(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
(** NB: This file is here only for compatibility with earlier version of [FSets] and [FMap]. Please use [Structures/Equalities.v] directly now. *)
(** * Examples of Decidable Type structures. *)
(** A particular case of [DecidableType] where the equality is the usual one of Coq. *)
Module Type UsualDecidableType := Equalities.UsualDecidableTypeOrig.
(** 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 := Equalities.MiniDecidableType.
Module Make_UDT (M:MiniDecidableType) <: UsualDecidableType
:= Equalities.Make_UDT M.
(** 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).

(* Specifications: *)
Lemma eq_refl : forall x : t, eq x x.
Lemma eq_sym : forall x y : t, eq x y -> eq y x.
Lemma eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.
Definition eq_dec : forall x y, { eq x y }+{ ~eq x y }.