X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fcsubt%2Fdrop.ma;fp=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fcsubt%2Fdrop.ma;h=23e56e0f3f74910dfbdad4c478b0a27c0355a27e;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

diff --git a/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/drop.ma b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubt/drop.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/fwd.ma".
+
+include "LambdaDelta-1/drop/fwd.ma".
+
+theorem csubt_drop_flat:
+ \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall 
+(c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 
+(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop n O c2 (CHead d2 (Flat f) u))))))))))))
+\def
+ \lambda (g: G).(\lambda (f: F).(\lambda (n: nat).(nat_ind (\lambda (n0: 
+nat).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: 
+C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) 
+u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 
+c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 
+(Flat f) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H 
+(CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let 
+H_x \def (csubt_gen_flat g d1 c2 u f H1) in (let H2 \def H_x in (ex2_ind C 
+(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) u))) (\lambda (e2: C).(csubt g 
+d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O 
+c2 (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x 
+(Flat f) u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Flat f) u) 
+(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop O O c (CHead d2 (Flat f) u))))) (ex_intro2 C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Flat f) u) (CHead d2 (Flat f) 
+u))) x H4 (drop_refl (CHead x (Flat f) u))) c2 H3)))) H2)))))))))) (\lambda 
+(n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) 
+\to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) 
+\to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 
+(CHead d2 (Flat f) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda 
+(H0: (csubt g c1 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall 
+(d1: C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Flat f) u)) \to (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead 
+d2 (Flat f) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u: 
+T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Flat f) u))).(and3_ind 
+(eq C (CHead d1 (Flat f) u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) 
+(CHead d2 (Flat f) u)))) (\lambda (_: (eq C (CHead d1 (Flat f) u) (CSort 
+n1))).(\lambda (H3: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 
+\def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee in nat return (\lambda 
+(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) 
+in (False_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop 
+(S n0) O (CSort n1) (CHead d2 (Flat f) u)))) H5))))) (drop_gen_sort n1 (S n0) 
+O (CHead d1 (Flat f) u) H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda 
+(H1: (csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop 
+(S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) 
+u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall 
+(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Flat f) 
+u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S 
+n0) O (CHead c3 k0 u) (CHead d2 (Flat f) u0))))))))) (\lambda (b: B).(\lambda 
+(u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead 
+c0 (Bind b) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g 
+d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g 
+d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead x 
+(Flat f) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0 (CHead d1 
+(Flat f) u0) u n0 H3)))))))) (\lambda (f0: F).(\lambda (u: T).(\lambda (d1: 
+C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f0) u) 
+(CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f0) 
+u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 
+x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Flat f0) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Flat f0) n0 c3 (CHead 
+x (Flat f) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f0) c0 (CHead d1 
+(Flat f) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: 
+T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) 
+u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S 
+n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Flat f) u))).(ex2_ind C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) 
+u))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
+(CHead c3 (Bind b) u2) (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H5: 
+(csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2 
+C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u2) (CHead d2 (Flat f) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x 
+(Flat f) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead 
+d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: 
+T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) 
+u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u 
+t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O (CHead c0 
+(Bind Abst) t) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g 
+d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind Abbr) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H5: (csubt 
+g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind Abbr) u) (CHead d2 (Flat f) u0))) x H5 (drop_drop (Bind Abbr) n0 c3 
+(CHead x (Flat f) u0) H6 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) 
+c0 (CHead d1 (Flat f) u0) t n0 H4))))))))))))) c1 c2 H0)))))) n))).
+
+theorem csubt_drop_abbr:
+ \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g 
+c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind 
+Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop 
+n O c2 (CHead d2 (Bind Abbr) u)))))))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: 
+C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: 
+T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) 
+u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 
+c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 
+(Bind Abbr) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H 
+(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in 
+(let H2 \def (csubt_gen_abbr g d1 c2 u H1) in (ex2_ind C (\lambda (e2: C).(eq 
+C c2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt g d1 e2)) (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 
+(Bind Abbr) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind Abbr) 
+u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abbr) u) 
+(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop O O c (CHead d2 (Bind Abbr) u))))) (ex_intro2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead 
+d2 (Bind Abbr) u))) x H4 (drop_refl (CHead x (Bind Abbr) u))) c2 H3)))) 
+H2))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: 
+C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 
+(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))))))))))).(\lambda 
+(c1: C).(\lambda (c2: C).(\lambda (H0: (csubt g c1 c2)).(csubt_ind g (\lambda 
+(c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c 
+(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u))))))))) (\lambda 
+(n1: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O 
+(CSort n1) (CHead d1 (Bind Abbr) u))).(and3_ind (eq C (CHead d1 (Bind Abbr) 
+u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) 
+u)))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n1))).(\lambda (H3: 
+(eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n0) 
+(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) 
+(CHead d2 (Bind Abbr) u)))) H5))))) (drop_gen_sort n1 (S n0) O (CHead d1 
+(Bind Abbr) u) H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: 
+(csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop (S 
+n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) 
+u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall 
+(d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind 
+Abbr) u0)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0))))))))) (\lambda 
+(b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop 
+(S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 
+(Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda 
+(x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x 
+(Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0))) x H4 
+(drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) (H c0 c3 H1 d1 
+u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abbr) u0) u n0 H3)))))))) 
+(\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda 
+(H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abbr) 
+u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S 
+n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) 
+u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: (drop (S 
+n0) O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
+Abbr) u0))) x H4 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) 
+(H2 d1 u0 (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abbr) u0) u n0 
+H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g 
+c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 
+(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda 
+(b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S n0) O (CHead c0 
+(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(ex2_ind C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u))) (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (x: C).(\lambda (H5: (csubt 
+g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abbr) u))).(ex_intro2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u2) (CHead d2 (Bind Abbr) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead 
+x (Bind Abbr) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 
+(CHead d1 (Bind Abbr) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda 
+(c3: C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: 
+C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead 
+d2 (Bind Abbr) u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g 
+c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O 
+(CHead c0 (Bind Abst) t) (CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) 
+u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) 
+O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: 
+C).(\lambda (H5: (csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind 
+Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0))) x H5 
+(drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abbr) u0) H6 u))))) (H c0 c3 H1 
+d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Bind Abbr) u0) t n0 
+H4))))))))))))) c1 c2 H0)))))) n)).
+
+theorem csubt_drop_abst:
+ \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g 
+c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind 
+Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n 
+O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))))))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: 
+C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (t: 
+T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) 
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
+(d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
+(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) (\lambda (c1: C).(\lambda 
+(c2: C).(\lambda (H: (csubt g c1 c2)).(\lambda (d1: C).(\lambda (t: 
+T).(\lambda (H0: (drop O O c1 (CHead d1 (Bind Abst) t))).(let H1 \def (eq_ind 
+C c1 (\lambda (c: C).(csubt g c c2)) H (CHead d1 (Bind Abst) t) 
+(drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in (let H2 \def 
+(csubt_gen_abst g d1 c2 t H1) in (or_ind (ex2 C (\lambda (e2: C).(eq C c2 
+(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 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 d1 e2))) (\lambda (e2: C).(\lambda 
+(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda 
+(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: 
+T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t))))) (\lambda (H3: (ex2 C (\lambda (e2: C).(eq C c2 (CHead 
+e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)))).(ex2_ind C (\lambda 
+(e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)) 
+(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 
+(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
+(\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) t))).(\lambda 
+(H5: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c: C).(or 
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c (CHead 
+d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c (CHead d2 (Bind Abbr) 
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_introl (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind 
+Abst) t) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x 
+(Bind Abst) t) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop O O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) t))) x H5 
+(drop_refl (CHead x (Bind Abst) t)))) c2 H4)))) H3)) (\lambda (H3: (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 d1 e2))) (\lambda (e2: C).(\lambda 
+(v2: T).(ty3 g e2 v2 t))))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (v2: 
+T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
+T).(csubt g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or 
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead 
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+d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) 
+x1))).(\lambda (H5: (csubt g d1 x0)).(\lambda (H6: (ty3 g x0 x1 t)).(eq_ind_r 
+C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: 
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+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u: T).(drop O O c (CHead d2 (Bind Abbr) u)))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_intror (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x0 (Bind Abbr) x1) 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
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+(Bind Abbr) x1) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: 
+T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt 
+g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x0 (Bind Abbr) 
+x1) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))) x0 x1 H5 (drop_refl (CHead x0 (Bind Abbr) x1)) H6)) c2 H4)))))) H3)) 
+H2))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: 
+C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n0 O c1 
+(CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda 
+(u: T).(ty3 g d2 u t))))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda 
+(H0: (csubt g c1 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall 
+(d1: C).(\forall (t: T).((drop (S n0) O c (CHead d1 (Bind Abst) t)) \to (or 
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c0 
+(CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t)))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (t: T).(\lambda (H1: 
+(drop (S n0) O (CSort n1) (CHead d1 (Bind Abst) t))).(and3_ind (eq C (CHead 
+d1 (Bind Abst) t) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CSort 
+n1) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))) (\lambda (_: (eq C (CHead d1 (Bind Abst) t) (CSort n1))).(\lambda (H3: 
+(eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n0) 
+(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2 
+C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort 
+n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
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+n1) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))) H5))))) (drop_gen_sort n1 (S n0) O (CHead d1 (Bind Abst) t) H1)))))) 
+(\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g c0 c3)).(\lambda 
+(H2: ((\forall (d1: C).(\forall (t: T).((drop (S n0) O c0 (CHead d1 (Bind 
+Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop 
+(S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 
+g d2 u t)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: 
+T).(\forall (d1: C).(\forall (t: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 
+(Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
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+c0 (Bind b) u) (CHead d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) 
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
+(d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda 
+(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (or (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) 
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
+c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
+T).(ty3 g d2 u0 t))))) (\lambda (H4: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 
+(Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C 
+T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind 
+Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda 
+(x: C).(\lambda (H5: (csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x 
+(Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) 
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind 
+Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex_intro2 
+C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
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+x (Bind Abst) t) H6 u)))))) H4)) (\lambda (H4: (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
+n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g 
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+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) 
+u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) (or (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda 
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+(CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda 
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+x1))).(\lambda (H7: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: 
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+T).(ty3 g d2 u0 t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: 
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+c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: 
+T).(ty3 g d2 u0 t))) x0 x1 H5 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) 
+x1) H6 u) H7))))))) H4)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind b) c0 (CHead 
+d1 (Bind Abst) t) u n0 H3)))))))) (\lambda (f: F).(\lambda (u: T).(\lambda 
+(d1: C).(\lambda (t: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f) u) 
+(CHead d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
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+C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (or (ex2 C (\lambda (d2: C).(csubt g 
+d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
+Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) 
+(\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead 
+d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) 
+(\lambda (H4: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop 
+(S n0) O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g 
+d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t))) (or 
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
+(CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
+(S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: 
+C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (x: C).(\lambda (H5: (csubt 
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+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
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+H4)) (\lambda (H4: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
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+C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
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+(drop_drop (Flat f) n0 c3 (CHead x0 (Bind Abbr) x1) H6 u) H7))))))) H4)) (H2 
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+