Lemma NR_to_hr_lt : ∏ x y : NonnegativeReals × NonnegativeReals, pr1 x + pr2 y < pr1 y + pr2 x <-> hr_lt_rel (NR_to_hr x) (NR_to_hr y). Proof. intros x y. split. - intros H. apply hinhpr ; exists 0 ; simpl. apply plusNonnegativeReals_ltcompat_l, H. - apply hinhuniv ; intros H. apply_pr2 (plusNonnegativeReals_ltcompat_l (pr1 H)). exact (pr2 H). 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.