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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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(** * Maximum and Minimum of two rational numbers *)
Local Open Scope Q_scope.
(** [Qmin] and [Qmax] are obtained the usual way from [Qcompare]. *)
Definition Qmax := gmax Qcompare.
Definition Qmin := gmin Qcompare.
Module QHasMinMax <: HasMinMax Q_as_OT.
Module QMM := GenericMinMax Q_as_OT.
Definition max := Qmax.
Definition min := Qmin.
Definition max_l := QMM.max_l.
Definition max_r := QMM.max_r.
Definition min_l := QMM.min_l.
Definition min_r := QMM.min_r.
End QHasMinMax.
Module Q.
(** We obtain hence all the generic properties of max and min. *)
Include MinMaxProperties Q_as_OT QHasMinMax.
(** * Properties specific to the [Q] domain *)
(** Compatibilities (consequences of monotonicity) *)

Lemma plus_max_distr_r : forall n m p, Qmax (n + p) (m + p) == Qmax n m + p.