intros x y z; unfold Rmax.
destruct (Rle_dec y z);
destruct (Rle_dec x y);
destruct (Rle_dec x z);
destruct (Rle_dec y z) ; try intuition.
contradict n.
apply Rle_trans with y ; auto.
contradict r.
apply Rlt_not_le, Rlt_trans with y ; apply Rnot_le_lt ; auto.
Qed.
(** Order *)

Lemma Rplus_le_Rmax : forall a b : R, a + b <= 2*Rmax a b.