Lemma hr_notlt_le : ∏ X Y, ¬ hr_lt_rel X Y <-> hr_le_rel Y X. Proof. intros x y. rewrite <- (hr_to_NR_bij x), <- (hr_to_NR_bij y). split ; intro H. - apply NR_to_hr_le. apply notlt_leNonnegativeReals. intro H0 ; apply H. apply NR_to_hr_lt. exact H0. - intro H0. refine (pr2 (notlt_leNonnegativeReals _ _) _ _). + refine (pr2 (NR_to_hr_le _ _) _). apply H. + apply_pr2 NR_to_hr_lt. exact H0. 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.