X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fcsubt%2Ffwd.ma;fp=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fcsubt%2Ffwd.ma;h=78ab95a2b01e618a9d6ced8807fbb152e28232c1;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

diff --git a/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/fwd.ma b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/fwd.ma
new file mode 100644
index 000000000..78ab95a2b
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/fwd.ma
@@ -0,0 +1,372 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/csubt/defs.ma".
+
+theorem 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)))))))
+\def
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda 
+(H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(insert_eq C (CHead e1 (Bind Abbr) 
+v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C (\lambda (e2: 
+C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g 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 (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: 
+T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u t)).(\lambda (H4: (eq C (CHead c1 
+(Bind Abst) t) (CHead e1 (Bind Abbr) v))).(let H5 \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) H4) 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))) 
+H5)))))))))) y c2 H0))) H))))).
+
+theorem 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))) (ex3_2 C T (\lambda 
+(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
+C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g 
+e2 v2 v1)))))))))
+\def
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda 
+(H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(insert_eq C (CHead e1 (Bind 
+Abst) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(or (ex2 C (\lambda 
+(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 
+e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind 
+Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: 
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (\lambda (y: C).(\lambda (H0: 
+(csubt g y c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead 
+e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c0 (CHead e2 (Bind 
+Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: 
+C).(\lambda (v2: T).(eq C c0 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
+C).(\lambda (_: T).(csubt g e1 e2))) (\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))) (ex3_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 (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))) (ex3_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 (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))) (ex3_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 (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))) (ex3_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 (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))) (ex3_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 (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))) (ex3_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 (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))) (ex3_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 (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))) (ex3_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 (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))) (ex3_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 (e2: C).(\lambda (v2: T).(ty3 g 
+e2 v2 v1)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c3 u 
+t)).(\lambda (H4: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) 
+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 Abst) t) (CHead e1 (Bind Abst) v1) H4) in ((let H6 \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) H4) in (\lambda (H7: (eq C c1 e1)).(let H8 
+\def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H3 v1 H6) in (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))) (ex3_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 (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H7) in 
+(let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H7) 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))) (ex3_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 (e2: 
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex3_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 (e2: 
+C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind 
+Abbr) u)) H10 H8))))))) H5)))))))))) y c2 H0))) H))))).
+
+theorem 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))))))))
+\def
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda 
+(f: F).(\lambda (H: (csubt g (CHead e1 (Flat f) v) c2)).(insert_eq C (CHead 
+e1 (Flat f) v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C 
+(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g 
+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 
+(\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 c3 u 
+t)).(\lambda (H4: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Flat f) v))).(let 
+H5 \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) 
+H4) 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))) H5)))))))))) y c2 
+H0))) H)))))).
+
+theorem 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))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))))))
+\def
+ \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda 
+(v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(insert_eq C 
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: 
+C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csubt g 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 (Bind 
+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: 
+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: 
+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 c3 u 
+t)).(\lambda (H4: (eq C (CHead c1 (Bind Abst) t) (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 Abst) t) (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 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) H4) 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 b1) v1) H4) in (\lambda (H8: (eq B Abst 
+b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind T t (\lambda (t0: 
+T).(ty3 g c3 u t0)) H3 v1 H7) in (let H11 \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 H12 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H9) 
+in (let H13 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) 
+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))))))) H11 Abst H8) in (ex2_3_intro B C T (\lambda (b2: 
+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)) H12)))))))) H6)) 
+H5)))))))))) y c2 H0))) H)))))).
+