fixed some regressions

[helm.git] / helm / software / matita / contribs / LAMBDA-TYPES / LambdaDelta-1 / csubc / fwd.ma

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/fwd.ma
index 18435fe3ec2549024593ac46ac73b5365bfe9d72..2ff7d012e395a68d1e7aca1b54f4397e3b688f2b 100644 (file)
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/fwd.ma
@@ -36,161 +36,307 @@ in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead
 _ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c2 k v) 
 (CHead c1 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: 
 (csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 
-c1)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: (sc3 g (asucc g a) c1 
-v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 w)).(\lambda (H5: (eq C (CHead 
-c1 (Bind Abst) v) (CSort n))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) v) 
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H5) in 
-(False_ind (eq C (CHead c2 (Bind Abbr) w) (CHead c1 (Bind Abst) v)) 
-H6)))))))))))) y x H0))) H)))).
+c1)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CSort 
+n))).(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 _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead 
+c2 (Bind b) u2) (CHead c1 (Bind Void) u1)) H5))))))))))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 
+(CSort n)) \to (eq C c2 c1)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: 
+(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 
+w)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) v) (CSort n))).(let H6 \def 
+(eq_ind C (CHead c1 (Bind Abst) v) (\lambda (ee: C).(match ee in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) 
+\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c2 (Bind Abbr) 
+w) (CHead c1 (Bind Abst) v)) H6)))))))))))) y x H0))) H)))).
 
 theorem csubc_gen_head_l:
  \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (k: 
-K).((csubc g (CHead c1 k v) x) \to (or (ex2 C (\lambda (c2: C).(eq C x (CHead 
-c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: 
+K).((csubc g (CHead c1 k v) x) \to (or3 (ex2 C (\lambda (c2: C).(eq C x 
+(CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: 
 C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2: 
 C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind Abbr) w))))) 
 (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda 
 (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
-(c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w)))))))))))
+(c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T 
+(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead c2 (Bind b) 
+v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 
+c2)))))))))))
 \def
  \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (k: 
 K).(\lambda (H: (csubc g (CHead c1 k v) x)).(insert_eq C (CHead c1 k v) 
-(\lambda (c: C).(csubc g c x)) (\lambda (_: C).(or (ex2 C (\lambda (c2: 
+(\lambda (c: C).(csubc g c x)) (\lambda (_: C).(or3 (ex2 C (\lambda (c2: 
 C).(eq C x (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A 
 (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
 (\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind 
 Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
-v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))))) 
-(\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g (\lambda (c: 
-C).(\lambda (c0: C).((eq C c (CHead c1 k v)) \to (or (ex2 C (\lambda (c2: 
-C).(eq C c0 (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) 
+(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead 
+c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k 
+(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 
+c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g (\lambda 
+(c: C).(\lambda (c0: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda 
+(c2: C).(eq C c0 (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C 
+T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
 (\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C c0 (CHead c2 (Bind 
 Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
-v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))))))) 
-(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c1 k 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 c1 k v) H1) in (False_ind (or (ex2 C (\lambda (c2: C).(eq C 
-(CSort n) (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
-(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C (CSort n) (CHead c2 
-(Bind Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g 
-c1 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) 
-c1 v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w)))))) 
-H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0 
-c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or (ex2 C (\lambda (c3: 
-C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
-(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind 
-Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
-c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
-v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 
-w))))))))).(\lambda (k0: K).(\lambda (v0: T).(\lambda (H3: (eq C (CHead c0 k0 
-v0) (CHead c1 k v))).(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 v0) (CHead c1 k v) 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 v0) 
-(CHead c1 k v) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in 
-C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) 
-\Rightarrow t])) (CHead c0 k0 v0) (CHead c1 k v) H3) in (\lambda (H7: (eq K 
-k0 k)).(\lambda (H8: (eq C c0 c1)).(eq_ind_r T v (\lambda (t: T).(or (ex2 C 
-(\lambda (c3: C).(eq C (CHead c2 k0 t) (CHead c3 k v))) (\lambda (c3: 
+v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) 
+(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C c0 
+(CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
+T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: 
+T).(csubc g c1 c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) 
+(CHead c1 k 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 c1 k v) H1) in (False_ind (or3 (ex2 C 
+(\lambda (c2: C).(eq C (CSort n) (CHead c2 k v))) (\lambda (c2: C).(csubc g 
+c1 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k 
+(Bind Abst))))) (\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C (CSort 
+n) (CHead c2 (Bind Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g c1 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g 
+(asucc g a) c1 v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g 
+a c2 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: 
+T).(eq C (CSort n) (CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: 
+C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c0: C).(\lambda (c2: 
+C).(\lambda (H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) 
+\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: 
 C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
 (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
-A).(eq C (CHead c2 k0 t) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: 
+A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: 
+T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda 
+(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (k0: 
+K).(\lambda (v0: T).(\lambda (H3: (eq C (CHead c0 k0 v0) (CHead c1 k 
+v))).(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 v0) (CHead c1 k v) 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 v0) (CHead c1 k v) H3) in 
+((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: 
+C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead 
+c0 k0 v0) (CHead c1 k v) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq 
+C c0 c1)).(eq_ind_r T v (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C 
+(CHead c2 k0 t) (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C 
+T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k0 t) (CHead 
+c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc 
+g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g 
+a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 
+w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C 
+(CHead c2 k0 t) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c1 c3))))))) (eq_ind_r K k (\lambda (k1: K).(or3 
+(ex2 C (\lambda (c3: C).(eq C (CHead c2 k1 v) (CHead c3 k v))) (\lambda (c3: 
+C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
+A).(eq C (CHead c2 k1 v) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: 
 C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: 
 C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))))) (eq_ind_r K k 
-(\lambda (k1: K).(or (ex2 C (\lambda (c3: C).(eq C (CHead c2 k1 v) (CHead c3 
-k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k1 v) (CHead c3 (Bind 
-Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
-c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
-v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))))) 
-(let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) \to (or 
-(ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g 
-c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k 
+C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda 
+(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k1 v) (CHead c3 
+(Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k 
+(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
+c3))))))) (let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) 
+\to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: 
+C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
+A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: 
+T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda 
+(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H8) in (let 
+H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H8) in 
+(or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) 
+(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: 
+T).(\lambda (_: A).(eq C (CHead c2 k v) (CHead c3 (Bind Abbr) w))))) (\lambda 
+(c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda 
+(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k v) (CHead c3 (Bind 
+b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
+c3))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) 
+(\lambda (c3: C).(csubc g c1 c3)) c2 (refl_equal C (CHead c2 k v)) H10)))) k0 
+H7) v0 H6)))) H5)) H4))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda 
+(H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 
+C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 
+c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k 
 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 
 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
 A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g 
 (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g 
-a c3 w)))))))) H2 c1 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc 
-g c c2)) H1 c1 H8) in (or_introl (ex2 C (\lambda (c3: C).(eq C (CHead c2 k v) 
-(CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: 
+a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: 
+T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (b: B).(\lambda (H3: (not 
+(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead 
+c0 (Bind Void) u1) (CHead c1 k v))).(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 (Bind Void) u1) (CHead c1 k v) H4) 
+in ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda 
+(_: C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) 
+\Rightarrow k0])) (CHead c0 (Bind Void) u1) (CHead c1 k v) 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 c0 
+(Bind Void) u1) (CHead c1 k v) H4) in (\lambda (H8: (eq K (Bind Void) 
+k)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind C c0 (\lambda (c: 
+C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead 
+c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: 
 C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k v) (CHead c3 (Bind Abbr) 
-w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) 
-(\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) 
-(ex_intro2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) (\lambda 
-(c3: C).(csubc g c1 c3)) c2 (refl_equal C (CHead c2 k v)) H10)))) k0 H7) v0 
-H6)))) H5)) H4))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: 
-(csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or (ex2 C 
+C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T 
+(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind 
+b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
+Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
+c3)))))))) H2 c1 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g c 
+c2)) H1 c1 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c1 
+(CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k0 v))) 
+(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: 
+T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda 
+(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0) 
+v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind 
+Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
+c3)))))))) H10 (Bind Void) H8) in (eq_ind K (Bind Void) (\lambda (k0: K).(or3 
+(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) u2) (CHead c3 k0 v))) 
+(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: 
+T).(\lambda (_: A).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T 
+(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) 
+u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K k0 (Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: 
+T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
+T).(csubc g c1 c3))))))) (or3_intro2 (ex2 C (\lambda (c3: C).(eq C (CHead c2 
+(Bind b) u2) (CHead c3 (Bind Void) v))) (\lambda (c3: C).(csubc g c1 c3))) 
+(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind 
+Void) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C 
+(CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda 
+(_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: 
+T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda 
+(c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) 
+v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) 
+(Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B 
+b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
+c3))))) (ex4_3_intro B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: 
+T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) (Bind Void))))) (\lambda 
+(b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))) b c2 u2 (refl_equal C 
+(CHead c2 (Bind b) u2)) (refl_equal K (Bind Void)) H3 H11)) k H8))))))) H6)) 
+H5))))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0 
+c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: 
+C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind 
+Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
+v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) 
+(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 
+(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
+T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
+T).(csubc g c1 c3))))))))).(\lambda (v0: T).(\lambda (a: A).(\lambda (H3: 
+(sc3 g (asucc g a) c0 v0)).(\lambda (w: T).(\lambda (H4: (sc3 g a c2 
+w)).(\lambda (H5: (eq C (CHead c0 (Bind Abst) v0) (CHead c1 k v))).(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 
+(Bind Abst) v0) (CHead c1 k v) H5) in ((let H7 \def (f_equal C K (\lambda (e: 
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind 
+Abst) | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind Abst) v0) (CHead c1 
+k v) H5) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return 
+(\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow 
+t])) (CHead c0 (Bind Abst) v0) (CHead c1 k v) H5) in (\lambda (H9: (eq K 
+(Bind Abst) k)).(\lambda (H10: (eq C c0 c1)).(let H11 \def (eq_ind T v0 
+(\lambda (t: T).(sc3 g (asucc g a) c0 t)) H3 v H8) in (let H12 \def (eq_ind C 
+c0 (\lambda (c: C).(sc3 g (asucc g a) c v)) H11 c1 H10) in (let H13 \def 
+(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C 
 (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) 
 (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind 
-Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 
-(Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g 
-c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) 
-c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 
-w))))))))).(\lambda (v0: T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a) 
-c0 v0)).(\lambda (w: T).(\lambda (H4: (sc3 g a c2 w)).(\lambda (H5: (eq C 
-(CHead c0 (Bind Abst) v0) (CHead c1 k v))).(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 (Bind Abst) v0) (CHead c1 k v) H5) 
-in ((let H7 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda 
-(_: C).K) with [(CSort _) \Rightarrow (Bind Abst) | (CHead _ k0 _) 
-\Rightarrow k0])) (CHead c0 (Bind Abst) v0) (CHead c1 k v) H5) in ((let H8 
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) 
-with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 
-(Bind Abst) v0) (CHead c1 k v) H5) in (\lambda (H9: (eq K (Bind Abst) 
-k)).(\lambda (H10: (eq C c0 c1)).(let H11 \def (eq_ind T v0 (\lambda (t: 
-T).(sc3 g (asucc g a) c0 t)) H3 v H8) in (let H12 \def (eq_ind C c0 (\lambda 
-(c: C).(sc3 g (asucc g a) c v)) H11 c1 H10) in (let H13 \def (eq_ind C c0 
-(\lambda (c: C).((eq C c (CHead c1 k v)) \to (or (ex2 C (\lambda (c3: C).(eq 
-C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+Abst))))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead 
+c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g 
+(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 
+g a0 c3 w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: 
+T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c1 c3)))))))) H2 c1 H10) in (let H14 \def (eq_ind 
+C c0 (\lambda (c: C).(csubc g c c2)) H1 c1 H10) in (let H15 \def (eq_ind_r K 
+k (\lambda (k0: K).((eq C c1 (CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: 
+C).(eq C c2 (CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) 
 (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind 
 Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) 
 c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 
-w0)))))))) H2 c1 H10) in (let H14 \def (eq_ind C c0 (\lambda (c: C).(csubc g 
-c c2)) H1 c1 H10) in (let H15 \def (eq_ind_r K k (\lambda (k0: K).((eq C c1 
-(CHead c1 k0 v)) \to (or (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k0 v))) 
-(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
-T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: 
-T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w0))))) (\lambda (c3: 
-C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) (\lambda 
-(c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0)))))))) H13 (Bind 
-Abst) H9) in (eq_ind K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (c3: 
-C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 k0 v))) (\lambda (c3: C).(csubc g 
-c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K 
-k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C 
-(CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) w0))))) (\lambda (c3: 
-C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) (\lambda 
-(c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))))) (or_intror 
-(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abst) 
-v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda 
-(_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: 
+w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C 
+c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
+T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
+T).(csubc g c1 c3)))))))) H13 (Bind Abst) H9) in (eq_ind K (Bind Abst) 
+(\lambda (k0: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind Abbr) w) 
+(CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda 
+(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: 
 C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 
 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g 
 c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g 
 a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 
-w0))))) (ex5_3_intro C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: 
-A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: 
+w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C 
+(CHead c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda 
+(_: C).(\lambda (_: T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c1 c3))))))) (or3_intro1 (ex2 C (\lambda (c3: 
+C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abst) v))) (\lambda (c3: 
+C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: 
 T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) 
 w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) 
 (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) 
-(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0)))) c2 w a 
-(refl_equal K (Bind Abst)) (refl_equal C (CHead c2 (Bind Abbr) w)) H14 H12 
-H4)) k H9))))))))) H7)) H6)))))))))))) y x H0))) H)))))).
+(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))) 
+(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead 
+c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c1 c3))))) (ex5_3_intro C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda 
+(c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) 
+(CHead c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g 
+(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 
+g a0 c3 w0)))) c2 w a (refl_equal K (Bind Abst)) (refl_equal C (CHead c2 
+(Bind Abbr) w)) H14 H12 H4)) k H9))))))))) H7)) H6)))))))))))) y x H0))) 
+H)))))).
 
 theorem csubc_gen_sort_r:
  \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g x (CSort n)) \to 
@@ -212,109 +358,236 @@ in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead
 _ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c1 k v) 
 (CHead c2 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: 
 (csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C c1 
-c2)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: (sc3 g (asucc g a) c1 
-v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 w)).(\lambda (H5: (eq C (CHead 
-c2 (Bind Abbr) w) (CSort n))).(let H6 \def (eq_ind C (CHead c2 (Bind Abbr) w) 
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) 
-\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H5) in 
-(False_ind (eq C (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr) w)) 
-H6)))))))))))) x y H0))) H)))).
+c2)))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CSort 
+n))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\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 (eq C (CHead c1 
+(Bind Void) u1) (CHead c2 (Bind b) u2)) H5))))))))))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 
+(CSort n)) \to (eq C c1 c2)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: 
+(sc3 g (asucc g a) c1 v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 
+w)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) w) (CSort n))).(let H6 \def 
+(eq_ind C (CHead c2 (Bind Abbr) w) (\lambda (ee: C).(match ee in C return 
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) 
+\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c1 (Bind Abst) 
+v) (CHead c2 (Bind Abbr) w)) H6)))))))))))) x y H0))) H)))).
 
 theorem csubc_gen_head_r:
  \forall (g: G).(\forall (c2: C).(\forall (x: C).(\forall (w: T).(\forall (k: 
-K).((csubc g x (CHead c2 k w)) \to (or (ex2 C (\lambda (c1: C).(eq C x (CHead 
-c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: 
+K).((csubc g x (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c1: C).(eq C x 
+(CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: 
 C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: 
 C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind Abst) v))))) 
 (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda 
 (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w)))))))))))
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T 
+(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind 
+Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
+b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) 
+(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 c2)))))))))))
 \def
  \lambda (g: G).(\lambda (c2: C).(\lambda (x: C).(\lambda (w: T).(\lambda (k: 
 K).(\lambda (H: (csubc g x (CHead c2 k w))).(insert_eq C (CHead c2 k w) 
-(\lambda (c: C).(csubc g x c)) (\lambda (_: C).(or (ex2 C (\lambda (c1: 
+(\lambda (c: C).(csubc g x c)) (\lambda (_: C).(or3 (ex2 C (\lambda (c1: 
 C).(eq C x (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A 
 (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
 (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind 
 Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
-v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))) 
-(\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g (\lambda (c: 
-C).(\lambda (c0: C).((eq C c0 (CHead c2 k w)) \to (or (ex2 C (\lambda (c1: 
-C).(eq C c (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) 
+(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead 
+c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K 
+k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 
+c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g (\lambda 
+(c: C).(\lambda (c0: C).((eq C c0 (CHead c2 k w)) \to (or3 (ex2 C (\lambda 
+(c1: C).(eq C c (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C 
+T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
 (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C c (CHead c1 (Bind 
 Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
-v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))))) 
-(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c2 k w))).(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 c2 k w) H1) in (False_ind (or (ex2 C (\lambda (c1: C).(eq C 
-(CSort n) (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
-(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C (CSort n) (CHead c1 
-(Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g 
-c1 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) 
-c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w)))))) 
-H2)))) (\lambda (c1: C).(\lambda (c0: C).(\lambda (H1: (csubc g c1 
-c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) \to (or (ex2 C (\lambda (c3: 
-C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
-(\lambda (c3: C).(\lambda (v: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind 
-Abst) v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 
-c2)))) (\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 
-v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 
-w))))))))).(\lambda (k0: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c0 k0 
-v) (CHead c2 k w))).(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 v) (CHead c2 k w) 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 v) (CHead c2 k 
-w) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return 
-(\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow 
-t])) (CHead c0 k0 v) (CHead c2 k w) H3) in (\lambda (H7: (eq K k0 
-k)).(\lambda (H8: (eq C c0 c2)).(eq_ind_r T w (\lambda (t: T).(or (ex2 C 
-(\lambda (c3: C).(eq C (CHead c1 k0 t) (CHead c3 k w))) (\lambda (c3: 
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) 
+(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C c (CHead 
+c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K 
+k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 
+c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c2 k 
+w))).(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 c2 k w) H1) in (False_ind (or3 (ex2 C (\lambda 
+(c1: C).(eq C (CSort n) (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) 
+(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind 
+Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C (CSort n) 
+(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g c1 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g 
+(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a 
+c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq 
+C (CSort n) (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: 
+C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) (\lambda (c1: C).(\lambda (c0: 
+C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) 
+\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: 
 C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
-(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: 
-A).(eq C (CHead c1 k0 t) (CHead c3 (Bind Abst) v0))))) (\lambda (c3: 
-C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: 
-C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))) (eq_ind_r K k 
-(\lambda (k1: K).(or (ex2 C (\lambda (c3: C).(eq C (CHead c1 k1 w) (CHead c3 
+(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: 
+A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: 
+T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda 
+(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (k0: 
+K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c0 k0 v) (CHead c2 k w))).(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 
+v) (CHead c2 k w) 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 v) (CHead c2 k w) H3) in ((let H6 \def 
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
+[(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 v) 
+(CHead c2 k w) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 
+c2)).(eq_ind_r T w (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead 
+c1 k0 t) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
+(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k0 t) 
+(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 
+g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 
+g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: 
+T).(eq C (CHead c1 k0 t) (CHead c3 (Bind Void) v1))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))) (eq_ind_r K k 
+(\lambda (k1: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k1 w) (CHead c3 
 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: 
 C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: 
 C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k1 w) (CHead c3 (Bind 
 Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 
 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) 
-c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 
-w))))))) (let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) 
-\to (or (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: 
-C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
-(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: 
-A).(eq C c1 (CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: 
-T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: 
-T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(sc3 g a c2 w)))))))) H2 c2 H8) in (let H10 \def (eq_ind C 
-c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H8) in (or_introl (ex2 C (\lambda 
-(c3: C).(eq C (CHead c1 k w) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 
-c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k 
-(Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C 
-(CHead c1 k w) (CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: 
-T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: 
-T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(sc3 g a c2 w))))) (ex_intro2 C (\lambda (c3: C).(eq C 
-(CHead c1 k w) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2)) c1 
-(refl_equal C (CHead c1 k w)) H10)))) k0 H7) v H6)))) H5)) H4))))))))) 
-(\lambda (c1: C).(\lambda (c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda 
-(H2: (((eq C c0 (CHead c2 k w)) \to (or (ex2 C (\lambda (c3: C).(eq C c1 
+c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) 
+(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead 
+c1 k1 w) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c3 c2))))))) (let H9 \def (eq_ind C c0 (\lambda 
+(c: C).((eq C c (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 
 (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: 
 C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: 
-C).(\lambda (v: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) 
+C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v0))))) 
 (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda 
-(c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))))).(\lambda (v: 
+(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C 
+T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind 
+Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
+b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) 
+(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 
+H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H8) 
+in (or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k w))) 
+(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: 
+T).(\lambda (_: A).(eq C (CHead c1 k w) (CHead c3 (Bind Abst) v0))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda 
+(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C 
+T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 k w) 
+(CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not 
+(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g 
+c3 c2))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k 
+w))) (\lambda (c3: C).(csubc g c3 c2)) c1 (refl_equal C (CHead c1 k w)) 
+H10)))) k0 H7) v H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c0: 
+C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) 
+\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: 
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: 
+A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: 
+T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda 
+(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (b: 
+B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (H4: (eq C (CHead c0 (Bind b) u2) (CHead c2 k w))).(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 (Bind b) 
+u2) (CHead c2 k w) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e 
+in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind b) | (CHead 
+_ k0 _) \Rightarrow k0])) (CHead c0 (Bind b) u2) (CHead c2 k w) 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 c0 
+(Bind b) u2) (CHead c2 k w) H4) in (\lambda (H8: (eq K (Bind b) k)).(\lambda 
+(H9: (eq C c0 c2)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead 
+c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda 
+(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: 
+T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: 
+C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda 
+(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) 
+v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
+b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 
+c2)))))))) H2 c2 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g 
+c1 c)) H1 c2 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c2 
+(CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k0 w))) 
+(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: 
+T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: 
+C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda 
+(_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) 
+v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind 
+b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 
+c2)))))))) H10 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2 
+C (\lambda (c3: C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 k0 w))) (\lambda 
+(c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: 
+T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst) 
+v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) 
+(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C 
+T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind 
+Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k0 (Bind b0))))) (\lambda (b0: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro2 (ex2 C (\lambda (c3: 
+C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind b) w))) (\lambda (c3: 
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K (Bind b) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: 
+T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst) 
+v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) 
+(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C 
+T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind 
+Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K (Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c3 c2))))) (ex4_3_intro B C T (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind Void) u1) (CHead 
+c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K 
+(Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not 
+(eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g 
+c3 c2)))) b c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) (refl_equal K 
+(Bind b)) H3 H11)) k H8))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda 
+(c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k 
+w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: 
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: 
+A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: 
+T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda 
+(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (v: 
 T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a) c1 v)).(\lambda (w0: 
 T).(\lambda (H4: (sc3 g a c0 w0)).(\lambda (H5: (eq C (CHead c0 (Bind Abbr) 
 w0) (CHead c2 k w))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C 
@@ -328,43 +601,61 @@ with [(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k0 _) \Rightarrow k0]))
 (CHead c2 k w) H5) in (\lambda (H9: (eq K (Bind Abbr) k)).(\lambda (H10: (eq 
 C c0 c2)).(let H11 \def (eq_ind T w0 (\lambda (t: T).(sc3 g a c0 t)) H4 w H8) 
 in (let H12 \def (eq_ind C c0 (\lambda (c: C).(sc3 g a c w)) H11 c2 H10) in 
-(let H13 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to (or 
+(let H13 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to (or3 
 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g 
 c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k 
 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 
 (CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
 A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 
 g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: 
-A).(sc3 g a0 c2 w)))))))) H2 c2 H10) in (let H14 \def (eq_ind C c0 (\lambda 
-(c: C).(csubc g c1 c)) H1 c2 H10) in (let H15 \def (eq_ind_r K k (\lambda 
-(k0: K).((eq C c2 (CHead c2 k0 w)) \to (or (ex2 C (\lambda (c3: C).(eq C c1 
-(CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda 
-(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: 
-C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v0))))) 
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda 
-(c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 v0)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w)))))))) H13 
-(Bind Abbr) H9) in (eq_ind K (Bind Abbr) (\lambda (k0: K).(or (ex2 C (\lambda 
-(c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 k0 w))) (\lambda (c3: 
+A).(sc3 g a0 c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda 
+(v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 H10) in (let H14 \def (eq_ind 
+C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H10) in (let H15 \def (eq_ind_r K 
+k (\lambda (k0: K).((eq C c2 (CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: 
+C).(eq C c1 (CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) 
+(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind 
+Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 
+c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) 
+c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 
+w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C 
+c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not 
+(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g 
+c3 c2)))))))) H13 (Bind Abbr) H9) in (eq_ind K (Bind Abbr) (\lambda (k0: 
+K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 k0 
+w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda 
+(_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda 
+(v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) 
+v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) 
+(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 
+v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) 
+(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead 
+c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro1 (ex2 C (\lambda (c3: 
+C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abbr) w))) (\lambda (c3: 
 C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
-(_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda 
-(_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) v0))))) (\lambda 
-(c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: 
-C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 v0)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))))) (or_intror (ex2 
-C (\lambda (c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abbr) w))) 
-(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
-T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: 
-C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 
-(Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g 
-c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g 
-a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 
-w))))) (ex5_3_intro C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq 
-K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: 
-A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) v0))))) (\lambda (c3: 
-C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: 
-C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 v0)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w)))) c1 v a (refl_equal 
-K (Bind Abbr)) (refl_equal C (CHead c1 (Bind Abst) v)) H14 H3 H12)) k 
-H9))))))))) H7)) H6)))))))))))) x y H0))) H)))))).
+(_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: 
+T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) 
+v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) 
+(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 
+v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) 
+(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead 
+c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K (Bind Abbr) (Bind b))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c3 c2))))) (ex5_3_intro C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda 
+(c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) 
+(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 
+g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: 
+A).(sc3 g a0 c2 w)))) c1 v a (refl_equal K (Bind Abbr)) (refl_equal C (CHead 
+c1 (Bind Abst) v)) H14 H3 H12)) k H9))))))))) H7)) H6)))))))))))) x y H0))) 
+H)))))).