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(*         *      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)         *)
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(** * An alternative (but equivalent) presentation for an Ordered Type inferface. *)
(** NB: [comparison], defined in [Datatypes.v] is [Eq|Lt|Gt] whereas [compare], defined in [OrderedType.v] is [EQ _ | LT _ | GT _ ] *)
Module Type OrderedTypeAlt.
Parameter t : Type.
Parameter compare : t -> t -> comparison.
Infix "?=" := compare (at level 70, no associativity).
Parameter compare_sym :
forall x y, (y?=x) = CompOpp (x?=y).
Parameter compare_trans :
forall c x y z, (x?=y) = c -> (y?=z) = c -> (x?=z) = c.
End OrderedTypeAlt.
(** From this new presentation to the original one. *)
Module OrderedType_from_Alt (O:OrderedTypeAlt) <: OrderedType.
Definition t := t.
Definition eq x y := (x?=y) = Eq.
Definition lt x y := (x?=y) = Lt.

Lemma eq_sym : forall x y, eq x y -> eq y x.
Definition eq_trans := (compare_trans Eq).