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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \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) *)
(************************************************************************)
Require Import QArith_base Orders QOrderedType GenericMinMax.
(** * 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_l : forall n m p, Qmax (p + n) (p + m) == p + Qmax n m.
Proof.
intros. apply max_monotone.
intros x x' Hx; rewrite Hx; auto with qarith.
intros x x' Hx. apply Qplus_le_compat; q_order.
Qed.
Lemma plus_max_distr_r : forall n m p, Qmax (n + p) (m + p) == Qmax n m + p.
Proof.
intros. rewrite (Qplus_comm n p), (Qplus_comm m p), (Qplus_comm _ p).
apply plus_max_distr_l.
Qed.
Lemma plus_min_distr_l : forall n m p, Qmin (p + n) (p + m) == p + Qmin n m.
Proof.
intros. apply min_monotone.
intros x x' Hx; rewrite Hx; auto with qarith.
intros x x' Hx. apply Qplus_le_compat; q_order.
Qed.
Lemma plus_min_distr_r : forall n m p, Qmin (n + p) (m + p) == Qmin n m + p.
Proof.
intros. rewrite (Qplus_comm n p), (Qplus_comm m p), (Qplus_comm _ p).
apply plus_min_distr_l.
Qed.
End Q.
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