--- /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 "LambdaDelta-1/csubst1/defs.ma".
+
+include "LambdaDelta-1/csubst0/fwd.ma".
+
+include "LambdaDelta-1/subst1/props.ma".
+
+theorem csubst1_gen_head:
+ \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall
+(v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T
+C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2:
+T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2:
+C).(csubst1 i v c1 c2))))))))))
+\def
+ \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
+(v: T).(\lambda (i: nat).(\lambda (H: (csubst1 (s k i) v (CHead c1 k u1)
+x)).(csubst1_ind (s k i) v (CHead c1 k u1) (\lambda (c: C).(ex3_2 T C
+(\lambda (u2: T).(\lambda (c2: C).(eq C c (CHead c2 k u2)))) (\lambda (u2:
+T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2:
+C).(csubst1 i v c1 c2))))) (ex3_2_intro T C (\lambda (u2: T).(\lambda (c2:
+C).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_:
+C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1
+c2))) u1 c1 (refl_equal C (CHead c1 k u1)) (subst1_refl i v u1) (csubst1_refl
+i v c1)) (\lambda (c2: C).(\lambda (H0: (csubst0 (s k i) v (CHead c1 k u1)
+c2)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i)
+(s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2))))
+(\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat
+(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda
+(j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda
+(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2:
+T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2 T C
+(\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2:
+T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
+C).(csubst1 i v c1 c3)))) (\lambda (H1: (ex3_2 T nat (\lambda (_: T).(\lambda
+(j: nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C
+c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1
+u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s
+k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2))))
+(\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))) (ex3_2 T C (\lambda
+(u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2:
+T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
+C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H2:
+(eq nat (s k i) (s k x1))).(\lambda (H3: (eq C c2 (CHead c1 k x0))).(\lambda
+(H4: (subst0 x1 v u1 x0)).(eq_ind_r C (CHead c1 k x0) (\lambda (c: C).(ex3_2
+T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda
+(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
+C).(csubst1 i v c1 c3))))) (let H5 \def (eq_ind_r nat x1 (\lambda (n:
+nat).(subst0 n v u1 x0)) H4 i (s_inj k i x1 H2)) in (ex3_2_intro T C (\lambda
+(u2: T).(\lambda (c3: C).(eq C (CHead c1 k x0) (CHead c3 k u2)))) (\lambda
+(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
+C).(csubst1 i v c1 c3))) x0 c1 (refl_equal C (CHead c1 k x0)) (subst1_single
+i v u1 x0 H5) (csubst1_refl i v c1))) c2 H3)))))) H1)) (\lambda (H1: (ex3_2 C
+nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda
+(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c1 c3))))).(ex3_2_ind C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0
+j v c1 c3))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3
+k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: C).(\lambda (x1:
+nat).(\lambda (H2: (eq nat (s k i) (s k x1))).(\lambda (H3: (eq C c2 (CHead
+x0 k u1))).(\lambda (H4: (csubst0 x1 v c1 x0)).(eq_ind_r C (CHead x0 k u1)
+(\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead
+c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda
+(_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H5 \def (eq_ind_r nat x1
+(\lambda (n: nat).(csubst0 n v c1 x0)) H4 i (s_inj k i x1 H2)) in
+(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x0 k u1)
+(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2)))
+(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) u1 x0 (refl_equal C
+(CHead x0 k u1)) (subst1_refl i v u1) (csubst1_sing i v c1 x0 H5))) c2
+H3)))))) H1)) (\lambda (H1: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda
+(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2:
+T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))).(ex4_3_ind T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k
+j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3
+k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
+c3)))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k
+u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1:
+C).(\lambda (x2: nat).(\lambda (H2: (eq nat (s k i) (s k x2))).(\lambda (H3:
+(eq C c2 (CHead x1 k x0))).(\lambda (H4: (subst0 x2 v u1 x0)).(\lambda (H5:
+(csubst0 x2 v c1 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c: C).(ex3_2 T C
+(\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2:
+T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
+C).(csubst1 i v c1 c3))))) (let H6 \def (eq_ind_r nat x2 (\lambda (n:
+nat).(csubst0 n v c1 x1)) H5 i (s_inj k i x2 H2)) in (let H7 \def (eq_ind_r
+nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H4 i (s_inj k i x2 H2)) in
+(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x1 k x0)
+(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2)))
+(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 x1 (refl_equal C
+(CHead x1 k x0)) (subst1_single i v u1 x0 H7) (csubst1_sing i v c1 x1 H6))))
+c2 H3)))))))) H1)) (csubst0_gen_head k c1 c2 u1 v (s k i) H0)))) x H))))))).
+
projects / helm.git / blobdiff