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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 8933 2006-06-09 14:08:38Z herbelin $ *)
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 t : Set.
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 OrderedType can be seen as a DecidableType *)
Module OT_as_DT (O:OrderedType) <: DecidableType.
Module OF := OrderedTypeFacts O.
Definition t := O.t.
Definition eq := O.eq.
Definition eq_refl := O.eq_refl.
Definition eq_sym := O.eq_sym.
Definition eq_trans := O.eq_trans.
Definition eq_dec := OF.eq_dec.
End OT_as_DT.
(** (Usual) Decidable Type for [nat], [positive], [N], [Z] *)
Module Nat_as_DT <: UsualDecidableType := OT_as_DT (Nat_as_OT).
Module Positive_as_DT <: UsualDecidableType := OT_as_DT (Positive_as_OT).
Module N_as_DT <: UsualDecidableType := OT_as_DT (N_as_OT).
Module Z_as_DT <: UsualDecidableType := OT_as_DT (Z_as_OT).
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