include "basic_1/tlt/defs.ma".
-theorem wadd_le:
+lemma wadd_le:
\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n:
nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((le v w) \to
(\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
nat).(le (wadd f v n0) (wadd g w n0))) H0 (\lambda (n0: nat).(\lambda (_: (le
(wadd f v n0) (wadd g w n0))).(H n0))) n))))))).
-theorem wadd_lt:
+lemma wadd_lt:
\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n:
nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((lt v w) \to
(\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
(S (S v)) (S w) (le_n_S (S v) w H0)))) (\lambda (n0: nat).(\lambda (_: (le
(wadd f v n0) (wadd g w n0))).(H n0))) n))))))).
-theorem wadd_O:
+lemma wadd_O:
\forall (n: nat).(eq nat (wadd (\lambda (_: nat).O) O n) O)
\def
\lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (wadd (\lambda (_:
nat).O) O n0) O)) (refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat
(wadd (\lambda (_: nat).O) O n0) O)).(refl_equal nat O))) n).
-theorem weight_le:
+lemma weight_le:
\forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t)
(weight_map g t)))))
t1)) (le_plus_plus (weight_map f0 t0) (weight_map g t0) (weight_map f0 t1)
(weight_map g t1) (H f0 g H1) (H0 f0 g H1))))))))))) k)) t).
-theorem weight_eq:
+lemma weight_eq:
\forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
nat))).(((\forall (n: nat).(eq nat (f n) (g n)))) \to (eq nat (weight_map f
t) (weight_map g t)))))
(H n)))) (weight_le t g f (\lambda (n: nat).(eq_ind_r nat (g n) (\lambda (n0:
nat).(le (g n) n0)) (le_n (g n)) (f n) (H n)))))))).
-theorem weight_add_O:
+lemma weight_add_O:
\forall (t: T).(eq nat (weight_map (wadd (\lambda (_: nat).O) O) t)
(weight_map (\lambda (_: nat).O) t))
\def
\lambda (t: T).(weight_eq t (wadd (\lambda (_: nat).O) O) (\lambda (_:
nat).O) (\lambda (n: nat).(wadd_O n))).
-theorem weight_add_S:
+lemma weight_add_S:
\forall (t: T).(\forall (m: nat).(le (weight_map (wadd (\lambda (_: nat).O)
O) t) (weight_map (wadd (\lambda (_: nat).O) (S m)) t)))
\def
(weight v))).(\lambda (H0: (lt (weight v) (weight t))).(lt_trans (weight u)
(weight v) (weight t) H H0))))).
-theorem tlt_head_sx:
+lemma tlt_head_sx:
\forall (k: K).(\forall (u: T).(\forall (t: T).(tlt u (THead k u t))))
\def
\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt
(weight_map (\lambda (_: nat).O) t)) (le_plus_l (weight_map (\lambda (_:
nat).O) u) (weight_map (\lambda (_: nat).O) t)))))) k).
-theorem tlt_head_dx:
+lemma tlt_head_dx:
\forall (k: K).(\forall (u: T).(\forall (t: T).(tlt t (THead k u t))))
\def
\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt