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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.
+Set Implicit Arguments.
+Unset Strict Implicit.
+
+(** * Examples of Decidable Type structures. *)
+
+(** A particular case of [DecidableType] where
+    the equality is the usual one of Coq. *)
+
+Module Type UsualDecidableType.
+ Parameter Inline t : Type.
+ Definition eq := @eq t.
+ Program Instance eq_equiv : Equivalence eq.
+ Parameter eq_dec : forall x y, { eq x y }+{~eq x y }.
+End UsualDecidableType.
+
+(** a [UsualDecidableType] is in particular an [DecidableType]. *)
+
+Module UDT_to_DT (U:UsualDecidableType) <: DecidableType := U.
+
+(** an shortcut for easily building a UsualDecidableType *)
+
+Module Type MiniDecidableType.
+ Parameter Inline t : Type.
+ Parameter eq_dec : forall x y:t, { x=y }+{ x<>y }.
+End MiniDecidableType.
+
+Module Make_UDT (M:MiniDecidableType) <: UsualDecidableType.
+ Definition t:=M.t.
+ Definition eq := @eq t.
+ Program Instance eq_equiv : Equivalence eq.
+ Definition eq_dec := M.eq_dec.
+End Make_UDT.
+
+(** From two decidable types, we can build a new DecidableType
+   over their cartesian product. *)
+
+Module PairDecidableType(D1 D2:DecidableType) <: DecidableType.
+
+ Definition t := prod D1.t D2.t.
+
+ Definition eq x y := D1.eq (fst x) (fst y) /\ D2.eq (snd x) (snd y).
+
+ Instance eq_equiv : Equivalence eq.
+ Proof.
+ constructor.
+ intros (x1,x2); red; simpl; auto.
+ intros (x1,x2) (y1,y2); unfold eq; simpl; intuition.
+ intros (x1,x2) (y1,y2) (z1,z2); unfold eq; simpl; intuition eauto.
+ Qed.
+
+ 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); intuition.
+ Defined.
+
+End PairDecidableType.
+
+(** Similarly for pairs of UsualDecidableType *)
+
+Module PairUsualDecidableType(D1 D2:UsualDecidableType) <: UsualDecidableType.
+ Definition t := prod D1.t D2.t.
+ 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.