X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fsubst0%2Ftlt.ma;fp=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fsubst0%2Ftlt.ma;h=0fc817dcd9443a62c80ba43d06752c07e511a016;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

diff --git a/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma
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--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma
@@ -0,0 +1,456 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/subst0/defs.ma".
+
+include "LambdaDelta-1/lift/props.ma".
+
+include "LambdaDelta-1/lift/tlt.ma".
+
+theorem subst0_weight_le:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d 
+u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+d) O u)) (g d)) \to (le (weight_map f z) (weight_map g t))))))))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda 
+(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda 
+(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+n) O t0)) (g n)) \to (le (weight_map f t2) (weight_map g t1)))))))))) 
+(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: 
+((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda 
+(H1: (lt (weight_map f (lift (S i) O v)) (g i))).(le_S_n (weight_map f (lift 
+(S i) O v)) (weight_map g (TLRef i)) (le_S (S (weight_map f (lift (S i) O 
+v))) (weight_map g (TLRef i)) H1)))))))) (\lambda (v: T).(\lambda (u2: 
+T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 
+u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+i) O v)) (g i)) \to (le (weight_map f u2) (weight_map g u1)))))))).(\lambda 
+(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead 
+k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind 
+(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t0)) (weight_map g 
+(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: 
+((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g 
+m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S 
+(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus 
+(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) 
+(le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S 
+(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f 
+g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map 
+g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S 
+(weight_map g u1)) (le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 
+H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt 
+(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) 
+(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) 
+t0)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) 
+t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd 
+g O) (\lambda (n: nat).(wadd_le f g H2 O O (le_n O) n))))))))) (\lambda (f: 
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus 
+(weight_map g u1) (weight_map (wadd g O) t0)) (le_plus_plus (weight_map f u2) 
+(weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) (H1 f 
+g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g 
+H2 O O (le_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 
+m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g 
+i))).(le_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g 
+u1) (weight_map g t0)) (le_plus_plus (weight_map f0 u2) (weight_map g u1) 
+(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g 
+H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: 
+T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 
+t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+(s k0 i)) O v)) (g (s k0 i))) \to (le (weight_map f t2) (weight_map g 
+t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: 
+((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map 
+f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead k0 u0 t2)) 
+(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda 
+(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: 
+nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) 
+\to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: 
+T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to 
+(le (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 
+t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda 
+(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: 
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le 
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: 
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f 
+u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) 
+t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f (S 
+(weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) 
+(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S 
+(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) 
+(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le 
+u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: 
+nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S 
+i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) 
+(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda 
+(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f 
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f 
+m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus 
+(weight_map g u0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u0) 
+(weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) 
+(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f 
+g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda 
+(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) 
+(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: 
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g 
+t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat 
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: 
+(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u0) 
+(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) 
+t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f O) 
+t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd g 
+O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat (weight_map 
+f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f O) 
+(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f)))))))))))))))) b)) 
+(\lambda (_: F).(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda 
+(i: nat).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H1: ((\forall (f0: ((nat 
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g 
+m)))) \to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (le (weight_map f0 
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 
+m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g 
+i))).(le_n_S (plus (weight_map f0 u0) (weight_map f0 t2)) (plus (weight_map g 
+u0) (weight_map g t1)) (le_plus_plus (weight_map f0 u0) (weight_map g u0) 
+(weight_map f0 t2) (weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 
+H3))))))))))))))) k)) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall 
+(f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f 
+m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le 
+(weight_map f u2) (weight_map g u1)))))))).(\lambda (k: K).(K_ind (\lambda 
+(k0: K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to 
+(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s 
+k0 i))) \to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (f: 
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
+(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map 
+f (THead k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: 
+B).(B_ind (\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s 
+(Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat 
+\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f 
+(lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f 
+t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: 
+((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map 
+f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2)) 
+(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: 
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le 
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f 
+m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f 
+u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) 
+t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S 
+(weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f 
+g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1))) 
+(\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2)) (S (weight_map g u1)) 
+(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat 
+(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 
+(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) 
+(lift_weight_add_O (S (weight_map f u2)) v (S i) f))))))))))))) (\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: 
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S 
+i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat 
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le 
+(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus 
+(weight_map g u1) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u2) 
+(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f 
+g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O O 
+(le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: 
+nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) 
+(lift_weight_add_O O v (S i) f))))))))))))) (\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f 
+t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat 
+\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: 
+(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) 
+(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) 
+t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) 
+t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) 
+(\lambda (m: nat).(wadd_le f g H4 O O (le_n O) m)) (eq_ind nat (weight_map f 
+(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) 
+(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) b)) 
+(\lambda (_: F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1 
+t2)).(\lambda (H3: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift 
+(S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g 
+t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H5: 
+(lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f0 u2) 
+(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) (le_plus_plus 
+(weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) (weight_map g t1) (H1 
+f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t z H))))).
+
+theorem subst0_weight_lt:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d 
+u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+d) O u)) (g d)) \to (lt (weight_map f z) (weight_map g t))))))))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda 
+(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda 
+(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+n) O t0)) (g n)) \to (lt (weight_map f t2) (weight_map g t1)))))))))) 
+(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: 
+((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda 
+(H1: (lt (weight_map f (lift (S i) O v)) (g i))).H1)))))) (\lambda (v: 
+T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i 
+v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda 
+(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead 
+k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind 
+(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t0)) (weight_map g 
+(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: 
+((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g 
+m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S 
+(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus 
+(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) 
+(lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S 
+(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f 
+g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map 
+g u1))) (\lambda (n: nat).(wadd_lt f g H2 (S (weight_map f u2)) (S 
+(weight_map g u1)) (lt_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 
+H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt 
+(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) 
+(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) 
+t0)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f 
+O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) 
+(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd 
+f O n) (wadd g O n) (wadd_le f g H2 O O (le_n O) n))))))))))) (\lambda (f: 
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus 
+(weight_map g u1) (weight_map (wadd g O) t0)) (lt_le_plus_plus (weight_map f 
+u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) 
+(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(le_S_n 
+(wadd f O n) (wadd g O n) (le_n_S (wadd f O n) (wadd g O n) (wadd_le f g H2 O 
+O (le_n O) n))))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 
+m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g 
+i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g 
+u1) (weight_map g t0)) (lt_le_plus_plus (weight_map f0 u2) (weight_map g u1) 
+(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g 
+H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: 
+T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 
+t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+(s k0 i)) O v)) (g (s k0 i))) \to (lt (weight_map f t2) (weight_map g 
+t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: 
+((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map 
+f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead k0 u0 t2)) 
+(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda 
+(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: 
+nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) 
+\to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: 
+T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to 
+(lt (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 
+t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda 
+(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: 
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt 
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: 
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f 
+u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) 
+t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f 
+(S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) 
+(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S 
+(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) 
+(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le 
+u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: 
+nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S 
+i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) 
+(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda 
+(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f 
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f 
+m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus 
+(weight_map g u0) (weight_map (wadd g O) t1)) (le_lt_plus_plus (weight_map f 
+u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) 
+(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f 
+g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda 
+(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) 
+(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: 
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g 
+t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat 
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: 
+(lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u0) 
+(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) 
+t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f 
+O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd 
+g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat 
+(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 
+(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i) 
+f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2: 
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1 
+t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift 
+(S i) O v)) (g i)) \to (lt (weight_map f0 t2) (weight_map g 
+t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat 
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda 
+(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map 
+f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1)) 
+(le_lt_plus_plus (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2) 
+(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k)) 
+(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda 
+(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall 
+(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt 
+(weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map 
+g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1: 
+T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (lt 
+(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead 
+k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind 
+(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v 
+t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (lt (weight_map f t2) 
+(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat 
+\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f 
+(lift (S i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t2)) 
+(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (H2: (subst0 (S i) v t1 t2)).(\lambda (_: ((\forall (f: 
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt 
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f 
+m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f 
+u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) 
+t1)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f 
+(S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 
+f g H4 H5) (subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f u2))) 
+(wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_lt f g H4 (S 
+(weight_map f u2)) (S (weight_map g u1)) (lt_n_S (weight_map f u2) 
+(weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat (weight_map f (lift (S i) O 
+v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f 
+u2))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u2)) v (S i) 
+f))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v 
+t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g 
+t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt 
+(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) 
+(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) 
+t1)) (lt_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) 
+t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) 
+(\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m) 
+(wadd g O m) (wadd_le f g H4 O O (le_n O) m)))) (eq_ind nat (weight_map f 
+(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) 
+(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) (\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: 
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S 
+i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat 
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le 
+(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus 
+(weight_map g u1) (weight_map (wadd g O) t1)) (lt_plus_plus (weight_map f u2) 
+(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f 
+g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O m) 
+(wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_n O) 
+m)))) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g 
+i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v 
+(S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall (f0: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g m)))) 
+\to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (lt (weight_map f0 t2) 
+(weight_map g t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat 
+\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda 
+(H5: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map 
+f0 u2) (weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) 
+(lt_plus_plus (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) 
+(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t 
+z H))))).
+
+theorem subst0_tlt_head:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt 
+(THead (Bind Abbr) u z) (THead (Bind Abbr) u t)))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t 
+z)).(lt_n_S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z)) (plus 
+(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S 
+(weight_map (\lambda (_: nat).O) u))) t)) (le_lt_plus_plus (weight_map 
+(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z) (weight_map 
+(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (le_n 
+(weight_map (\lambda (_: nat).O) u)) (subst0_weight_lt u t z O H (wadd 
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (wadd (\lambda 
+(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (\lambda (m: nat).(le_n 
+(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u)) m))) 
+(eq_ind nat (weight_map (\lambda (_: nat).O) (lift O O u)) (\lambda (n: 
+nat).(lt n (S (weight_map (\lambda (_: nat).O) u)))) (eq_ind_r T u (\lambda 
+(t0: T).(lt (weight_map (\lambda (_: nat).O) t0) (S (weight_map (\lambda (_: 
+nat).O) u)))) (le_n (S (weight_map (\lambda (_: nat).O) u))) (lift O O u) 
+(lift_r u O)) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda 
+(_: nat).O) u))) (lift (S O) O u)) (lift_weight_add_O (S (weight_map (\lambda 
+(_: nat).O) u)) u O (\lambda (_: nat).O))))))))).
+
+theorem subst0_tlt:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt z 
+(THead (Bind Abbr) u t)))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t 
+z)).(tlt_trans (THead (Bind Abbr) u z) z (THead (Bind Abbr) u t) (tlt_head_dx 
+(Bind Abbr) u z) (subst0_tlt_head u t z H))))).
+