--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/csubst1/fwd".
+
+include "csubst1/defs.ma".
+
+include "csubst0/fwd.ma".
+
+include "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)).(let H0 \def (match H in csubst1 return (\lambda (c: C).(\lambda (_:
+(csubst1 ? ? ? c)).((eq C c 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))))))) with
+[csubst1_refl \Rightarrow (\lambda (H0: (eq C (CHead c1 k u1) x)).(eq_ind C
+(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)) x H0)) | (csubst1_sing c2
+H0) \Rightarrow (\lambda (H1: (eq C c2 x)).(eq_ind C x (\lambda (c:
+C).((csubst0 (s k i) v (CHead c1 k u1) c) \to (ex3_2 T C (\lambda (u2:
+T).(\lambda (c3: C).(eq C x (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: (csubst0 (s k i) v (CHead c1 k u1) x)).(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 x (CHead c1 k u2)))) (\lambda (u2:
+T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C x (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 x (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 x (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 (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat
+(s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k
+u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))))).(ex3_2_ind T
+nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda
+(u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2:
+T).(\lambda (j: nat).(subst0 j v u1 u2))) (ex3_2 T C (\lambda (u2:
+T).(\lambda (c3: C).(eq C x (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 (H4: (eq nat (s k i) (s k
+x1))).(\lambda (H5: (eq C x (CHead c1 k x0))).(\lambda (H6: (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 H7 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v u1 x0))
+H6 i (s_inj k i x1 H4)) 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 H7)
+(csubst1_refl i v c1))) x H5)))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda
+(_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda
+(_: nat).(eq C x (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 x
+(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 x (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 (H4: (eq nat (s k i) (s k x1))).(\lambda (H5: (eq C x (CHead x0
+k u1))).(\lambda (H6: (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 H7 \def (eq_ind_r nat x1
+(\lambda (n: nat).(csubst0 n v c1 x0)) H6 i (s_inj k i x1 H4)) 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 H7))) x
+H5)))))) H3)) (\lambda (H3: (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 x (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 x (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 x (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 (H4: (eq nat (s k i) (s k x2))).(\lambda (H5:
+(eq C x (CHead x1 k x0))).(\lambda (H6: (subst0 x2 v u1 x0)).(\lambda (H7:
+(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 H8 \def (eq_ind_r nat x2 (\lambda (n:
+nat).(csubst0 n v c1 x1)) H7 i (s_inj k i x2 H4)) in (let H9 \def (eq_ind_r
+nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H6 i (s_inj k i x2 H4)) 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 H9) (csubst1_sing i v c1 x1 H8))))
+x H5)))))))) H3)) (csubst0_gen_head k c1 x u1 v (s k i) H2))) c2 (sym_eq C c2
+x H1) H0))]) in (H0 (refl_equal C x))))))))).
+
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