X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fpr2%2Fclen.ma;h=ab6e6ef402e4dd497c77cb0388f5611244c4823a;hb=89519c7b52e06304a94019dd528925300380cdc0;hp=27275fa3a10fb54d5926a107b59231d87ac45cb2;hpb=831af787465e1bff886e22ee14b68c8f1bb0177c;p=helm.git

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/clen.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/clen.ma
index 27275fa3a..ab6e6ef40 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/clen.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/clen.ma
@@ -14,11 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pr2/clen".
+include "LambdaDelta-1/pr2/props.ma".
 
-include "pr2/props.ma".
-
-include "clen/getl.ma".
+include "LambdaDelta-1/clen/getl.ma".
 
 theorem pr2_gen_ctail:
  \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall 
@@ -28,54 +26,55 @@ T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0
 \def
  \lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda 
 (t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c) 
-(\lambda (c0: C).(pr2 c0 t1 t2)) (or (pr2 c t1 t2) (ex3 T (\lambda (_: T).(eq 
-K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 (clen 
-c) u t t2)))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda 
-(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CTail k u c)) \to (or 
-(pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t3: 
-T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3 t0)))))))) (\lambda 
-(c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda 
-(_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4) (ex3 T (\lambda (_: 
-T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(subst0 
-(clen c) u t t4))) (pr2_free c t3 t4 H1))))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind 
-Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 
-t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 t)).(\lambda (H4: (eq C c0 
-(CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead 
-d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let H_x \def (getl_gen_tail k 
-Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind (ex2 C (\lambda (e: C).(eq 
-C d (CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) u0)))) 
-(ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k 
-(Bind Abbr))) (\lambda (_: nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort 
-n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda 
-(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda 
-(H7: (ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c 
-(CHead e (Bind Abbr) u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u 
-e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) 
-(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) 
-(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq 
-C d (CTail k u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) 
-u0))).(or_introl (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) 
-(\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) 
-(pr2_delta c x u0 i H9 t3 t4 H2 t H3))))) H7)) (\lambda (H7: (ex4 nat 
-(\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind 
-Abbr))) (\lambda (_: nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort 
-n))))).(ex4_ind nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: 
-nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) (\lambda (n: 
-nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k 
-(Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) 
-u t0 t)))) (\lambda (x0: nat).(\lambda (H8: (eq nat i (clen c))).(\lambda 
-(H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T u u0)).(\lambda (_: (eq C d 
-(CSort x0))).(let H12 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u0 t4 
-t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0 (\lambda (t0: T).(subst0 
-(clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or 
-(pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind Abbr))) (\lambda (t0: 
-T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))))) (or_intror (pr2 
-c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: 
-T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (ex3_intro T 
-(\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) 
-(\lambda (t0: T).(subst0 (clen c) u t0 t)) t4 (refl_equal K (Bind Abbr)) H2 
-H13)) k H9)))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))).
+(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(or (pr2 c t1 t2) (ex3 T 
+(\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda 
+(t: T).(subst0 (clen c) u t t2))))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 
+(CTail k u c)) \to (or (pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind 
+Abbr))) (\lambda (t3: T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3 
+t0)))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4) 
+(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t)) 
+(\lambda (t: T).(subst0 (clen c) u t t4))) (pr2_free c t3 t4 H1))))))) 
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda 
+(H1: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 
+t)).(\lambda (H4: (eq C c0 (CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda 
+(c1: C).(getl i c1 (CHead d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let 
+H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind 
+(ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c 
+(CHead e (Bind Abbr) u0)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) 
+(\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) 
+(\lambda (n: nat).(eq C d (CSort n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_: 
+T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: 
+T).(subst0 (clen c) u t0 t)))) (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d 
+(CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) 
+u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: 
+C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) (ex3 T (\lambda (_: 
+T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: 
+T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq C d (CTail k 
+u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(or_introl (pr2 c 
+t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 
+t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (pr2_delta c x u0 i H9 t3 t4 
+H2 t H3))))) H7)) (\lambda (H7: (ex4 nat (\lambda (_: nat).(eq nat i (clen 
+c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) 
+(\lambda (n: nat).(eq C d (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq 
+nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: 
+nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3 
+T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) 
+(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x0: nat).(\lambda (H8: 
+(eq nat i (clen c))).(\lambda (H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T 
+u u0)).(\lambda (_: (eq C d (CSort x0))).(let H12 \def (eq_ind nat i (\lambda 
+(n: nat).(subst0 n u0 t4 t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0 
+(\lambda (t0: T).(subst0 (clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind 
+Abbr) (\lambda (k0: K).(or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind 
+Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 
+t))))) (or_intror (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind 
+Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 
+t))) (ex3_intro T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda 
+(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)) t4 
+(refl_equal K (Bind Abbr)) H2 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1 
+t2 H0))) H)))))).
 
 theorem pr2_gen_cbind:
  \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
@@ -83,100 +82,71 @@ theorem pr2_gen_cbind:
 (THead (Bind b) v t2)))))))
 \def
  \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(let H0 \def (match H 
-in pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: 
-(pr2 c0 t t0)).((eq C c0 (CHead c (Bind b) v)) \to ((eq T t t1) \to ((eq T t0 
-t2) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))))))) with 
-[(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Bind b) 
-v))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead 
-c (Bind b) v) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 
-t3) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)))))) (\lambda (H4: 
-(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to 
-(pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))) (\lambda (H5: (eq T t3 
-t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) v 
-t1) (THead (Bind b) v t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead 
-(Bind b) v t1) (THead (Bind b) v t2) (pr0_comp v v (pr0_refl v) t1 t2 H6 
-(Bind b)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 
-(CHead c (Bind b) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t3 H1 t H2) 
-\Rightarrow (\lambda (H3: (eq C c0 (CHead c (Bind b) v))).(\lambda (H4: (eq T 
-t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) v) (\lambda 
-(c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) 
-u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead (Bind b) v t1) 
-(THead (Bind b) v t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 
-(\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Bind b) v) (CHead d 
-(Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead 
-(Bind b) v t1) (THead (Bind b) v t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind 
-T t2 (\lambda (t4: T).((getl i (CHead c (Bind b) v) (CHead d (Bind Abbr) u)) 
-\to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c (THead (Bind b) v t1) 
-(THead (Bind b) v t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) v) (CHead 
-d (Bind Abbr) u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3 
-t2)).(let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) v i H8) in (let 
-H11 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d (Bind Abbr) u) 
-(CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda 
-(j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead (Bind b) v t1) 
-(THead (Bind b) v t2)) (\lambda (H12: (land (eq nat i O) (eq C (CHead d (Bind 
-Abbr) u) (CHead c (Bind b) v)))).(and_ind (eq nat i O) (eq C (CHead d (Bind 
-Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t1) (THead (Bind b) v 
-t2)) (\lambda (H13: (eq nat i O)).(\lambda (H14: (eq C (CHead d (Bind Abbr) 
-u) (CHead c (Bind b) v))).(let H15 \def (f_equal C C (\lambda (e: C).(match e 
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) 
-\Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H14) in ((let 
-H16 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) 
-with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K 
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H14) in 
-((let H17 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead 
-d (Bind Abbr) u) (CHead c (Bind b) v) H14) in (\lambda (H18: (eq B Abbr 
-b)).(\lambda (_: (eq C d c)).(let H20 \def (eq_ind nat i (\lambda (n: 
-nat).(subst0 n u t3 t2)) H10 O H13) in (let H21 \def (eq_ind T u (\lambda 
-(t4: T).(subst0 O t4 t3 t2)) H20 v H17) in (eq_ind B Abbr (\lambda (b0: 
-B).(pr2 c (THead (Bind b0) v t1) (THead (Bind b0) v t2))) (pr2_free c (THead 
-(Bind Abbr) v t1) (THead (Bind Abbr) v t2) (pr0_delta v v (pr0_refl v) t1 t3 
-H9 t2 H21)) b H18)))))) H16)) H15)))) H12)) (\lambda (H12: (ex2 nat (\lambda 
-(j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) 
-u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: 
-nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b) v t1) (THead 
-(Bind b) v t2)) (\lambda (x: nat).(\lambda (H13: (eq nat i (S x))).(\lambda 
-(H14: (getl x c (CHead d (Bind Abbr) u))).(let H15 \def (f_equal nat nat 
-(\lambda (e: nat).e) i (S x) H13) in (let H16 \def (eq_ind nat i (\lambda (n: 
-nat).(subst0 n u t3 t2)) H10 (S x) H15) in (pr2_head_2 c v t1 t2 (Bind b) 
-(pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead c (Bind 
-b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H14) t1 t3 H9 t2 
-H16))))))) H12)) H11)))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 
-(sym_eq C c0 (CHead c (Bind b) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal 
-C (CHead c (Bind b) v)) (refl_equal T t1) (refl_equal T t2)))))))).
+(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(insert_eq C (CHead c 
+(Bind b) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead 
+(Bind b) v t1) (THead (Bind b) v t2))) (\lambda (y: C).(\lambda (H0: (pr2 y 
+t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 
+(CHead c (Bind b) v)) \to (pr2 c (THead (Bind b) v t) (THead (Bind b) v 
+t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v))).(pr2_free c (THead 
+(Bind b) v t3) (THead (Bind b) v t4) (pr0_comp v v (pr0_refl v) t3 t4 H1 
+(Bind b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: 
+(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) v))).(let H5 \def 
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead 
+c (Bind b) v) H4) in (let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) 
+v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d 
+(Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S 
+j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead 
+(Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C 
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(land_ind (eq nat i O) (eq C 
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3) 
+(THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead 
+d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \def (f_equal C C (\lambda 
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d 
+| (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) 
+v) H9) in ((let H11 \def (f_equal C B (\lambda (e: C).(match e in C return 
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) 
+(CHead c (Bind b) v) H9) in ((let H12 \def (f_equal C T (\lambda (e: 
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) 
+H9) in (\lambda (H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def 
+(eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 O H8) in (let H16 \def 
+(eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H15 v H12) in (eq_ind B Abbr 
+(\lambda (b0: B).(pr2 c (THead (Bind b0) v t3) (THead (Bind b0) v t))) 
+(pr2_free c (THead (Bind Abbr) v t3) (THead (Bind Abbr) v t) (pr0_delta v v 
+(pr0_refl v) t3 t4 H2 t H16)) b H13)))))) H11)) H10)))) H7)) (\lambda (H7: 
+(ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c 
+(CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j))) 
+(\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b) 
+v t3) (THead (Bind b) v t)) (\lambda (x: nat).(\lambda (H8: (eq nat i (S 
+x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let H10 \def (f_equal 
+nat nat (\lambda (e: nat).e) i (S x) H8) in (let H11 \def (eq_ind nat i 
+(\lambda (n: nat).(subst0 n u t4 t)) H3 (S x) H10) in (pr2_head_2 c v t3 t 
+(Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead 
+c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H9) t3 t4 H2 t 
+H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))).
 
 theorem pr2_gen_cflat:
  \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
 (t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2))))))
 \def
  \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(let H0 \def (match H 
-in pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: 
-(pr2 c0 t t0)).((eq C c0 (CHead c (Flat f) v)) \to ((eq T t t1) \to ((eq T t0 
-t2) \to (pr2 c t1 t2)))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow 
-(\lambda (H1: (eq C c0 (CHead c (Flat f) v))).(\lambda (H2: (eq T t0 
-t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Flat f) v) (\lambda (_: 
-C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c t1 t2))))) 
-(\lambda (H4: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to 
-((pr0 t t3) \to (pr2 c t1 t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 
-(\lambda (t: T).((pr0 t1 t) \to (pr2 c t1 t2))) (\lambda (H6: (pr0 t1 
-t2)).(pr2_free c t1 t2 H6)) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) 
-c0 (sym_eq C c0 (CHead c (Flat f) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i 
-H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Flat f) 
-v))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead 
-c (Flat f) v) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 
-(CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c 
-t1 t2))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T 
-t t2) \to ((getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) u)) \to ((pr0 t4 
-t3) \to ((subst0 i u t3 t) \to (pr2 c t1 t2)))))) (\lambda (H7: (eq T t 
-t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Flat f) v) (CHead d 
-(Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c t1 
-t2))))) (\lambda (H8: (getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) 
-u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3 t2)).(let H_y 
-\def (getl_gen_flat f c (CHead d (Bind Abbr) u) v i H8) in (pr2_delta c d u i 
-H_y t1 t3 H9 t2 H10))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 
-(sym_eq C c0 (CHead c (Flat f) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal 
-C (CHead c (Flat f) v)) (refl_equal T t1) (refl_equal T t2)))))))).
+(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(insert_eq C (CHead c 
+(Flat f) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c t1 t2)) 
+(\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to (pr2 
+c t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Flat f) v))).(pr2_free c t3 t4 
+H1)))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: 
+(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) v))).(let H5 \def 
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead 
+c (Flat f) v) H4) in (let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u) 
+v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3)))))))))))))) y t1 t2 H0))) 
+H)))))).