X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FBase-1%2Fblt%2Fprops.ma;fp=matita%2Fcontribs%2FLAMBDA-TYPES%2FBase-1%2Fblt%2Fprops.ma;h=751dcee9c59db4727038f5315f0b7f4269985b49;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

diff --git a/matita/contribs/LAMBDA-TYPES/Base-1/blt/props.ma b/matita/contribs/LAMBDA-TYPES/Base-1/blt/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 "Base-1/blt/defs.ma".
+
+theorem lt_blt:
+ \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to 
+(eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0 
+\def (match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat 
+n O) \to (eq bool (blt y O) true)))) with [le_n \Rightarrow (\lambda (H0: (eq 
+nat (S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e in 
+nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) 
+\Rightarrow True])) I O H0) in (False_ind (eq bool (blt y O) true) H1))) | 
+(le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind 
+nat (S m) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) 
+with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind 
+((le (S y) m) \to (eq bool (blt y O) true)) H2)) H0))]) in (H0 (refl_equal 
+nat O))))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to 
+(eq bool (blt y n) true))))).(\lambda (y: nat).(nat_ind (\lambda (n0: 
+nat).((lt n0 (S n)) \to (eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S 
+n))).(refl_equal bool true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) 
+\to (eq bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m 
+n)]) true)))).(\lambda (H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) 
+y)))) x).
+
+theorem le_bge:
+ \forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to 
+(eq bool (blt y n) false)))) (\lambda (y: nat).(\lambda (_: (le O 
+y)).(refl_equal bool false))) (\lambda (n: nat).(\lambda (H: ((\forall (y: 
+nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(nat_ind 
+(\lambda (n0: nat).((le (S n) n0) \to (eq bool (blt n0 (S n)) false))) 
+(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 in le return (\lambda (n0: 
+nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to (eq bool (blt O (S n)) 
+false)))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def 
+(eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_: 
+nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in 
+(False_ind (eq bool (blt O (S n)) false) H2))) | (le_S m H1) \Rightarrow 
+(\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: 
+nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False 
+| (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S n) m) \to (eq bool 
+(blt O (S n)) false)) H3)) H1))]) in (H1 (refl_equal nat O)))) (\lambda (n0: 
+nat).(\lambda (_: (((le (S n) n0) \to (eq bool (blt n0 (S n)) 
+false)))).(\lambda (H1: (le (S n) (S n0))).(H n0 (le_S_n n n0 H1))))) y)))) 
+x).
+
+theorem blt_lt:
+ \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt 
+y n) true) \to (lt y n)))) (\lambda (y: nat).(\lambda (H: (eq bool (blt y O) 
+true)).(let H0 \def (match H in eq return (\lambda (b: bool).(\lambda (_: (eq 
+? ? b)).((eq bool b true) \to (lt y O)))) with [refl_equal \Rightarrow 
+(\lambda (H0: (eq bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O) 
+(\lambda (e: bool).(match e in bool return (\lambda (_: bool).Prop) with 
+[true \Rightarrow False | false \Rightarrow True])) I true H0) in (False_ind 
+(lt y O) H1)))]) in (H0 (refl_equal bool true))))) (\lambda (n: nat).(\lambda 
+(H: ((\forall (y: nat).((eq bool (blt y n) true) \to (lt y n))))).(\lambda 
+(y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt n0 (S n)) true) \to (lt 
+n0 (S n)))) (\lambda (_: (eq bool true true)).(le_S_n (S O) (S n) (le_n_S (S 
+O) (S n) (le_n_S O n (le_O_n n))))) (\lambda (n0: nat).(\lambda (_: (((eq 
+bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true) 
+\to (lt n0 (S n))))).(\lambda (H1: (eq bool (blt n0 n) true)).(lt_n_S n0 n (H 
+n0 H1))))) y)))) x).
+
+theorem bge_le:
+ \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt 
+y n) false) \to (le n y)))) (\lambda (y: nat).(\lambda (_: (eq bool (blt y O) 
+false)).(le_O_n y))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq 
+bool (blt y n) false) \to (le n y))))).(\lambda (y: nat).(nat_ind (\lambda 
+(n0: nat).((eq bool (blt n0 (S n)) false) \to (le (S n) n0))) (\lambda (H0: 
+(eq bool (blt O (S n)) false)).(let H1 \def (match H0 in eq return (\lambda 
+(b: bool).(\lambda (_: (eq ? ? b)).((eq bool b false) \to (le (S n) O)))) 
+with [refl_equal \Rightarrow (\lambda (H1: (eq bool (blt O (S n)) 
+false)).(let H2 \def (eq_ind bool (blt O (S n)) (\lambda (e: bool).(match e 
+in bool return (\lambda (_: bool).Prop) with [true \Rightarrow True | false 
+\Rightarrow False])) I false H1) in (False_ind (le (S n) O) H2)))]) in (H1 
+(refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_: (((eq bool (blt n0 
+(S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool (blt (S n0) (S n)) 
+false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0 
+H1))))))) y)))) x).
+