new ng freescale, no external dependencies

[helm.git] / helm / software / matita / contribs / ng_assembly / freescale / bool_lemmas.ma

diff --git a/helm/software/matita/contribs/ng_assembly/freescale/bool_lemmas.ma b/helm/software/matita/contribs/ng_assembly/freescale/bool_lemmas.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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* ********************************************************************** *)
+(*                           Progetto FreeScale                           *)
+(*                                                                        *)
+(* Sviluppato da:                                                         *)
+(*   Cosimo Oliboni, oliboni@cs.unibo.it                                  *)
+(*                                                                        *)
+(* Questo materiale fa parte della tesi:                                  *)
+(*   "Formalizzazione Interattiva dei Microcontroller a 8bit FreeScale"   *)
+(*                                                                        *)
+(*                    data ultima modifica 15/11/2007                     *)
+(* ********************************************************************** *)
+
+include "freescale/theory.ma".
+include "freescale/bool.ma".
+
+(* ******** *)
+(* BOOLEANI *)
+(* ******** *)
+
+ndefinition boolRelation : Type → Type ≝
+λA:Type.A → A → bool. 
+
+ndefinition boolSymmetric: ∀A:Type.∀R:boolRelation A.Prop ≝
+λA.λR.∀x,y:A.R x y = R y x.
+
+ntheorem bool_destruct_true_false : true ≠ false.
+ nnormalize;
+ #H;
+ nchange with (match true with [ true ⇒ False | false ⇒ True]);
+ nrewrite > H;
+ nnormalize;
+ napply I.
+nqed.
+
+ntheorem bool_destruct_false_true : false ≠ true.
+ nnormalize;
+ #H;
+ nchange with (match true with [ true ⇒ False | false ⇒ True]);
+ nrewrite < H;
+ nnormalize;
+ napply I.
+nqed.
+
+nlemma bsymmetric_eqbool : boolSymmetric bool eq_bool.
+ nnormalize;
+ #b1; #b2;
+ nelim b1;
+ nelim b2;
+ nnormalize;
+ napply (refl_eq ??).
+nqed.
+
+nlemma bsymmetric_andbool : boolSymmetric bool and_bool.
+ nnormalize;
+ #b1; #b2;
+ nelim b1;
+ nelim b2;
+ nnormalize;
+ napply (refl_eq ??).
+nqed.
+
+nlemma bsymmetric_orbool : boolSymmetric bool or_bool.
+ nnormalize;
+ #b1; #b2;
+ nelim b1;
+ nelim b2;
+ nnormalize;
+ napply (refl_eq ??).
+nqed.
+
+nlemma bsymmetric_xorbool : boolSymmetric bool xor_bool.
+ nnormalize;
+ #b1; #b2;
+ nelim b1;
+ nelim b2;
+ nnormalize;
+ napply (refl_eq ??).
+nqed.
+
+nlemma eqbool_to_eq : ∀b1,b2:bool.(eq_bool b1 b2 = true) → (b1 = b2).
+ #b1; #b2;
+ nelim b1;
+ nelim b2;
+ nnormalize;
+ #H;
+ ##[ ##2,3: nelim (bool_destruct_false_true H) ##]
+ napply (refl_eq ??).
+nqed.
+
+nlemma eq_to_eqbool : ∀b1,b2.b1 = b2 → eq_bool b1 b2 = true.
+ #b1; #b2;
+ nelim b1;
+ nelim b2;
+ nnormalize;
+ #H;
+ ##[ ##2: nelim (bool_destruct_true_false H)
+ ##| ##3: nelim (bool_destruct_false_true H) ##]
+ napply (refl_eq ??).
+nqed.
+
+nlemma andb_true_true: ∀b1,b2.(b1 ⊗ b2) = true → b1 = true.
+ #b1; #b2;
+ nelim b1;
+ nelim b2;
+ nnormalize;
+ #H;
+ ##[ ##3,4: nelim (bool_destruct_false_true H) ##]
+ napply (refl_eq ??).
+nqed.
+
+nlemma andb_true_true_r: ∀b1,b2.(b1 ⊗ b2) = true → b2 = true.
+ #b1; #b2;
+ nelim b1;
+ nelim b2;
+ nnormalize;
+ #H;
+ ##[ ##2,3,4: nelim (bool_destruct_false_true H) ##]
+ napply (refl_eq ??).
+nqed.
+
+nlemma orb_false_false : ∀b1,b2:bool.(b1 ⊕ b2) = false → b1 = false.
+ #b1; #b2;
+ nelim b1;
+ nelim b2;
+ nnormalize;
+ #H;
+ ##[ ##1,2: nelim (bool_destruct_true_false H) ##]
+ napply (refl_eq ??).
+nqed.
+
+nlemma orb_false_false_r : ∀b1,b2:bool.(b1 ⊕ b2) = false → b2 = false.
+ #b1; #b2;
+ nelim b1;
+ nelim b2;
+ nnormalize;
+ #H;
+ ##[ ##1,3: nelim (bool_destruct_true_false H) ##]
+ napply (refl_eq ??).
+nqed.