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[helm.git] / matitaB / matita / contribs / ng_assembly / num / quatern_lemmas.ma

diff --git a/matitaB/matita/contribs/ng_assembly/num/quatern_lemmas.ma b/matitaB/matita/contribs/ng_assembly/num/quatern_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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it              *)
+(*   Sviluppo: 2008-2010                                                  *)
+(*                                                                        *)
+(* ********************************************************************** *)
+
+include "num/quatern.ma".
+include "num/bool_lemmas.ma".
+
+(* ********** *)
+(* QUATERNARI *)
+(* ********** *)
+
+(*
+ndefinition quatern_destruct_aux ≝
+Πn1,n2:quatern.ΠP:Prop.n1 = n2 →
+ match eq_qu n1 n2 with [ true ⇒ P → P | false ⇒ P ].
+
+ndefinition quatern_destruct : quatern_destruct_aux.
+ #n1; #n2; #P; #H;
+ nrewrite < H;
+ nelim n1;
+ nnormalize;
+ napply (λx.x).
+nqed.
+*)
+
+nlemma eq_to_eqqu : ∀n1,n2.n1 = n2 → eq_qu n1 n2 = true.
+ #n1; #n2; #H;
+ nrewrite > H;
+ nelim n2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma neqqu_to_neq : ∀n1,n2.eq_qu n1 n2 = false → n1 ≠ n2.
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_qu n1 n2 = true) …);
+ ##[ ##1: napply (eq_to_eqqu n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue … H)
+ ##]
+nqed.
+
+nlemma eqqu_to_eq : ∀n1,n2.eq_qu n1 n2 = true → n1 = n2.
+ #n1; #n2;
+ ncases n1;
+ ncases n2;
+ nnormalize;
+ ##[ ##1,6,11,16: #H; napply refl_eq
+ ##| ##*: #H; ndestruct (*napply (bool_destruct … H)*)
+ ##]
+nqed.
+
+nlemma neq_to_neqqu : ∀n1,n2.n1 ≠ n2 → eq_qu n1 n2 = false.
+ #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_qu n1 n2));
+ napply (not_to_not (eq_qu n1 n2 = true) (n1 = n2) ? H);
+ napply (eqqu_to_eq n1 n2).
+nqed.
+
+nlemma decidable_qu : ∀x,y:quatern.decidable (x = y).
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_qu x y = true) (eq_qu x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqqu_to_eq … H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqqu_to_neq … H))
+ ##]
+nqed.
+
+nlemma symmetric_eqqu : symmetricT quatern bool eq_qu.
+ #n1; #n2;
+ napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (decidable_qu n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqqu n1 n2 H);
+          napply (symmetric_eq ? (eq_qu n2 n1) false);
+          napply (neq_to_neqqu n2 n1 (symmetric_neq ? n1 n2 H))
+ ##]
+nqed.