# 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.

# 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.