# Library Coq.QArith.Qminmax

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.
Lemma plus_max_distr_r : forall n m p, Qmax (n + p) (m + p) == Qmax n m + p.
Lemma plus_min_distr_l : forall n m p, Qmin (p + n) (p + m) == p + Qmin n m.
Lemma plus_min_distr_r : forall n m p, Qmin (n + p) (m + p) == Qmin n m + p.
End Q.