Library Coq.Reals.Rminmax
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.
Proof.
unfold Rmax. intros.
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.
Proof.
unfold Rmax. intros.
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.
Proof.
unfold Rmin. intros.
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.
Proof.
unfold Rmin. intros.
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.
Lemma plus_max_distr_l : forall n m p, Rmax (p + n) (p + m) = p + Rmax n m.
Proof.
intros. apply max_monotone.
intros x 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.
Proof.
intros. rewrite (Rplus_comm n p), (Rplus_comm m p), (Rplus_comm _ p).
apply plus_max_distr_l.
Qed.
Lemma plus_min_distr_l : forall n m p, Rmin (p + n) (p + m) = p + Rmin n m.
Proof.
intros. apply min_monotone.
intros x 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.
Proof.
intros. rewrite (Rplus_comm n p), (Rplus_comm m p), (Rplus_comm _ p).
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).
Proof.
intros. symmetry. apply min_max_antimonotone.
do 3 red. intros; apply Rge_le. apply Ropp_le_ge_contravar; auto.
Qed.
Lemma opp_min_distr : forall n m : R, - (Rmin n m) = Rmax (- n) (- m).
Proof.
intros. symmetry. apply max_min_antimonotone.
do 3 red. intros; apply Rge_le. 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.
Proof.
unfold Rminus. intros. rewrite opp_min_distr. apply plus_max_distr_l.
Qed.
Lemma minus_max_distr_r : forall n m p, Rmax (n - p) (m - p) = Rmax n m - p.
Proof.
unfold Rminus. intros. apply plus_max_distr_r.
Qed.
Lemma minus_min_distr_l : forall n m p, Rmin (p - n) (p - m) = p - Rmax n m.
Proof.
unfold Rminus. intros. rewrite opp_max_distr. apply plus_min_distr_l.
Qed.
Lemma minus_min_distr_r : forall n m p, Rmin (n - p) (m - p) = Rmin n m - p.
Proof.
unfold Rminus. intros. apply plus_min_distr_r.
Qed.
End R.