(* 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
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
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
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
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))
\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)
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
\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
\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))
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
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
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
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)))
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projects / helm.git / blobdiff