# Library Coq.Reals.Rminmax

Require Import Orders Rbase Rbasic_fun ROrderedType GenericMinMax.

# Maximum and Minimum of two real numbers

Local Open Scope R_scope.
The functions Rmax and Rmin implement indeed a maximum and a minimum
Lemma Rmax_l : forall x y, y <= x -> Rmax x y = x.
Lemma Rmax_r : forall x y, x <= y -> Rmax x y = y.
Lemma Rmin_l : forall x y, x <= y -> Rmin x y = x.
Lemma Rmin_r : forall x y, y <= x -> Rmin x y = y.
Module RHasMinMax <: HasMinMax R_as_OT.
Definition max := Rmax.
Definition min := Rmin.
Definition max_l := Rmax_l.
Definition max_r := Rmax_r.
Definition min_l := Rmin_l.
Definition min_r := Rmin_r.
End RHasMinMax.
Module R.
We obtain hence all the generic properties of max and min.
Include UsualMinMaxProperties R_as_OT RHasMinMax.

# Properties specific to the R domain

Compatibilities (consequences of monotonicity)
Lemma plus_max_distr_l : forall n m p, Rmax (p + n) (p + m) = p + Rmax n m.
Lemma plus_max_distr_r : forall n m p, Rmax (n + p) (m + p) = Rmax n m + p.
Lemma plus_min_distr_l : forall n m p, Rmin (p + n) (p + m) = p + Rmin n m.
Lemma plus_min_distr_r : forall n m p, Rmin (n + p) (m + p) = Rmin n m + p.
Anti-monotonicity swaps the role of min and max
Lemma opp_max_distr : forall n m : R, -( Rmax n m ) = Rmin (- n) (- m).
Lemma opp_min_distr : forall n m : R, - ( Rmin n m ) = Rmax (- n) (- m).
Lemma minus_max_distr_l : forall n m p, Rmax (p - n) (p - m) = p - Rmin n m.
Lemma minus_max_distr_r : forall n m p, Rmax (n - p) (m - p) = Rmax n m - p.
Lemma minus_min_distr_l : forall n m p, Rmin (p - n) (p - m) = p - Rmax n m.
Lemma minus_min_distr_r : forall n m p, Rmin (n - p) (m - p) = Rmin n m - p.
End R.