(* This file was automatically generated: do not edit *********************)
-include "Basic-1/tlt/defs.ma".
+include "basic_1/T/fwd.ma".
-theorem wadd_le:
+include "basic_1/tlt/defs.ma".
+
+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).(\lambda (H0: (le v w)).(\lambda (n: nat).(nat_ind (\lambda (n0:
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))))))).
-(* COMMENTS
-Initial nodes: 81
-END *)
-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))))))))
\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H:
((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w:
nat).(\lambda (H0: (lt v w)).(\lambda (n: nat).(nat_ind (\lambda (n0:
-nat).(le (wadd f v n0) (wadd g w n0))) (le_S_n v w (le_S (S v) w H0))
-(\lambda (n0: nat).(\lambda (_: (le (wadd f v n0) (wadd g w n0))).(H n0)))
-n))))))).
-(* COMMENTS
-Initial nodes: 95
-END *)
+nat).(le (wadd f v n0) (wadd g w n0))) (le_S_n v w (le_S_n (S v) (S w) (le_S
+(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).
-(* COMMENTS
-Initial nodes: 53
-END *)
-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)))))
nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
\to (le (weight_map f t0) (weight_map g t0)))))) (\lambda (n: nat).(\lambda
(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall
-(n0: nat).(le (f n0) (g n0))))).(le_n (weight_map g (TSort n))))))) (\lambda
-(n: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda
-(H: ((\forall (n0: nat).(le (f n0) (g n0))))).(H n))))) (\lambda (k:
-K).(K_ind (\lambda (k0: K).(\forall (t0: T).(((\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
-\to (le (weight_map f t0) (weight_map g t0)))))) \to (\forall (t1:
+(n0: nat).(le (f n0) (g n0))))).(le_O_n (weight_map g (TSort n)))))))
+(\lambda (n: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H: ((\forall (n0: nat).(le (f n0) (g n0))))).(H n)))))
+(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t0: T).(((\forall (f: ((nat
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g
+n)))) \to (le (weight_map f t0) (weight_map g t0)))))) \to (\forall (t1:
T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g t1))))))
\to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
(plus (weight_map g t0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map
f t0) (weight_map g t0) (weight_map (wadd f O) t1) (weight_map (wadd g O) t1)
(H f g H1) (H0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O
-(le_n O) n)))))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
-\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda
-(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
-(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g
-t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+(le_O_n O) n)))))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g
+n)))) \to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1:
+T).(\lambda (H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t1)
+(weight_map g t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus
(weight_map f t0) (weight_map (wadd f O) t1)) (plus (weight_map g t0)
(weight_map (wadd g O) t1)) (le_plus_plus (weight_map f t0) (weight_map g t0)
(weight_map (wadd f O) t1) (weight_map (wadd g O) t1) (H f g H1) (H0 (wadd f
-O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O (le_n O) n))))))))))))
+O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O (le_O_n O) n))))))))))))
b)) (\lambda (_: F).(\lambda (t0: T).(\lambda (H: ((\forall (f0: ((nat \to
nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f0 n) (g n))))
\to (le (weight_map f0 t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda
(weight_map f0 t0) (weight_map f0 t1)) (plus (weight_map g t0) (weight_map g
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).
-(* COMMENTS
-Initial nodes: 1309
-END *)
-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)))))
nat).(eq_ind_r nat (g n) (\lambda (n0: nat).(le n0 (g n))) (le_n (g n)) (f n)
(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)))))))).
-(* COMMENTS
-Initial nodes: 121
-END *)
-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))).
-(* COMMENTS
-Initial nodes: 23
-END *)
-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
\lambda (t: T).(\lambda (m: nat).(weight_le t (wadd (\lambda (_: nat).O) O)
(wadd (\lambda (_: nat).O) (S m)) (\lambda (n: nat).(wadd_le (\lambda (_:
-nat).O) (\lambda (_: nat).O) (\lambda (_: nat).(le_n O)) O (S m) (le_S O m
-(le_O_n m)) n)))).
-(* COMMENTS
-Initial nodes: 61
-END *)
+nat).O) (\lambda (_: nat).O) (\lambda (_: nat).(le_O_n O)) O (S m) (le_O_n (S
+m)) n)))).
theorem tlt_trans:
\forall (v: T).(\forall (u: T).(\forall (t: T).((tlt u v) \to ((tlt v t) \to
\lambda (v: T).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (lt (weight u)
(weight v))).(\lambda (H0: (lt (weight v) (weight t))).(lt_trans (weight u)
(weight v) (weight t) H H0))))).
-(* COMMENTS
-Initial nodes: 43
-END *)
-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) u) (plus (weight_map (\lambda (_: nat).O) u)
(weight_map (\lambda (_: nat).O) t)) (le_plus_l (weight_map (\lambda (_:
nat).O) u) (weight_map (\lambda (_: nat).O) t)))))) k).
-(* COMMENTS
-Initial nodes: 379
-END *)
-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
(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t))
(le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_:
nat).O) t)))))) k).
-(* COMMENTS
-Initial nodes: 659
-END *)
-
-theorem tlt_wf__q_ind:
- \forall (P: ((T \to Prop))).(((\forall (n: nat).((\lambda (P0: ((T \to
-Prop))).(\lambda (n0: nat).(\forall (t: T).((eq nat (weight t) n0) \to (P0
-t))))) P n))) \to (\forall (t: T).(P t)))
-\def
- let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t:
-T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to
-Prop))).(\lambda (H: ((\forall (n: nat).(\forall (t: T).((eq nat (weight t)
-n) \to (P t)))))).(\lambda (t: T).(H (weight t) t (refl_equal nat (weight
-t)))))).
-(* COMMENTS
-Initial nodes: 61
-END *)
-
-theorem tlt_wf_ind:
- \forall (P: ((T \to Prop))).(((\forall (t: T).(((\forall (v: T).((tlt v t)
-\to (P v)))) \to (P t)))) \to (\forall (t: T).(P t)))
-\def
- let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t:
-T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to
-Prop))).(\lambda (H: ((\forall (t: T).(((\forall (v: T).((lt (weight v)
-(weight t)) \to (P v)))) \to (P t))))).(\lambda (t: T).(tlt_wf__q_ind
-(\lambda (t0: T).(P t0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (t0:
-T).(P t0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
-\to (Q (\lambda (t0: T).(P t0)) m))))).(\lambda (t0: T).(\lambda (H1: (eq nat
-(weight t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall
-(m: nat).((lt m n1) \to (\forall (t1: T).((eq nat (weight t1) m) \to (P
-t1)))))) H0 (weight t0) H1) in (H t0 (\lambda (v: T).(\lambda (H3: (lt
-(weight v) (weight t0))).(H2 (weight v) H3 v (refl_equal nat (weight
-v))))))))))))) t)))).
-(* COMMENTS
-Initial nodes: 179
-END *)
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