Lemma hr_plus_lecompat_l : ∏ x y z : hr_commring, hr_le_rel y z <-> hr_le_rel (y + x)%ring (z + x)%ring. Proof. intros x y z ; split ; intro Hle. - apply hr_notlt_le. apply_pr2_in hr_notlt_le Hle. intro Hlt ; apply Hle. apply_pr2 (hr_plus_ltcompat_l x). exact Hlt. - apply hr_notlt_le. apply_pr2_in hr_notlt_le Hle. intro Hlt ; apply Hle. apply hr_plus_ltcompat_l. exact Hlt. Qed.
Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.OrderTheory.Lattice.Lattice. Require Import UniMath.RealNumbers.Prelim. Require Import UniMath.RealNumbers.Sets. Require Import UniMath.RealNumbers.NonnegativeRationals.