X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fcsubc%2Ffwd.ma;fp=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fcsubc%2Ffwd.ma;h=18435fe3ec2549024593ac46ac73b5365bfe9d72;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

diff --git a/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/fwd.ma b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/fwd.ma
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+++ b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/fwd.ma
@@ -0,0 +1,370 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/csubc/defs.ma".
+
+theorem csubc_gen_sort_l:
+ \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g (CSort n) x) \to 
+(eq C x (CSort n)))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g 
+(CSort n) x)).(insert_eq C (CSort n) (\lambda (c: C).(csubc g c x)) (\lambda 
+(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g 
+(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) 
+(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def 
+(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with 
+[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0) 
+(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort 
+n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C 
+c2 c1)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c1 k v) 
+(CSort n))).(let H4 \def (eq_ind C (CHead c1 k v) (\lambda (ee: C).(match ee 
+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)))).
+
+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 (_: 
+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)))))))))))
+\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: 
+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))))) 
+(\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: 
+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))))))) (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 
+(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 (_: 
+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 
+(\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))))) 
+(\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: 
+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: 
+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 
+(eq C x (CSort n)))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g x 
+(CSort n))).(insert_eq C (CSort n) (\lambda (c: C).(csubc g x c)) (\lambda 
+(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g 
+(\lambda (c: C).(\lambda (c0: C).((eq C c0 (CSort n)) \to (eq C c c0)))) 
+(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def 
+(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with 
+[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0) 
+(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort 
+n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C 
+c1 c2)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c2 k v) 
+(CSort n))).(let H4 \def (eq_ind C (CHead c2 k v) (\lambda (ee: C).(match ee 
+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)))).
+
+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 (_: 
+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)))))))))))
+\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: 
+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))))) 
+(\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: 
+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 
+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 
+(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 (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 
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
+\Rightarrow c])) (CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H7 
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) 
+with [(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k0 _) \Rightarrow k0])) 
+(CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H8 \def (f_equal C T 
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow w0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abbr) w0) 
+(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 
+(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: 
+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)))))).
+