X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fsubst0%2Ftlt.ma;h=0fc817dcd9443a62c80ba43d06752c07e511a016;hb=89519c7b52e06304a94019dd528925300380cdc0;hp=87c1fec52564055146bb4e09ffc339255912c346;hpb=d0982534aee06a30f91a06d2f3e82834b132a3d3;p=helm.git

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma
index 87c1fec52..0fc817dcd 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "subst0/defs.ma".
+include "LambdaDelta-1/subst0/defs.ma".
 
-include "lift/props.ma".
+include "LambdaDelta-1/lift/props.ma".
 
-include "lift/tlt.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 
@@ -52,7 +52,7 @@ i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t0)) (weight_map 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)) 
-(plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S 
+(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 
@@ -61,20 +61,20 @@ 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)) (plus_le_compat (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)) (plus_le_compat (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 
+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)) (plus_le_compat (weight_map f0 u2) (weight_map g u1) 
+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 
@@ -102,8 +102,8 @@ t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
 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)) (plus_le_compat (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) 
+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 
@@ -118,8 +118,8 @@ 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)) (plus_le_compat (weight_map f 
-u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) 
+(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)) 
@@ -132,41 +132,40 @@ 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)) (plus_le_compat (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)) 
-(plus_le_compat (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 
+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 i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2)) 
+(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) 
@@ -176,9 +175,9 @@ 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)) (plus_le_compat (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))) 
+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 
@@ -191,10 +190,10 @@ 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)) (plus_le_compat (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: 
+(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 
@@ -204,8 +203,8 @@ 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)) (plus_le_compat (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) 
+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)) 
@@ -216,10 +215,9 @@ nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift
 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)) 
-(plus_le_compat (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))))).
+(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 
@@ -251,7 +249,7 @@ i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t0)) (weight_map 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)) 
-(plus_lt_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S 
+(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 
@@ -260,23 +258,23 @@ 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)) (plus_lt_le_compat (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) 
+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)) (plus_lt_le_compat (weight_map 
-f u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) 
+(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)) (plus_lt_le_compat (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 
+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 
@@ -303,8 +301,8 @@ t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
 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)) (plus_le_lt_compat (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) 
+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 
@@ -319,8 +317,8 @@ 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)) (plus_le_lt_compat (weight_map 
-f u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) 
+(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)) 
@@ -333,9 +331,9 @@ 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)) (plus_le_lt_compat (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 
+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: 
@@ -347,7 +345,7 @@ 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)) 
-(plus_le_lt_compat (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2) 
+(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 
@@ -377,10 +375,10 @@ 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)) (plus_lt_le_compat (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 
+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 
@@ -393,8 +391,8 @@ 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)) (plus_lt_compat (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) 
+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) 
@@ -406,13 +404,13 @@ 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)) (plus_lt_compat (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: 
+(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) 
@@ -420,7 +418,7 @@ nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g m))))
 \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)) 
-(plus_lt_compat (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) 
+(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))))).
 
@@ -432,7 +430,7 @@ theorem subst0_tlt_head:
 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)) (plus_le_lt_compat (weight_map 
+(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