
-x0 k u1) (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1
-u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) u1 x0
-(refl_equal C (CHead x0 k u1)) (subst1_refl i v u1) (csubst1_sing i v c1 x0
-H6)))) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda
-(_: C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2:
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_:
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))).(ex4_3_ind T C
-nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k
-j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3
-k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
-c3)))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k
-u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_:
-T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1:
-C).(\lambda (x2: nat).(\lambda (H3: (eq nat (s k i) (s k x2))).(\lambda (H4:
-(eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v u1 x0)).(\lambda (H6:
-(csubst0 x2 v c1 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c: C).(ex3_2 T C
-(\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2:
-T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
-C).(csubst1 i v c1 c3))))) (let H_y \def (s_inj k i x2 H3) in (let H7 \def
-(eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c1 x1)) H6 i H_y) in (let H8
-\def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H5 i H_y) in
-(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x1 k x0)
-(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2)))
-(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 x1 (refl_equal C
-(CHead x1 k x0)) (subst1_single i v u1 x0 H8) (csubst1_sing i v c1 x1 H7)))))
-c2 H4)))))))) H2)) H1)))) x H))))))).
-
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