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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: DecidableTypeEx.v 11699 2008-12-18 11:49:08Z letouzey $ *)
-
-Require Import DecidableType OrderedType OrderedTypeEx.
-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.
- Definition eq_refl := @refl_equal t.
- Definition eq_sym := @sym_eq t.
- Definition eq_trans := @trans_eq t.
- 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.
- Definition eq_refl := @refl_equal t.
- Definition eq_sym := @sym_eq t.
- Definition eq_trans := @trans_eq t.
- Definition eq_dec := M.eq_dec.
-End Make_UDT.
-
-(** An OrderedType can now directly be seen as a DecidableType *)
-
-Module OT_as_DT (O:OrderedType) <: DecidableType := O.
-
-(** (Usual) Decidable Type for [nat], [positive], [N], [Z] *)
-
-Module Nat_as_DT <: UsualDecidableType := Nat_as_OT.
-Module Positive_as_DT <: UsualDecidableType := Positive_as_OT.
-Module N_as_DT <: UsualDecidableType := N_as_OT.
-Module Z_as_DT <: UsualDecidableType := Z_as_OT.
-
-(** 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).
-
- Lemma eq_refl : forall x : t, eq x x.
- Proof. 
- intros (x1,x2); red; simpl; auto.
- Qed.
-
- Lemma eq_sym : forall x y : t, eq x y -> eq y x.
- Proof. 
- intros (x1,x2) (y1,y2); unfold eq; simpl; intuition.
- Qed.
-
- Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.
- Proof. 
- 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.
- Definition eq_refl := @refl_equal t.
- Definition eq_sym := @sym_eq t.
- Definition eq_trans := @trans_eq t.
- 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.