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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.forall n m p : Q, Qmax (p + n) (p + m) == p + Qmax n m
Proof.forall n m p : Q, Qmax (p + n) (p + m) == p + Qmax n m
 intros.n, m, p:Q
Qmax (p + n) (p + m) == p + Qmax n m apply max_monotone.n, m, p:Q
Proper (Qeq ==> Qeq) (Qplus p)
n, m, p:Q
Proper (Qle ==> Qle) (Qplus p)
 intros x x' Hx; rewrite Hx; auto with qarith.n, m, p:Q
Proper (Qle ==> Qle) (Qplus p)
 intros x x' Hx.n, m, p, x, x':Q
Hx:x <= x'
p + x <= p + x' 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.forall n m p : Q, Qmax (n + p) (m + p) == Qmax n m + p
Proof.forall n m p : Q, Qmax (n + p) (m + p) == Qmax n m + p
 intros.n, m, p:Q
Qmax (n + p) (m + p) == Qmax n m + p rewrite (Qplus_comm n p), (Qplus_comm m p), (Qplus_comm _ p).n, m, p:Q
Qmax (p + n) (p + m) == p + Qmax n m
 apply plus_max_distr_l.
Qed.

Lemma plus_min_distr_l : forall n m p, Qmin (p + n) (p + m) == p + Qmin n m.forall n m p : Q, Qmin (p + n) (p + m) == p + Qmin n m
Proof.forall n m p : Q, Qmin (p + n) (p + m) == p + Qmin n m
 intros.n, m, p:Q
Qmin (p + n) (p + m) == p + Qmin n m apply min_monotone.n, m, p:Q
Proper (Qeq ==> Qeq) (Qplus p)
n, m, p:Q
Proper (Qle ==> Qle) (Qplus p)
 intros x x' Hx; rewrite Hx; auto with qarith.n, m, p:Q
Proper (Qle ==> Qle) (Qplus p)
 intros x x' Hx.n, m, p, x, x':Q
Hx:x <= x'
p + x <= p + x' 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.forall n m p : Q, Qmin (n + p) (m + p) == Qmin n m + p
Proof.forall n m p : Q, Qmin (n + p) (m + p) == Qmin n m + p
 intros.n, m, p:Q
Qmin (n + p) (m + p) == Qmin n m + p rewrite (Qplus_comm n p), (Qplus_comm m p), (Qplus_comm _ p).n, m, p:Q
Qmin (p + n) (p + m) == p + Qmin n m
 apply plus_min_distr_l.
Qed.

End Q.