X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fcsubt%2Ffwd.ma;h=eb571467a6a2b66956f9a2b6121822a50e2087d6;hb=57ae1762497a5f3ea75740e2908e04adb8642cc2;hp=63a3eca4756e65687a279d93038d498f02e22db5;hpb=88a68a9c334646bc17314d5327cd3b790202acd6;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/fwd.ma
index 63a3eca47..eb571467a 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/csubt/fwd.ma
@@ -14,9 +14,25 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/csubt/defs.ma".
+include "basic_1/csubt/defs.ma".
 
-theorem csubt_gen_abbr:
+implied rec lemma csubt_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (n: 
+nat).(P (CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csubt 
+g c1 c2) \to ((P c1 c2) \to (\forall (k: K).(\forall (u: T).(P (CHead c1 k u) 
+(CHead c2 k u))))))))) (f1: (\forall (c1: C).(\forall (c2: C).((csubt g c1 
+c2) \to ((P c1 c2) \to (\forall (b: B).((not (eq B b Void)) \to (\forall (u1: 
+T).(\forall (u2: T).(P (CHead c1 (Bind Void) u1) (CHead c2 (Bind b) 
+u2))))))))))) (f2: (\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to ((P 
+c1 c2) \to (\forall (u: T).(\forall (t: T).((ty3 g c1 u t) \to ((ty3 g c2 u 
+t) \to (P (CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u))))))))))) (c: C) 
+(c0: C) (c1: csubt g c c0) on c1: P c c0 \def match c1 with [(csubt_sort n) 
+\Rightarrow (f n) | (csubt_head c2 c3 c4 k u) \Rightarrow (f0 c2 c3 c4 
+((csubt_ind g P f f0 f1 f2) c2 c3 c4) k u) | (csubt_void c2 c3 c4 b n u1 u2) 
+\Rightarrow (f1 c2 c3 c4 ((csubt_ind g P f f0 f1 f2) c2 c3 c4) b n u1 u2) | 
+(csubt_abst c2 c3 c4 u t t0 t1) \Rightarrow (f2 c2 c3 c4 ((csubt_ind g P f f0 
+f1 f2) c2 c3 c4) u t t0 t1)].
+
+lemma csubt_gen_abbr:
  \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csubt g 
 (CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 
 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))))
@@ -29,64 +45,57 @@ C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))))
 C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda 
 (e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 
 e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind 
-Abbr) v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C 
-return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) 
-\Rightarrow False])) I (CHead e1 (Bind Abbr) v) H1) in (False_ind (ex2 C 
-(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abbr) v))) (\lambda (e2: 
-C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: 
-(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 
-C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: 
-C).(csubt g e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C 
-(CHead c1 k u) (CHead e1 (Bind Abbr) v))).(let H4 \def (f_equal C C (\lambda 
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 
-| (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) 
-in ((let H5 \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 c1 k u) (CHead e1 (Bind Abbr) v) 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 c1 k u) (CHead e1 (Bind 
-Abbr) v) H3) in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 
-e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k 
-t) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K 
-(Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) 
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def 
-(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C 
-(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt 
-g e1 e2))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g 
-c c3)) H1 e1 H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) 
-v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)) c3 
-(refl_equal C (CHead c3 (Bind Abbr) v)) H10))) k H7) u H6)))) H5)) 
-H4))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 
-c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda 
-(e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 
-e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: 
-T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 
-(Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda 
-(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: 
-B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void 
-\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) 
-v) H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) 
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5))))))))))) 
-(\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: 
-(((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u: 
+Abbr) v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 
+(Bind Abbr) v) H1) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CSort n) 
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H2)))) (\lambda 
+(c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C 
+c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
+(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (k: 
+K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind Abbr) 
+v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 
+(Bind Abbr) v) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k 
+u) (CHead e1 (Bind Abbr) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
+(CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) in (\lambda (H7: (eq K k (Bind 
+Abbr))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C 
+(\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 (Bind Abbr) v))) (\lambda 
+(e2: C).(csubt g e1 e2)))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C 
+(\lambda (e2: C).(eq C (CHead c3 k0 v) (CHead e2 (Bind Abbr) v))) (\lambda 
+(e2: C).(csubt g e1 e2)))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c 
+(CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
+(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))))) H2 e1 H8) in (let H10 
+\def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (ex_intro2 C 
+(\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) v) (CHead e2 (Bind Abbr) v))) 
+(\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Bind Abbr) v)) 
+H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3: 
+C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind 
+Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) 
+(\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B 
+b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 
+(Bind Void) u1) (CHead e1 (Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1 
+(Bind Void) u1) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False 
+| (CHead _ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow (match b0 
+with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow 
+True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H4) in 
+(False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) (CHead e2 
+(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5))))))))))) (\lambda 
+(c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C 
+c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
+(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u: 
 T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (_: (ty3 g c3 u 
 t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abbr) 
 v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match 
-ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | 
-(CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
-[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr 
-\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat 
-_) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H5) in (False_ind (ex2 C 
-(\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v))) 
-(\lambda (e2: C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H))))).
-(* COMMENTS
-Initial nodes: 1111
-END *)
+ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k 
+with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow False | Abst 
+\Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) 
+I (CHead e1 (Bind Abbr) v) H5) in (False_ind (ex2 C (\lambda (e2: C).(eq C 
+(CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g 
+e1 e2))) H6))))))))))) y c2 H0))) H))))).
 
-theorem csubt_gen_abst:
+lemma csubt_gen_abst:
  \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g 
 (CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead 
 e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda 
@@ -109,114 +118,107 @@ C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c0
 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: 
 C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (n: nat).(\lambda (H1: 
 (eq C (CSort n) (CHead e1 (Bind Abst) v1))).(let H2 \def (eq_ind C (CSort n) 
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abst) 
-v1) H1) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2 
-(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: 
-C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda 
-(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: 
-T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))) 
-H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 
-c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C 
-(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt 
-g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 
-(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) 
-(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda 
-(v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: 
-(eq C (CHead c1 k u) (CHead e1 (Bind Abst) v1))).(let H4 \def (f_equal C C 
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 
-(Bind Abst) v1) H3) in ((let H5 \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 c1 k u) (CHead e1 (Bind Abst) v1) 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 c1 k u) 
-(CHead e1 (Bind Abst) v1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda 
-(H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(or (ex2 C (\lambda (e2: 
-C).(eq C (CHead c3 k t) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g 
-e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) 
-(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 
-e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: 
-C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (eq_ind_r K (Bind Abst) (\lambda 
-(k0: K).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind 
-Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: 
-C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abbr) v2)))) 
-(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda 
-(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
-v1)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind 
-Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) 
-v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda 
-(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
+(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow True | (CHead _ _ _) 
+\Rightarrow False])) I (CHead e1 (Bind Abst) v1) H1) in (False_ind (or (ex2 C 
+(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abst) v1))) (\lambda (e2: 
+C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
 T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) 
-(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let 
-H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_introl 
-(ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) 
-v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda 
-(v2: T).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda 
-(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: 
-T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) 
-(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind 
-Abst) v1))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 
-(Bind Abst) v1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
-C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 
+(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))) H2)))) (\lambda (c1: 
+C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 
 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: 
 C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
 C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g 
 e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
-v1)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: 
-T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 
-(Bind Abst) v1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda 
-(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: 
-B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void 
-\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst) 
-v1) H4) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) 
-u2) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T 
-(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2 
+v1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) 
+(CHead e1 (Bind Abst) v1))).(let H4 \def (f_equal C C (\lambda (e: C).(match 
+e with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k 
+u) (CHead e1 (Bind Abst) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
+(CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in ((let H6 \def (f_equal C T 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) 
+\Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in (\lambda 
+(H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 
+(\lambda (t: T).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 
+(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: 
+C).(\lambda (v2: T).(eq C (CHead c3 k t) (CHead e2 (Bind Abbr) v2)))) 
+(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda 
+(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
+v1)))))) (eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (e2: 
+C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt 
+g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 
+v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 
+e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: 
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (let H9 \def (eq_ind C c1 (\lambda 
+(c: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: 
+C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) 
+(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) 
+v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: 
+C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 
+g e2 v2 v1))))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: 
+C).(csubt g c c3)) H1 e1 H8) in (or_introl (ex2 C (\lambda (e2: C).(eq C 
+(CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt 
+g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind 
+Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
+T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) 
+(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex_intro2 C (\lambda 
+(e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda 
+(e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Bind Abst) v1)) H10)))) 
+k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda 
+(_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to 
+(or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda 
+(e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C 
+c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 
+e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: 
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (b: B).(\lambda (_: (not 
+(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead 
+c1 (Bind Void) u1) (CHead e1 (Bind Abst) v1))).(let H5 \def (eq_ind C (CHead 
+c1 (Bind Void) u1) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow 
+False | (CHead _ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow (match 
+b0 with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow 
+True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst) v1) H4) in 
+(False_ind (or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) (CHead e2 
+(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: 
+C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Bind Abbr) v2)))) 
+(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda 
+(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
+v1))))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt 
+g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C 
+(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt 
+g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 
 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) 
 (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda 
-(v2: T).(ty3 g e2 v2 v1))))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: 
-C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind 
-Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) 
-v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda 
-(v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
-T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) 
-(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (u: 
-T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u 
-t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) 
-v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) 
-(CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H5) in ((let H7 \def 
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind 
-Abst) t) (CHead e1 (Bind Abst) v1) H5) in (\lambda (H8: (eq C c1 e1)).(let H9 
-\def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H4 v1 H7) in (let H10 \def 
-(eq_ind T t (\lambda (t0: T).(ty3 g c1 u t0)) H3 v1 H7) in (let H11 \def 
-(eq_ind C c1 (\lambda (c: C).(ty3 g c u v1)) H10 e1 H8) in (let H12 \def 
-(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2 
-C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: 
-C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 
+(v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: 
+(ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 
+(Bind Abst) t) (CHead e1 (Bind Abst) v1))).(let H6 \def (f_equal C C (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow 
+c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H5) in ((let H7 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow t | (CHead 
+_ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) 
+H5) in (\lambda (H8: (eq C c1 e1)).(let H9 \def (eq_ind T t (\lambda (t0: 
+T).(ty3 g c3 u t0)) H4 v1 H7) in (let H10 \def (eq_ind T t (\lambda (t0: 
+T).(ty3 g c1 u t0)) H3 v1 H7) in (let H11 \def (eq_ind C c1 (\lambda (c: 
+C).(ty3 g c u v1)) H10 e1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c: 
+C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C 
+c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T 
+(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) 
+(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda 
+(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
+v1))))))) H2 e1 H8) in (let H13 \def (eq_ind C c1 (\lambda (c: C).(csubt g c 
+c3)) H1 e1 H8) in (or_intror (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind 
+Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) 
+(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) 
 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 
 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: 
-C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H8) in (let H13 \def (eq_ind 
-C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (or_intror (ex2 C (\lambda 
-(e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda 
-(e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
-C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g 
-e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) 
-(ex4_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind 
-Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt 
-g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: 
-C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind 
-Abbr) u)) H13 H11 H9))))))))) H6))))))))))) y c2 H0))) H))))).
-(* COMMENTS
-Initial nodes: 2362
-END *)
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex4_2_intro C T (\lambda (e2: 
+C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) 
+v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: 
+C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 
+g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H13 H11 
+H9))))))))) H6))))))))))) y c2 H0))) H))))).
 
-theorem csubt_gen_flat:
+lemma csubt_gen_flat:
  \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).(\forall 
 (f: F).((csubt g (CHead e1 (Flat f) v) c2) \to (ex2 C (\lambda (e2: C).(eq C 
 c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))))))
@@ -229,60 +231,54 @@ e1 e2)))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda
 (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda 
 (e2: C).(eq C c0 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 
 e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Flat f) 
-v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return 
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) 
-\Rightarrow False])) I (CHead e1 (Flat f) v) H1) in (False_ind (ex2 C 
-(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Flat f) v))) (\lambda (e2: 
-C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: 
-(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2 C 
+v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with [(CSort 
+_) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Flat f) 
+v) H1) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2 (Flat 
+f) v))) (\lambda (e2: C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda 
+(c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 
+(Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) 
+(\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (k: K).(\lambda (u: 
+T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Flat f) v))).(let H4 \def 
+(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead 
+c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in ((let H5 
+\def (f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow k | 
+(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in 
+((let H6 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Flat 
+f) v) H3) in (\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 
+e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k 
+t) (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K 
+(Flat f) (\lambda (k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) 
+(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def 
+(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C 
 (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g 
-e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k 
-u) (CHead e1 (Flat f) v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e 
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) 
-\Rightarrow c])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in ((let H5 \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 c1 k u) 
-(CHead e1 (Flat f) v) 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 c1 k u) (CHead e1 (Flat f) v) H3) in 
-(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v 
-(\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 (Flat 
-f) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K (Flat f) (\lambda 
-(k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) (CHead e2 (Flat f) v))) 
-(\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def (eq_ind C c1 (\lambda (c: 
-C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 
-(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))) H2 e1 H8) in 
-(let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in 
-(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Flat f) v) (CHead e2 (Flat f) 
-v))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Flat f) 
-v)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3: 
-C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f) 
-v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda 
-(e2: C).(csubt g e1 e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b 
-Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind 
-Void) u1) (CHead e1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c1 (Bind 
-Void) u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with 
-[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return 
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
-False])])) I (CHead e1 (Flat f) v) H4) in (False_ind (ex2 C (\lambda (e2: 
-C).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Flat f) v))) (\lambda (e2: 
-C).(csubt g e1 e2))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda 
-(_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Flat f) v)) \to (ex2 
-C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g 
-e1 e2)))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u 
-t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) 
-(CHead e1 (Flat f) v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) 
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
-False])])) I (CHead e1 (Flat f) v) H5) in (False_ind (ex2 C (\lambda (e2: 
-C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Flat f) v))) (\lambda (e2: 
-C).(csubt g e1 e2))) H6))))))))))) y c2 H0))) H)))))).
-(* COMMENTS
-Initial nodes: 1103
-END *)
+e1 e2))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c 
+c3)) H1 e1 H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Flat f) v) 
+(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C 
+(CHead c3 (Flat f) v)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 
+(CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat 
+f) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (b: B).(\lambda (_: 
+(not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C 
+(CHead c1 (Bind Void) u1) (CHead e1 (Flat f) v))).(let H5 \def (eq_ind C 
+(CHead c1 (Bind Void) u1) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e1 (Flat f) v) 
+H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) (CHead 
+e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))) H5))))))))))) (\lambda 
+(c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C 
+c1 (CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
+(Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u: T).(\lambda 
+(t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda 
+(H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Flat f) v))).(let H6 \def 
+(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e1 (Flat f) v) 
+H5) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) 
+(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))) H6))))))))))) y c2 
+H0))) H)))))).
 
-theorem csubt_gen_bind:
+lemma csubt_gen_bind:
  \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
 (v1: T).((csubt g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: 
 B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
@@ -299,85 +295,80 @@ b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
 T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
 C).(\lambda (_: T).(csubt g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq 
 C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) 
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) 
-v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda 
-(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda 
-(e2: C).(\lambda (_: T).(csubt g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda 
-(c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 
-(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda 
-(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (k: K).(\lambda (u: 
-T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def 
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) 
-(CHead e1 (Bind b1) v1) H3) in ((let H5 \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 c1 k u) (CHead e1 (Bind b1) v1) 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 c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind 
-b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) 
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
-T).(csubt g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) 
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
-T).(csubt g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c 
-(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
-C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 H8) in (let 
-H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in 
-(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda 
-(e2: C).(\lambda (_: T).(csubt g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3 
-(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
+(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow True | (CHead _ _ _) 
+\Rightarrow False])) I (CHead e1 (Bind b1) v1) H1) in (False_ind (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g 
+e1 e2))))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 
+c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind 
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 
+e2)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) 
+(CHead e1 (Bind b1) v1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k 
+u) (CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
+(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in ((let H6 \def (f_equal C T 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) 
+\Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: 
+(eq K k (Bind b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: 
+T).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c3 k t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csubt g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: 
+K).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c3 k0 v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csubt g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda 
+(c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 
+H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) 
+in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq 
+C (CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))) b1 c3 v1 (refl_equal C 
+(CHead c3 (Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
 C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 
 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
 C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
 B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (b: 
 B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
 T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) 
-v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda 
-(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) 
-(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def 
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with 
-[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return 
-(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
-Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7 
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
-with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 
-(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void 
-b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c: 
-C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 
-H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H9) 
-in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind 
-b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Void) u1) 
+(CHead e1 (Bind b1) v1) H4) in ((let H6 \def (f_equal C B (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Void])])) 
+(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u1 | (CHead 
+_ _ t) \Rightarrow t])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) 
+in (\lambda (H8: (eq B Void b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def 
+(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C 
+T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind 
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 
+e2))))))) H2 e1 H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csubt g c 
+c3)) H1 e1 H9) in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 
+(CHead e1 (Bind b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H10 Void H8) in 
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c3 (Bind b) u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda 
+(e2: C).(\lambda (_: T).(csubt g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 
+(Bind b) u2)) H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: 
+C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind 
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
 T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csubt g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) 
-u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
-T).(csubt g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2)) 
-H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: 
-(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 
-B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g 
-e1 e2)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u 
-t)).(\lambda (H4: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) 
-t) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match 
-e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ 
-_) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in 
-((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: 
-C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k 
-in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) 
-\Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in 
-((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead 
-c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abst 
-b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda (t0: 
-T).(ty3 g c3 u t0)) H4 v1 H8) in (let H12 \def (eq_ind T t (\lambda (t0: 
+C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (u: T).(\lambda (t: 
+T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u t)).(\lambda (H5: 
+(eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(let H6 \def 
+(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead 
+c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) 
+in ((let H7 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 (Bind Abst) t) 
+(CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) 
+(CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B 
+Abst b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda 
+(t0: T).(ty3 g c3 u t0)) H4 v1 H8) in (let H12 \def (eq_ind T t (\lambda (t0: 
 T).(ty3 g c1 u t0)) H3 v1 H8) in (let H13 \def (eq_ind C c1 (\lambda (c: 
 C).(ty3 g c u v1)) H12 e1 H10) in (let H14 \def (eq_ind C c1 (\lambda (c: 
 C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
@@ -392,7 +383,4 @@ B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2
 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g 
 e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H15)))))))))) 
 H7)) H6))))))))))) y c2 H0))) H)))))).
-(* COMMENTS
-Initial nodes: 1899
-END *)