X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fcsubst0%2Ffwd.ma;h=a267513d139456aba64a928835fb84103467a41b;hb=57ae1762497a5f3ea75740e2908e04adb8642cc2;hp=9b3de2983a93f8056bb192511a9ebbdbb46cf87b;hpb=88a68a9c334646bc17314d5327cd3b790202acd6;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/fwd.ma
index 9b3de2983..a267513d1 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst0/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubst0/fwd.ma
@@ -14,9 +14,27 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/csubst0/defs.ma".
+include "basic_1/csubst0/defs.ma".
 
-theorem csubst0_gen_sort:
+include "basic_1/C/fwd.ma".
+
+implied rec lemma csubst0_ind (P: (nat \to (T \to (C \to (C \to Prop))))) (f: 
+(\forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
+(u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(P (s k i) v (CHead c k u1) 
+(CHead c k u2)))))))))) (f0: (\forall (k: K).(\forall (i: nat).(\forall (c1: 
+C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to ((P i v c1 c2) 
+\to (\forall (u: T).(P (s k i) v (CHead c1 k u) (CHead c2 k u))))))))))) (f1: 
+(\forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
+(u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst0 i 
+v c1 c2) \to ((P i v c1 c2) \to (P (s k i) v (CHead c1 k u1) (CHead c2 k 
+u2))))))))))))) (n: nat) (t: T) (c: C) (c0: C) (c1: csubst0 n t c c0) on c1: 
+P n t c c0 \def match c1 with [(csubst0_snd k i v u1 u2 s0 c2) \Rightarrow (f 
+k i v u1 u2 s0 c2) | (csubst0_fst k i c2 c3 v c4 u) \Rightarrow (f0 k i c2 c3 
+v c4 ((csubst0_ind P f f0 f1) i v c2 c3 c4) u) | (csubst0_both k i v u1 u2 s0 
+c2 c3 c4) \Rightarrow (f1 k i v u1 u2 s0 c2 c3 c4 ((csubst0_ind P f f0 f1) i 
+v c2 c3 c4))].
+
+lemma csubst0_gen_sort:
  \forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0 
 i v (CSort n) x) \to (\forall (P: Prop).P)))))
 \def
@@ -28,27 +46,22 @@ C).(\lambda (H0: (csubst0 i v y x)).(csubst0_ind (\lambda (_: nat).(\lambda
 (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda 
 (u2: T).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H2: (eq 
 C (CHead c k u1) (CSort n))).(let H3 \def (eq_ind C (CHead c k u1) (\lambda 
-(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H2) in 
-(False_ind P H3)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (_: (csubst0 i0 v0 c1 
-c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (u: T).(\lambda 
-(H3: (eq C (CHead c1 k u) (CSort n))).(let H4 \def (eq_ind C (CHead c1 k u) 
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
+(ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) 
+\Rightarrow True])) I (CSort n) H2) in (False_ind P H3)))))))))) (\lambda (k: 
+K).(\lambda (i0: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v0: 
+T).(\lambda (_: (csubst0 i0 v0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to 
+P))).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) (CSort n))).(let H4 
+\def (eq_ind C (CHead c1 k u) (\lambda (ee: C).(match ee with [(CSort _) 
 \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in 
 (False_ind P H4))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: 
 T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i0 v0 u1 
 u2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubst0 i0 v0 c1 
 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (H4: (eq C (CHead 
 c1 k u1) (CSort n))).(let H5 \def (eq_ind C (CHead c1 k u1) (\lambda (ee: 
-C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow 
-False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind P 
-H5))))))))))))) i v y x H0))) H)))))).
-(* COMMENTS
-Initial nodes: 355
-END *)
+C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
+True])) I (CSort n) H4) in (False_ind P H5))))))))))))) i v y x H0))) H)))))).
 
-theorem csubst0_gen_head:
+lemma csubst0_gen_head:
  \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall 
 (v: T).(\forall (i: nat).((csubst0 i v (CHead c1 k u1) x) \to (or3 (ex3_2 T 
 nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: 
@@ -89,100 +102,98 @@ T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j t c1 c2))))))))))) (\lambda
 (k0: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: 
 T).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (c: C).(\lambda (H2: (eq C 
 (CHead c k0 u0) (CHead c1 k u1))).(let H3 \def (f_equal C C (\lambda (e: 
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c | 
-(CHead c0 _ _) \Rightarrow c0])) (CHead c k0 u0) (CHead c1 k u1) H2) in ((let 
-H4 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) 
-with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c k0 
-u0) (CHead c1 k u1) H2) in ((let H5 \def (f_equal C T (\lambda (e: C).(match 
-e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ 
-t) \Rightarrow t])) (CHead c k0 u0) (CHead c1 k u1) H2) in (\lambda (H6: (eq 
-K k0 k)).(\lambda (H7: (eq C c c1)).(eq_ind_r C c1 (\lambda (c0: C).(or3 
-(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) 
-(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c1 k u3)))) 
-(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat 
-(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c2: 
-C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k u1)))) (\lambda (c2: 
-C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda 
-(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k 
-u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 
-u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 
-c2))))))) (let H8 \def (eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 
-u1 H5) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_: 
-T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda 
-(_: nat).(eq C (CHead c1 k1 u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda 
+C).(match e with [(CSort _) \Rightarrow c | (CHead c0 _ _) \Rightarrow c0])) 
+(CHead c k0 u0) (CHead c1 k u1) H2) in ((let H4 \def (f_equal C K (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow 
+k1])) (CHead c k0 u0) (CHead c1 k u1) H2) in ((let H5 \def (f_equal C T 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
+\Rightarrow t])) (CHead c k0 u0) (CHead c1 k u1) H2) in (\lambda (H6: (eq K 
+k0 k)).(\lambda (H7: (eq C c c1)).(eq_ind_r C c1 (\lambda (c0: C).(or3 (ex3_2 
+T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda 
+(u3: T).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c1 k u3)))) (\lambda 
+(u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: 
+C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c2: C).(\lambda 
+(_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda 
+(j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u3: T).(\lambda 
+(c2: C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k u3))))) (\lambda 
+(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: 
+T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2))))))) (let H8 \def 
+(eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H5) in (eq_ind_r K k 
+(\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat 
+(s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k1 
+u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 
+u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k 
+j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k1 u2) (CHead c2 k 
+u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C 
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k 
+j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k1 
+u2) (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: 
+nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: 
+nat).(csubst0 j v0 c1 c2))))))) (or3_intro0 (ex3_2 T nat (\lambda (_: 
+T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda 
+(_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda 
 (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
-nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C 
-(CHead c1 k1 u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: 
+nat).(eq nat (s k i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C 
+(CHead c1 k u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: 
 nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda 
-(c2: C).(\lambda (_: nat).(eq C (CHead c1 k1 u2) (CHead c2 k u3))))) (\lambda 
+C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda 
+(c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u3))))) (\lambda 
 (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: 
-T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2))))))) (or3_intro0 
-(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) 
-(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) 
-(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat 
-(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c2: 
-C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u1)))) (\lambda (c2: 
-C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda 
-(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k 
-u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 
-u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 
-c2))))) (ex3_2_intro T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) 
-(s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 
-k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3))) u2 i0 
-(refl_equal nat (s k i0)) (refl_equal C (CHead c1 k u2)) H8)) k0 H6)) c 
-H7)))) H4)) H3)))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (c0: 
-C).(\lambda (c2: C).(\lambda (v0: T).(\lambda (H1: (csubst0 i0 v0 c0 
-c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda 
-(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2: T).(\lambda (_: 
-nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
-v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k 
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda 
-(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: 
+T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2))))) (ex3_2_intro T 
+nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda 
+(u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) (\lambda 
+(u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3))) u2 i0 (refl_equal nat (s k 
+i0)) (refl_equal C (CHead c1 k u2)) H8)) k0 H6)) c H7)))) H4)) H3)))))))))) 
+(\lambda (k0: K).(\lambda (i0: nat).(\lambda (c0: C).(\lambda (c2: 
+C).(\lambda (v0: T).(\lambda (H1: (csubst0 i0 v0 c0 c2)).(\lambda (H2: (((eq 
+C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: 
+nat).(eq nat i0 (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead 
+c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 
+C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda 
+(j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat i0 (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 v0 u1 u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda (u: T).(\lambda 
+(H3: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(let H4 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
+\Rightarrow c])) (CHead c0 k0 u) (CHead c1 k u1) H3) in ((let H5 \def 
+(f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow k0 | (CHead 
+_ k1 _) \Rightarrow k1])) (CHead c0 k0 u) (CHead c1 k u1) H3) in ((let H6 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
+(CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u) (CHead c1 k u1) H3) in 
+(\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 c1)).(eq_ind_r T u1 
+(\lambda (t: T).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat 
+(s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 k0 t) 
+(CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) 
+(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) 
+(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k0 t) (CHead c3 k u1)))) 
+(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat 
+(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k 
+j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k0 
+t) (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: 
+nat).(subst0 j v0 u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v0 c1 c3))))))) (let H9 \def (eq_ind C c0 (\lambda (c: 
+C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda 
+(j: nat).(eq nat i0 (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 
+(CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) 
+(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda 
+(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: 
 T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (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 v0 u1 u2)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda 
-(u: T).(\lambda (H3: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(let H4 \def 
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u) 
-(CHead c1 k u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in 
-C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) 
-\Rightarrow k1])) (CHead c0 k0 u) (CHead c1 k u1) H3) in ((let H6 \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 c0 k0 u) 
-(CHead c1 k u1) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 
-c1)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex3_2 T nat (\lambda (_: 
-T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda 
-(_: nat).(eq C (CHead c2 k0 t) (CHead c1 k u2)))) (\lambda (u2: T).(\lambda 
-(j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
-nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C 
-(CHead c2 k0 t) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))))))) H2 c1 H8) 
+in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubst0 i0 v0 c c2)) H1 c1 H8) 
+in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda 
+(j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq 
+C (CHead c2 k1 u1) (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: 
+nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
+nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C 
+(CHead c2 k1 u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: 
 nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u2: T).(\lambda 
-(c3: C).(\lambda (_: nat).(eq C (CHead c2 k0 t) (CHead c3 k u2))))) (\lambda 
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (let H9 \def 
-(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat 
-(\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u2: 
-T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda 
-(j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
-nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead 
-c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 
-T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (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 v0 u1 
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 
-c3)))))))) H2 c1 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubst0 
-i0 v0 c c2)) H1 c1 H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat 
-(\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u2: 
-T).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c1 k u2)))) (\lambda (u2: 
-T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: 
-C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda 
-(_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda 
-(j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
 C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u2: T).(\lambda 
 (c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u2))))) (\lambda 
 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: 
@@ -215,59 +226,55 @@ T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u3))))) (\lambda
 (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: 
 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda 
 (H4: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C 
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k 
-u1) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return 
-(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) 
-\Rightarrow k1])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H7 \def 
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) 
-(CHead c1 k u1) H4) in (\lambda (H8: (eq K k0 k)).(\lambda (H9: (eq C c0 
-c1)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to 
-(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) 
-(\lambda (u3: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: 
-T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: 
-C).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 
-j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
-nat).(eq nat i0 (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda 
-(j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v0 c1 c3)))))))) H3 c1 H9) in (let H11 \def (eq_ind C c0 
-(\lambda (c: C).(csubst0 i0 v0 c c2)) H2 c1 H9) in (let H12 \def (eq_ind T u0 
-(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H7) in (eq_ind_r K k (\lambda (k1: 
-K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k 
-j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c1 k 
-u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat 
-(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: 
-C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k u1)))) (\lambda (c3: 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
+\Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H6 \def 
+(f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow k0 | (CHead 
+_ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H7 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u0 | 
+(CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in 
+(\lambda (H8: (eq K k0 k)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind 
+C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda 
+(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u3: T).(\lambda (_: 
+nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j 
+v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k 
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u3: 
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u3))))) (\lambda 
+(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))))))) H3 c1 H9) 
+in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubst0 i0 v0 c c2)) H2 c1 H9) 
+in (let H12 \def (eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H7) 
+in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda 
+(j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq 
+C (CHead c2 k1 u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: 
+nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
+nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C 
+(CHead c2 k1 u2) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda 
+(c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k u3))))) (\lambda 
+(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (or3_intro2 
+(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) 
+(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c1 k u3)))) 
+(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat 
+(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: 
+C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u1)))) (\lambda (c3: 
 C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda 
-(u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k 
-u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 
-u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 
-c3))))))) (or3_intro2 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat 
-(s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k u2) 
-(CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) 
-(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) 
-(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u1)))) 
-(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat 
-(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k 
-j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k 
-u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: 
-nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v0 c1 c3))))) (ex4_3_intro T C nat (\lambda (_: T).(\lambda 
-(_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: 
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda 
+(u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k 
 u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 
 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 
-c3)))) u2 c2 i0 (refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u2)) H12 
-H11)) k0 H8))))))) H6)) H5))))))))))))) i v y x H0))) H))))))).
-(* COMMENTS
-Initial nodes: 4039
-END *)
+c3))))) (ex4_3_intro T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
+nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda 
+(_: nat).(eq C (CHead c2 k u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda 
+(_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) u2 c2 i0 (refl_equal nat (s k 
+i0)) (refl_equal C (CHead c2 k u2)) H12 H11)) k0 H8))))))) H6)) 
+H5))))))))))))) i v y x H0))) H))))))).
 
-theorem csubst0_gen_S_bind_2:
+lemma csubst0_gen_S_bind_2:
  \forall (b: B).(\forall (x: C).(\forall (c2: C).(\forall (v: T).(\forall 
 (v2: T).(\forall (i: nat).((csubst0 (S i) v x (CHead c2 (Bind b) v2)) \to 
 (or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x 
@@ -277,186 +284,190 @@ C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_:
 T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 
 (Bind b) v1))))))))))))
 \def
- \lambda (b: B).(\lambda (x: C).(\lambda (c2: C).(\lambda (v: T).(\lambda 
-(v2: T).(\lambda (i: nat).(\lambda (H: (csubst0 (S i) v x (CHead c2 (Bind b) 
-v2))).(insert_eq C (CHead c2 (Bind b) v2) (\lambda (c: C).(csubst0 (S i) v x 
-c)) (\lambda (_: C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda 
-(v1: T).(eq C x (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i 
-v c1 c2)) (\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T 
+ \lambda (b: B).(\lambda (x: C).(C_ind (\lambda (c: C).(\forall (c2: 
+C).(\forall (v: T).(\forall (v2: T).(\forall (i: nat).((csubst0 (S i) v c 
+(CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) 
+(\lambda (v1: T).(eq C c (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: 
+C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C c (CHead c1 (Bind b) v2)))) 
+(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda 
+(c1: C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
+T).(eq C c (CHead c1 (Bind b) v1)))))))))))) (\lambda (n: nat).(\lambda (c2: 
+C).(\lambda (v: T).(\lambda (v2: T).(\lambda (i: nat).(\lambda (H: (csubst0 
+(S i) v (CSort n) (CHead c2 (Bind b) v2))).(csubst0_gen_sort (CHead c2 (Bind 
+b) v2) v (S i) n H (or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda 
+(v1: T).(eq C (CSort n) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: 
+C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C (CSort n) (CHead c1 (Bind b) 
+v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) 
+(\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: 
+C).(\lambda (v1: T).(eq C (CSort n) (CHead c1 (Bind b) v1))))))))))))) 
+(\lambda (c: C).(\lambda (_: ((\forall (c2: C).(\forall (v: T).(\forall (v2: 
+T).(\forall (i: nat).((csubst0 (S i) v c (CHead c2 (Bind b) v2)) \to (or3 
+(ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C c (CHead 
+c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: 
+C).(eq C c (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: 
+T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
+c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C c (CHead c1 (Bind b) 
+v1))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: 
+T).(\lambda (v2: T).(\lambda (i: nat).(\lambda (H0: (csubst0 (S i) v (CHead c 
+k t) (CHead c2 (Bind b) v2))).(let H1 \def (csubst0_gen_head k c (CHead c2 
+(Bind b) v2) t v (S i) H0) in (or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda 
+(j: nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C 
+(CHead c2 (Bind b) v2) (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
+nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq 
+nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 (Bind 
+b) v2) (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
+c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq 
+nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq 
+C (CHead c2 (Bind b) v2) (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: 
+C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3))))) (or3 (ex2 T (\lambda (v1: 
+T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k t) (CHead c2 (Bind 
+b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C 
+(CHead c k t) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda 
+(v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
+c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c k t) (CHead c1 (Bind 
+b) v1)))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq 
+nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 (Bind 
+b) v2) (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t 
+u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k 
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 (Bind b) v2) (CHead 
+c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or3 (ex2 T 
+(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k t) 
+(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) 
+(\lambda (c1: C).(eq C (CHead c k t) (CHead c1 (Bind b) v2)))) (ex3_2 C T 
 (\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: 
 C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
-T).(eq C x (CHead c1 (Bind b) v1))))))) (\lambda (y: C).(\lambda (H0: 
-(csubst0 (S i) v x y)).(insert_eq nat (S i) (\lambda (n: nat).(csubst0 n v x 
-y)) (\lambda (_: nat).((eq C y (CHead c2 (Bind b) v2)) \to (or3 (ex2 T 
-(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x (CHead c2 (Bind 
+T).(eq C (CHead c k t) (CHead c1 (Bind b) v1)))))) (\lambda (x0: T).(\lambda 
+(x1: nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C (CHead 
+c2 (Bind b) v2) (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 
+\def (f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | 
+(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) v2) (CHead c k x0) H4) in 
+((let H7 \def (f_equal C K (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow (Bind b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 (Bind b) 
+v2) (CHead c k x0) H4) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow v2 | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 
+(Bind b) v2) (CHead c k x0) H4) in (\lambda (H9: (eq K (Bind b) k)).(\lambda 
+(H10: (eq C c2 c)).(let H11 \def (eq_ind_r T x0 (\lambda (t0: T).(subst0 x1 v 
+t t0)) H5 v2 H8) in (eq_ind_r C c (\lambda (c0: C).(or3 (ex2 T (\lambda (v1: 
+T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k t) (CHead c0 (Bind 
+b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c0)) (\lambda (c1: C).(eq C 
+(CHead c k t) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda 
+(v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
+c0))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c k t) (CHead c1 (Bind 
+b) v1))))))) (let H12 \def (eq_ind_r K k (\lambda (k0: K).(eq nat (S i) (s k0 
+x1))) H3 (Bind b) H9) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2 T 
+(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k0 t) 
+(CHead c (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c)) (\lambda 
+(c1: C).(eq C (CHead c k0 t) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda 
+(_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: 
+T).(csubst0 i v c1 c))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c k0 
+t) (CHead c1 (Bind b) v1))))))) (let H13 \def (f_equal nat nat (\lambda (e: 
+nat).(match e with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S x1) 
+H12) in (let H14 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t v2)) 
+H11 i H13) in (or3_intro0 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) 
+(\lambda (v1: T).(eq C (CHead c (Bind b) t) (CHead c (Bind b) v1)))) (ex2 C 
+(\lambda (c1: C).(csubst0 i v c1 c)) (\lambda (c1: C).(eq C (CHead c (Bind b) 
+t) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: 
+T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 c))) 
+(\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c (Bind b) t) (CHead c1 (Bind 
+b) v1))))) (ex_intro2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: 
+T).(eq C (CHead c (Bind b) t) (CHead c (Bind b) v1))) t H14 (refl_equal C 
+(CHead c (Bind b) t)))))) k H9)) c2 H10))))) H7)) H6))))))) H2)) (\lambda 
+(H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j)))) 
+(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 (Bind b) v2) (CHead c3 k 
+t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C 
+nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3: 
+C).(\lambda (_: nat).(eq C (CHead c2 (Bind b) v2) (CHead c3 k t)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or3 (ex2 T (\lambda (v1: 
+T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k t) (CHead c2 (Bind 
 b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C 
-x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: 
-T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
-c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind b) v1)))))))) 
-(\lambda (y0: nat).(\lambda (H1: (csubst0 y0 v x y)).(csubst0_ind (\lambda 
-(n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n (S i)) 
-\to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: 
-T).(subst0 i t v1 v2)) (\lambda (v1: T).(eq C c (CHead c2 (Bind b) v1)))) 
-(ex2 C (\lambda (c1: C).(csubst0 i t c1 c2)) (\lambda (c1: C).(eq C c (CHead 
-c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i t v1 
-v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i t c1 c2))) (\lambda (c1: 
-C).(\lambda (v1: T).(eq C c (CHead c1 (Bind b) v1)))))))))))) (\lambda (k: 
-K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat 
-(s k i0) (S i))).(\lambda (H4: (eq C (CHead c k u2) (CHead c2 (Bind b) 
-v2))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow c | (CHead c0 _ _) \Rightarrow c0])) 
-(CHead c k u2) (CHead c2 (Bind b) v2) H4) in ((let H6 \def (f_equal C K 
-(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) 
-\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c k u2) (CHead c2 
-(Bind b) v2) H4) in ((let H7 \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 c k u2) (CHead c2 (Bind b) v2) H4) in (\lambda (H8: 
-(eq K k (Bind b))).(\lambda (H9: (eq C c c2)).(eq_ind_r C c2 (\lambda (c0: 
-C).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C 
-(CHead c0 k u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i 
-v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v2)))) 
-(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda 
-(c1: C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1: 
-T).(eq C (CHead c0 k u1) (CHead c1 (Bind b) v1))))))) (let H10 \def (eq_ind T 
-u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H7) in (let H11 \def (eq_ind K 
-k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H3 (Bind b) H8) in (eq_ind_r K 
-(Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) 
-(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c2 (Bind b) v1)))) (ex2 C 
-(\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda (c1: C).(eq C (CHead c2 k0 
-u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: 
-T).(subst0 i v0 v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v0 c1 
-c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c2 k0 u1) (CHead c1 
-(Bind b) v1))))))) (let H12 \def (f_equal nat nat (\lambda (e: nat).(match e 
-in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) 
-\Rightarrow n])) (S i0) (S i) H11) in (let H13 \def (eq_ind nat i0 (\lambda 
-(n: nat).(subst0 n v0 u1 v2)) H10 i H12) in (or3_intro0 (ex2 T (\lambda (v1: 
-T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind b) u1) (CHead 
-c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v0 c1 c2)) (\lambda 
-(c1: C).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v2)))) (ex3_2 C T 
-(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c1: 
-C).(\lambda (_: T).(csubst0 i v0 c1 c2))) (\lambda (c1: C).(\lambda (v1: 
-T).(eq C (CHead c2 (Bind b) u1) (CHead c1 (Bind b) v1))))) (ex_intro2 T 
-(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c2 (Bind 
-b) u1) (CHead c2 (Bind b) v1))) u1 H13 (refl_equal C (CHead c2 (Bind b) 
-u1)))))) k H8))) c H9)))) H6)) H5))))))))))) (\lambda (k: K).(\lambda (i0: 
-nat).(\lambda (c1: C).(\lambda (c0: C).(\lambda (v0: T).(\lambda (H2: 
-(csubst0 i0 v0 c1 c0)).(\lambda (H3: (((eq nat i0 (S i)) \to ((eq C c0 (CHead 
-c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) 
-(\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: 
-C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) 
-(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda 
-(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: 
-T).(eq C c1 (CHead c3 (Bind b) v1)))))))))).(\lambda (u: T).(\lambda (H4: (eq 
-nat (s k i0) (S i))).(\lambda (H5: (eq C (CHead c0 k u) (CHead c2 (Bind b) 
-v2))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) 
-(CHead c0 k u) (CHead c2 (Bind b) v2) H5) in ((let H7 \def (f_equal C K 
-(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) 
-\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead c2 
-(Bind b) v2) H5) 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 c0 k u) (CHead c2 (Bind b) v2) H5) in (\lambda (H9: 
-(eq K k (Bind b))).(\lambda (H10: (eq C c0 c2)).(eq_ind_r T v2 (\lambda (t: 
-T).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C 
-(CHead c1 k t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i 
-v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k t) (CHead c3 (Bind b) v2)))) 
-(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda 
-(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: 
-T).(eq C (CHead c1 k t) (CHead c3 (Bind b) v1))))))) (let H11 \def (eq_ind C 
-c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C c (CHead c2 (Bind b) v2)) 
-\to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C 
-c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) 
-(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: 
-C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: 
-T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead 
-c3 (Bind b) v1))))))))) H3 c2 H10) in (let H12 \def (eq_ind C c0 (\lambda (c: 
-C).(csubst0 i0 v0 c1 c)) H2 c2 H10) in (let H13 \def (eq_ind K k (\lambda 
-(k0: K).(eq nat (s k0 i0) (S i))) H4 (Bind b) H9) in (eq_ind_r K (Bind b) 
-(\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda 
-(v1: T).(eq C (CHead c1 k0 v2) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: 
-C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 k0 v2) (CHead c3 
-(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 
-v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: 
-C).(\lambda (v1: T).(eq C (CHead c1 k0 v2) (CHead c3 (Bind b) v1))))))) (let 
-H14 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return (\lambda 
-(_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n])) (S i0) (S i) 
-H13) in (let H15 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n (S i)) \to 
-((eq C c2 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i 
-v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C 
-(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 
-(Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 
-v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: 
-C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H11 i H14) in 
-(let H16 \def (eq_ind nat i0 (\lambda (n: nat).(csubst0 n v0 c1 c2)) H12 i 
-H14) in (or3_intro1 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda 
-(v1: T).(eq C (CHead c1 (Bind b) v2) (CHead c2 (Bind b) v1)))) (ex2 C 
-(\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind 
-b) v2) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: 
-T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 
-c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind b) v2) (CHead 
-c3 (Bind b) v1))))) (ex_intro2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) 
-(\lambda (c3: C).(eq C (CHead c1 (Bind b) v2) (CHead c3 (Bind b) v2))) c1 H16 
-(refl_equal C (CHead c1 (Bind b) v2))))))) k H9)))) u H8)))) H7)) 
-H6)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda 
-(u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c1: 
-C).(\lambda (c0: C).(\lambda (H3: (csubst0 i0 v0 c1 c0)).(\lambda (H4: (((eq 
-nat i0 (S i)) \to ((eq C c0 (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda 
-(v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) 
-v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C 
-c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: 
-T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: T).(csubst0 i v0 c3 
-c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind b) 
-v1)))))))))).(\lambda (H5: (eq nat (s k i0) (S i))).(\lambda (H6: (eq C 
-(CHead c0 k u2) (CHead c2 (Bind b) v2))).(let H7 \def (f_equal C C (\lambda 
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 
-| (CHead c _ _) \Rightarrow c])) (CHead c0 k u2) (CHead c2 (Bind b) v2) H6) 
-in ((let H8 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda 
-(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
-(CHead c0 k u2) (CHead c2 (Bind b) v2) H6) in ((let H9 \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 c0 k u2) (CHead c2 
-(Bind b) v2) H6) in (\lambda (H10: (eq K k (Bind b))).(\lambda (H11: (eq C c0 
-c2)).(let H12 \def (eq_ind C c0 (\lambda (c: C).((eq nat i0 (S i)) \to ((eq C 
-c (CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 
-v2)) (\lambda (v1: T).(eq C c1 (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: 
-C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) 
-(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda 
-(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: 
-T).(eq C c1 (CHead c3 (Bind b) v1))))))))) H4 c2 H11) in (let H13 \def 
-(eq_ind C c0 (\lambda (c: C).(csubst0 i0 v0 c1 c)) H3 c2 H11) in (let H14 
-\def (eq_ind T u2 (\lambda (t: T).(subst0 i0 v0 u1 t)) H2 v2 H9) in (let H15 
-\def (eq_ind K k (\lambda (k0: K).(eq nat (s k0 i0) (S i))) H5 (Bind b) H10) 
-in (eq_ind_r K (Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: T).(subst0 
-i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 k0 u1) (CHead c2 (Bind b) 
-v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) (\lambda (c3: C).(eq C 
-(CHead c1 k0 u1) (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: 
-C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: 
-T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 
-k0 u1) (CHead c3 (Bind b) v1))))))) (let H16 \def (f_equal nat nat (\lambda 
-(e: nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i0 
-| (S n) \Rightarrow n])) (S i0) (S i) H15) in (let H17 \def (eq_ind nat i0 
-(\lambda (n: nat).((eq nat n (S i)) \to ((eq C c2 (CHead c2 (Bind b) v2)) \to 
-(or3 (ex2 T (\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C c1 
-(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 c2)) 
-(\lambda (c3: C).(eq C c1 (CHead c3 (Bind b) v2)))) (ex3_2 C T (\lambda (_: 
-C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: C).(\lambda (_: 
-T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead 
-c3 (Bind b) v1))))))))) H12 i H16) in (let H18 \def (eq_ind nat i0 (\lambda 
-(n: nat).(csubst0 n v0 c1 c2)) H13 i H16) in (let H19 \def (eq_ind nat i0 
-(\lambda (n: nat).(subst0 n v0 u1 v2)) H14 i H16) in (or3_intro2 (ex2 T 
-(\lambda (v1: T).(subst0 i v0 v1 v2)) (\lambda (v1: T).(eq C (CHead c1 (Bind 
-b) u1) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c3: C).(csubst0 i v0 c3 
-c2)) (\lambda (c3: C).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v2)))) 
-(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda 
-(c3: C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: 
-T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1))))) (ex3_2_intro C T 
-(\lambda (_: C).(\lambda (v1: T).(subst0 i v0 v1 v2))) (\lambda (c3: 
-C).(\lambda (_: T).(csubst0 i v0 c3 c2))) (\lambda (c3: C).(\lambda (v1: 
-T).(eq C (CHead c1 (Bind b) u1) (CHead c3 (Bind b) v1)))) c1 u1 H19 H18 
-(refl_equal C (CHead c1 (Bind b) u1)))))))) k H10)))))))) H8)) 
-H7)))))))))))))) y0 v x y H1))) H0))) H))))))).
-(* COMMENTS
-Initial nodes: 3878
-END *)
+(CHead c k t) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda 
+(v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
+c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c k t) (CHead c1 (Bind 
+b) v1)))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S i) 
+(s k x1))).(\lambda (H4: (eq C (CHead c2 (Bind b) v2) (CHead x0 k 
+t))).(\lambda (H5: (csubst0 x1 v c x0)).(let H6 \def (f_equal C C (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow 
+c0])) (CHead c2 (Bind b) v2) (CHead x0 k t) H4) in ((let H7 \def (f_equal C K 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow (Bind b) | (CHead _ k0 
+_) \Rightarrow k0])) (CHead c2 (Bind b) v2) (CHead x0 k t) H4) in ((let H8 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow v2 | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) v2) (CHead x0 k t) H4) in 
+(\lambda (H9: (eq K (Bind b) k)).(\lambda (H10: (eq C c2 x0)).(let H11 \def 
+(eq_ind_r C x0 (\lambda (c0: C).(csubst0 x1 v c c0)) H5 c2 H10) in (eq_ind_r 
+T t (\lambda (t0: T).(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 t0)) 
+(\lambda (v1: T).(eq C (CHead c k t) (CHead c2 (Bind b) v1)))) (ex2 C 
+(\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C (CHead c k t) 
+(CHead c1 (Bind b) t0)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 
+i v v1 t0))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda 
+(c1: C).(\lambda (v1: T).(eq C (CHead c k t) (CHead c1 (Bind b) v1))))))) 
+(let H12 \def (eq_ind_r K k (\lambda (k0: K).(eq nat (S i) (s k0 x1))) H3 
+(Bind b) H9) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: 
+T).(subst0 i v v1 t)) (\lambda (v1: T).(eq C (CHead c k0 t) (CHead c2 (Bind 
+b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C 
+(CHead c k0 t) (CHead c1 (Bind b) t)))) (ex3_2 C T (\lambda (_: C).(\lambda 
+(v1: T).(subst0 i v v1 t))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
+c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c k0 t) (CHead c1 (Bind 
+b) v1))))))) (let H13 \def (f_equal nat nat (\lambda (e: nat).(match e with 
+[O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S x1) H12) in (let H14 \def 
+(eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c c2)) H11 i H13) in 
+(or3_intro1 (ex2 T (\lambda (v1: T).(subst0 i v v1 t)) (\lambda (v1: T).(eq C 
+(CHead c (Bind b) t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: 
+C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C (CHead c (Bind b) t) (CHead c1 
+(Bind b) t)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 
+t))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: 
+C).(\lambda (v1: T).(eq C (CHead c (Bind b) t) (CHead c1 (Bind b) v1))))) 
+(ex_intro2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C 
+(CHead c (Bind b) t) (CHead c1 (Bind b) t))) c H14 (refl_equal C (CHead c 
+(Bind b) t)))))) k H9)) v2 H8))))) H7)) H6))))))) H2)) (\lambda (H2: (ex4_3 T 
+C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k 
+j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 
+(Bind b) v2) (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: 
+C).(\lambda (_: nat).(eq C (CHead c2 (Bind b) v2) (CHead c3 k u2))))) 
+(\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) 
+(\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or3 
+(ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k 
+t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) 
+(\lambda (c1: C).(eq C (CHead c k t) (CHead c1 (Bind b) v2)))) (ex3_2 C T 
+(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: 
+C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
+T).(eq C (CHead c k t) (CHead c1 (Bind b) v1)))))) (\lambda (x0: T).(\lambda 
+(x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S i) (s k x2))).(\lambda 
+(H4: (eq C (CHead c2 (Bind b) v2) (CHead x1 k x0))).(\lambda (H5: (subst0 x2 
+v t x0)).(\lambda (H6: (csubst0 x2 v c x1)).(let H7 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead c2 (Bind b) v2) (CHead x1 k x0) H4) in ((let H8 \def 
+(f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow (Bind b) | 
+(CHead _ k0 _) \Rightarrow k0])) (CHead c2 (Bind b) v2) (CHead x1 k x0) H4) 
+in ((let H9 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow v2 | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) v2) 
+(CHead x1 k x0) H4) in (\lambda (H10: (eq K (Bind b) k)).(\lambda (H11: (eq C 
+c2 x1)).(let H12 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 x2 v c c0)) H6 
+c2 H11) in (let H13 \def (eq_ind_r T x0 (\lambda (t0: T).(subst0 x2 v t t0)) 
+H5 v2 H9) in (let H14 \def (eq_ind_r K k (\lambda (k0: K).(eq nat (S i) (s k0 
+x2))) H3 (Bind b) H10) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2 T 
+(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k0 t) 
+(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) 
+(\lambda (c1: C).(eq C (CHead c k0 t) (CHead c1 (Bind b) v2)))) (ex3_2 C T 
+(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: 
+C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
+T).(eq C (CHead c k0 t) (CHead c1 (Bind b) v1))))))) (let H15 \def (f_equal 
+nat nat (\lambda (e: nat).(match e with [O \Rightarrow i | (S n) \Rightarrow 
+n])) (S i) (S x2) H14) in (let H16 \def (eq_ind_r nat x2 (\lambda (n: 
+nat).(csubst0 n v c c2)) H12 i H15) in (let H17 \def (eq_ind_r nat x2 
+(\lambda (n: nat).(subst0 n v t v2)) H13 i H15) in (or3_intro2 (ex2 T 
+(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c (Bind b) 
+t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) 
+(\lambda (c1: C).(eq C (CHead c (Bind b) t) (CHead c1 (Bind b) v2)))) (ex3_2 
+C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: 
+C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
+T).(eq C (CHead c (Bind b) t) (CHead c1 (Bind b) v1))))) (ex3_2_intro C T 
+(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: 
+C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
+T).(eq C (CHead c (Bind b) t) (CHead c1 (Bind b) v1)))) c t H17 H16 
+(refl_equal C (CHead c (Bind b) t))))))) k H10))))))) H8)) H7))))))))) H2)) 
+H1))))))))))) x)).