X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Ftlt%2Fprops.ma;fp=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Ftlt%2Fprops.ma;h=72b13d733f8628b6a33c31dac064244daa8f3a60;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

diff --git a/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "LambdaDelta-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))))))).
+
+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))))))).
+
+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).
+
+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).
+
+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)))))))).
+
+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))).
+
+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)))).
+
+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))))).
+
+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).
+
+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).
+
+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)))))).
+
+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)))).
+