+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 "Basic-1/tlt/defs.ma".
-
-theorem 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))))))))
-\def
- \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: (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:
- \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))))))))
-\def
- \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 *)
-
-theorem 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:
- \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)))))
-\def
- \lambda (t: T).(T_ind (\lambda (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)))))) (\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:
-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
-(n: nat).(le (f n) (g n)))) \to (le (weight_map f (THead k0 t0 t1))
-(weight_map g (THead k0 t0 t1))))))))))) (\lambda (b: B).(B_ind (\lambda (b0:
-B).(\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 (n: nat).(le (f n) (g n))))
-\to (le (match b0 with [Abbr \Rightarrow (S (plus (weight_map f t0)
-(weight_map (wadd f (S (weight_map f t0))) t1))) | Abst \Rightarrow (S (plus
-(weight_map f t0) (weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus
-(weight_map f t0) (weight_map (wadd f O) t1)))]) (match b0 with [Abbr
-\Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g (S (weight_map g
-t0))) t1))) | Abst \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g
-O) t1))) | Void \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g O)
-t1)))])))))))))) (\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 (S (weight_map f t0))) t1)) (plus
-(weight_map g t0) (weight_map (wadd g (S (weight_map g t0))) t1))
-(le_plus_plus (weight_map f t0) (weight_map g t0) (weight_map (wadd f (S
-(weight_map f t0))) t1) (weight_map (wadd g (S (weight_map g t0))) t1) (H f g
-H1) (H0 (wadd f (S (weight_map f t0))) (wadd g (S (weight_map g t0)))
-(\lambda (n: nat).(wadd_le f g H1 (S (weight_map f t0)) (S (weight_map g t0))
-(le_n_S (weight_map f t0) (weight_map g t0) (H f g H1)) 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)))))))))))) (\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))))))))))))
-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
-(H0: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
-(n: nat).(le (f0 n) (g n)))) \to (le (weight_map f0 t1) (weight_map g
-t1))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to
-nat))).(\lambda (H1: ((\forall (n: nat).(le (f0 n) (g n))))).(le_n_S (plus
-(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:
- \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)))))
-\def
- \lambda (t: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
-nat))).(\lambda (H: ((\forall (n: nat).(eq nat (f n) (g n))))).(le_antisym
-(weight_map f t) (weight_map g t) (weight_le t f g (\lambda (n:
-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:
- \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:
- \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 *)
-
-theorem tlt_trans:
- \forall (v: T).(\forall (u: T).(\forall (t: T).((tlt u v) \to ((tlt v t) \to
-(tlt u t)))))
-\def
- \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:
- \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) (weight_map (\lambda (_: nat).O) (THead
-k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall
-(t: T).(lt (weight_map (\lambda (_: nat).O) u) (match b0 with [Abbr
-\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst
-\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
-(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda
-(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda
-(u: T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus
-(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S
-(weight_map (\lambda (_: nat).O) u))) t)) (le_plus_l (weight_map (\lambda (_:
-nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_:
-nat).O) u))) t))))) (\lambda (u: T).(\lambda (t: T).(le_n_S (weight_map
-(\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) (weight_map
-(wadd (\lambda (_: nat).O) O) t)) (le_plus_l (weight_map (\lambda (_: nat).O)
-u) (weight_map (wadd (\lambda (_: nat).O) O) t))))) (\lambda (u: T).(\lambda
-(t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda
-(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)) (le_plus_l
-(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O)
-t))))) b)) (\lambda (_: F).(\lambda (u: T).(\lambda (t: T).(le_n_S
-(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:
- \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) t) (weight_map (\lambda (_: nat).O) (THead
-k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall
-(t: T).(lt (weight_map (\lambda (_: nat).O) t) (match b0 with [Abbr
-\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst
-\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
-(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda
-(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda
-(u: T).(\lambda (t: T).(lt_le_trans (weight_map (\lambda (_: nat).O) t) (S
-(weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda (_:
-nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_:
-nat).O) u))) t))) (lt_n_Sn (weight_map (\lambda (_: nat).O) t)) (le_n_S
-(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u)
-(weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O)
-u))) t)) (le_trans (weight_map (\lambda (_: nat).O) t) (weight_map (wadd
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (plus
-(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S
-(weight_map (\lambda (_: nat).O) u))) t)) (eq_ind nat (weight_map (wadd
-(\lambda (_: nat).O) O) t) (\lambda (n: nat).(le n (weight_map (wadd (\lambda
-(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) (weight_add_S t
-(weight_map (\lambda (_: nat).O) u)) (weight_map (\lambda (_: nat).O) t)
-(weight_add_O t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map
-(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t)))))))
-(\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map (\lambda (_:
-nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: nat).O) t) (S (plus
-(weight_map (\lambda (_: nat).O) u) n)))) (le_n_S (weight_map (\lambda (_:
-nat).O) t) (plus (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))) (weight_map (wadd (\lambda (_: nat).O) O) t)
-(weight_add_O t)))) (\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map
-(\lambda (_: nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_:
-nat).O) t) (S (plus (weight_map (\lambda (_: nat).O) u) n)))) (le_n_S
-(weight_map (\lambda (_: nat).O) t) (plus (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))) (weight_map (wadd (\lambda
-(_: nat).O) O) t) (weight_add_O t)))) b)) (\lambda (_: F).(\lambda (u:
-T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) t) (plus
-(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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