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(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
(* $Id$ *)
Require Import DecidableType2 Morphisms RelationPairs.
Set Implicit Arguments.
Unset Strict Implicit.
(** * Examples of Decidable Type structures. *)
(** From two decidable types, we can build a new DecidableType
over their cartesian product. *)
Module PairDecidableType(D1 D2:DecidableType) <: DecidableType.
Definition t := (D1.t * D2.t)%type.
Definition eq := (D1.eq * D2.eq)%signature.
Instance eq_equiv : Equivalence eq.
Definition eq_dec : forall x y, { eq x y }+{ ~eq x y }.
Proof.
intros (x1,x2) (y1,y2); unfold eq; simpl.
destruct (D1.eq_dec x1 y1); destruct (D2.eq_dec x2 y2);
compute; intuition.
Defined.
End PairDecidableType.
(** Similarly for pairs of UsualDecidableType *)
Module PairUsualDecidableType(D1 D2:UsualDecidableType) <: UsualDecidableType.
Definition t := (D1.t * D2.t)%type.
Definition eq := @eq t.
Program Instance eq_equiv : Equivalence eq.
Definition eq_dec : forall x y, { eq x y }+{ ~eq x y }.
Proof.
intros (x1,x2) (y1,y2);
destruct (D1.eq_dec x1 y1); destruct (D2.eq_dec x2 y2);
unfold eq, D1.eq, D2.eq in *; simpl;
(left; f_equal; auto; fail) ||
(right; intro H; injection H; auto).
Defined.
End PairUsualDecidableType.
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