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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: BoolEq.v,v 1.1.2.1 2004/07/16 19:31:25 herbelin Exp $ i*)
(* Cuihtlauac Alvarado - octobre 2000 *)
(** Properties of a boolean equality *)
Require Export Bool.
Section Bool_eq_dec.
Variable A : Set.
Variable beq : A -> A -> bool.
Variable beq_refl : (x:A)true=(beq x x).
Variable beq_eq : (x,y:A)true=(beq x y)->x=y.
Definition beq_eq_true : (x,y:A)x=y->true=(beq x y).
Proof.
Intros x y H.
Case H.
Apply beq_refl.
Defined.
Definition beq_eq_not_false : (x,y:A)x=y->~false=(beq x y).
Proof.
Intros x y e.
Rewrite <- beq_eq_true; Trivial; Discriminate.
Defined.
Definition beq_false_not_eq : (x,y:A)false=(beq x y)->~x=y.
Proof.
Exact [x,y:A; H:(false=(beq x y)); e:(x=y)](beq_eq_not_false x y e H).
Defined.
Definition exists_beq_eq : (x,y:A){b:bool | b=(beq x y)}.
Proof.
Intros.
Exists (beq x y).
Constructor.
Defined.
Definition not_eq_false_beq : (x,y:A)~x=y->false=(beq x y).
Proof.
Intros x y H.
Symmetry.
Apply not_true_is_false.
Intro.
Apply H.
Apply beq_eq.
Symmetry.
Assumption.
Defined.
Definition eq_dec : (x,y:A){x=y}+{~x=y}.
Proof.
Intros x y; Case (exists_beq_eq x y).
Intros b; Case b; Intro H.
Left; Apply beq_eq; Assumption.
Right; Apply beq_false_not_eq; Assumption.
Defined.
End Bool_eq_dec.
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