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[helm.git] / helm / software / matita / contribs / ng_assembly / num / oct_lemmas.ma

diff --git a/helm/software/matita/contribs/ng_assembly/num/oct_lemmas.ma b/helm/software/matita/contribs/ng_assembly/num/oct_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/oct.ma".
-include "num/bool_lemmas.ma".
-
-(* ****** *)
-(* OTTALI *)
-(* ****** *)
-
-(*
-ndefinition oct_destruct_aux ≝
-Πn1,n2:oct.ΠP:Prop.n1 = n2 →
- match eq_oct n1 n2 with [ true ⇒ P → P | false ⇒ P ].
-
-ndefinition oct_destruct : oct_destruct_aux.
- #n1; #n2; #P; #H;
- nrewrite < H;
- nelim n1;
- nnormalize;
- napply (λx.x).
-nqed.
-*)
-
-nlemma eq_to_eqoct : ∀n1,n2.n1 = n2 → eq_oct n1 n2 = true.
- #n1; #n2; #H;
- nrewrite > H;
- nelim n2;
- nnormalize;
- napply refl_eq.
-nqed.
-
-nlemma neqoct_to_neq : ∀n1,n2.eq_oct n1 n2 = false → n1 ≠ n2.
- #n1; #n2; #H;
- napply (not_to_not (n1 = n2) (eq_oct n1 n2 = true) …);
- ##[ ##1: napply (eq_to_eqoct n1 n2)
- ##| ##2: napply (eqfalse_to_neqtrue … H)
- ##]
-nqed.
-
-nlemma eqoct_to_eq : ∀n1,n2.eq_oct n1 n2 = true → n1 = n2.
- #n1; #n2;
- ncases n1;
- ncases n2;
- nnormalize;
- ##[ ##1,10,19,28,37,46,55,64: #H; napply refl_eq
- ##| ##*: #H; ndestruct (*napply (bool_destruct … H)*)
- ##]
-nqed.
-
-nlemma neq_to_neqoct : ∀n1,n2.n1 ≠ n2 → eq_oct n1 n2 = false.
- #n1; #n2; #H;
- napply (neqtrue_to_eqfalse (eq_oct n1 n2));
- napply (not_to_not (eq_oct n1 n2 = true) (n1 = n2) ? H);
- napply (eqoct_to_eq n1 n2).
-nqed.
-
-nlemma decidable_oct : ∀x,y:oct.decidable (x = y).
- #x; #y; nnormalize;
- napply (or2_elim (eq_oct x y = true) (eq_oct x y = false) ? (decidable_bexpr ?));
- ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqoct_to_eq … H))
- ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqoct_to_neq … H))
- ##]
-nqed.
-
-nlemma symmetric_eqoct : symmetricT oct bool eq_oct.
- #n1; #n2;
- napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (decidable_oct n1 n2));
- ##[ ##1: #H; nrewrite > H; napply refl_eq
- ##| ##2: #H; nrewrite > (neq_to_neqoct n1 n2 H);
-          napply (symmetric_eq ? (eq_oct n2 n1) false);
-          napply (neq_to_neqoct n2 n1 (symmetric_neq ? n1 n2 H))
- ##]
-nqed.