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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.forall x y : R, y <= x -> Rmax x y = x
Proof.forall x y : R, y <= x -> Rmax x y = x
 unfold Rmax.forall x y : R,
y <= x -> (if Rle_dec x y then y else x) = x intros.x, y:R
H:y <= x
(if Rle_dec x y then y else x) = x
 destruct Rle_dec as [H'|H']; [| apply Rnot_le_lt in H' ];
  unfold Rle in *; intuition.
Qed.

Lemma Rmax_r : forall x y, x<=y -> Rmax x y = y.forall x y : R, x <= y -> Rmax x y = y
Proof.forall x y : R, x <= y -> Rmax x y = y
 unfold Rmax.forall x y : R,
x <= y -> (if Rle_dec x y then y else x) = y intros.x, y:R
H:x <= y
(if Rle_dec x y then y else x) = y
 destruct Rle_dec as [H'|H']; [| apply Rnot_le_lt in H' ];
  unfold Rle in *; intuition.
Qed.

Lemma Rmin_l : forall x y, x<=y -> Rmin x y = x.forall x y : R, x <= y -> Rmin x y = x
Proof.forall x y : R, x <= y -> Rmin x y = x
 unfold Rmin.forall x y : R,
x <= y -> (if Rle_dec x y then x else y) = x intros.x, y:R
H:x <= y
(if Rle_dec x y then x else y) = x
 destruct Rle_dec as [H'|H']; [| apply Rnot_le_lt in H' ];
  unfold Rle in *; intuition.
Qed.

Lemma Rmin_r : forall x y, y<=x -> Rmin x y = y.forall x y : R, y <= x -> Rmin x y = y
Proof.forall x y : R, y <= x -> Rmin x y = y
 unfold Rmin.forall x y : R,
y <= x -> (if Rle_dec x y then x else y) = y intros.x, y:R
H:y <= x
(if Rle_dec x y then x else y) = y
 destruct Rle_dec as [H'|H']; [| apply Rnot_le_lt in H' ];
  unfold Rle in *; intuition.
Qed.

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.forall n m p : R, Rmax (p + n) (p + m) = p + Rmax n m
Proof.forall n m p : R, Rmax (p + n) (p + m) = p + Rmax n m
 intros.n, m, p:R
Rmax (p + n) (p + m) = p + Rmax n m apply max_monotone.n, m, p:R
Proper (Rle ==> Rle) (Rplus p)
 intros x y.n, m, p, x, y:R
x <= y -> p + x <= p + y apply Rplus_le_compat_l.
Qed.

Lemma plus_max_distr_r : forall n m p, Rmax (n + p) (m + p) = Rmax n m + p.forall n m p : R, Rmax (n + p) (m + p) = Rmax n m + p
Proof.forall n m p : R, Rmax (n + p) (m + p) = Rmax n m + p
 intros.n, m, p:R
Rmax (n + p) (m + p) = Rmax n m + p rewrite (Rplus_comm n p), (Rplus_comm m p), (Rplus_comm _ p).n, m, p:R
Rmax (p + n) (p + m) = p + Rmax n m
 apply plus_max_distr_l.
Qed.

Lemma plus_min_distr_l : forall n m p, Rmin (p + n) (p + m) = p + Rmin n m.forall n m p : R, Rmin (p + n) (p + m) = p + Rmin n m
Proof.forall n m p : R, Rmin (p + n) (p + m) = p + Rmin n m
 intros.n, m, p:R
Rmin (p + n) (p + m) = p + Rmin n m apply min_monotone.n, m, p:R
Proper (Rle ==> Rle) (Rplus p)
 intros x y.n, m, p, x, y:R
x <= y -> p + x <= p + y apply Rplus_le_compat_l.
Qed.

Lemma plus_min_distr_r : forall n m p, Rmin (n + p) (m + p) = Rmin n m + p.forall n m p : R, Rmin (n + p) (m + p) = Rmin n m + p
Proof.forall n m p : R, Rmin (n + p) (m + p) = Rmin n m + p
 intros.n, m, p:R
Rmin (n + p) (m + p) = Rmin n m + p rewrite (Rplus_comm n p), (Rplus_comm m p), (Rplus_comm _ p).n, m, p:R
Rmin (p + n) (p + m) = p + Rmin n m
 apply plus_min_distr_l.
Qed.

Anti-monotonicity swaps the role of min and max

Lemma opp_max_distr : forall n m : R, -(Rmax n m) = Rmin (- n) (- m).forall n m : R, - Rmax n m = Rmin (- n) (- m)
Proof.forall n m : R, - Rmax n m = Rmin (- n) (- m)
 intros.n, m:R
- Rmax n m = Rmin (- n) (- m) symmetry.n, m:R
Rmin (- n) (- m) = - Rmax n m apply min_max_antimonotone.n, m:R
Proper (Rle ==> Basics.flip Rle) Ropp
 do 3 red.n, m:R
forall x y : R, x <= y -> - y <= - x intros; apply Rge_le.n, m, x, y:R
H:x <= y
- x >= - y apply Ropp_le_ge_contravar; auto.
Qed.

Lemma opp_min_distr : forall n m : R, - (Rmin n m) = Rmax (- n) (- m).forall n m : R, - Rmin n m = Rmax (- n) (- m)
Proof.forall n m : R, - Rmin n m = Rmax (- n) (- m)
 intros.n, m:R
- Rmin n m = Rmax (- n) (- m) symmetry.n, m:R
Rmax (- n) (- m) = - Rmin n m apply max_min_antimonotone.n, m:R
Proper (Rle ==> Basics.flip Rle) Ropp
 do 3 red.n, m:R
forall x y : R, x <= y -> - y <= - x intros; apply Rge_le.n, m, x, y:R
H:x <= y
- x >= - y apply Ropp_le_ge_contravar; auto.
Qed.

Lemma minus_max_distr_l : forall n m p, Rmax (p - n) (p - m) = p - Rmin n m.forall n m p : R, Rmax (p - n) (p - m) = p - Rmin n m
Proof.forall n m p : R, Rmax (p - n) (p - m) = p - Rmin n m
 unfold Rminus.forall n m p : R,
Rmax (p + - n) (p + - m) = p + - Rmin n m intros.n, m, p:R
Rmax (p + - n) (p + - m) = p + - Rmin n m rewrite opp_min_distr.n, m, p:R
Rmax (p + - n) (p + - m) = p + Rmax (- n) (- m) apply plus_max_distr_l.
Qed.

Lemma minus_max_distr_r : forall n m p, Rmax (n - p) (m - p) = Rmax n m - p.forall n m p : R, Rmax (n - p) (m - p) = Rmax n m - p
Proof.forall n m p : R, Rmax (n - p) (m - p) = Rmax n m - p
 unfold Rminus.forall n m p : R,
Rmax (n + - p) (m + - p) = Rmax n m + - p intros.n, m, p:R
Rmax (n + - p) (m + - p) = Rmax n m + - p apply plus_max_distr_r.
Qed.

Lemma minus_min_distr_l : forall n m p, Rmin (p - n) (p - m) = p - Rmax n m.forall n m p : R, Rmin (p - n) (p - m) = p - Rmax n m
Proof.forall n m p : R, Rmin (p - n) (p - m) = p - Rmax n m
 unfold Rminus.forall n m p : R,
Rmin (p + - n) (p + - m) = p + - Rmax n m intros.n, m, p:R
Rmin (p + - n) (p + - m) = p + - Rmax n m rewrite opp_max_distr.n, m, p:R
Rmin (p + - n) (p + - m) = p + Rmin (- n) (- m) apply plus_min_distr_l.
Qed.

Lemma minus_min_distr_r : forall n m p, Rmin (n - p) (m - p) = Rmin n m - p.forall n m p : R, Rmin (n - p) (m - p) = Rmin n m - p
Proof.forall n m p : R, Rmin (n - p) (m - p) = Rmin n m - p
 unfold Rminus.forall n m p : R,
Rmin (n + - p) (m + - p) = Rmin n m + - p intros.n, m, p:R
Rmin (n + - p) (m + - p) = Rmin n m + - p apply plus_min_distr_r.
Qed.

End R.