Lemma NonnegativeRationals_to_NonnegativeReals_lt : ∏ x y : NonnegativeRationals, (x < y)%NRat <-> NonnegativeRationals_to_NonnegativeReals x < NonnegativeRationals_to_NonnegativeReals y. Proof. intros x y ; split. - intros Hxy. apply hinhpr. exists x. split ; simpl. + now apply isirrefl_ltNonnegativeRationals. + exact Hxy. - apply hinhuniv ; simpl ; intros q. eapply istrans_le_lt_ltNonnegativeRationals, (pr2 (pr2 q)). apply notlt_geNonnegativeRationals. exact (pr1 (pr2 q)). Qed.
Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.RealNumbers.Sets. Require Import UniMath.OrderTheory.Lattice.Lattice. Require Import UniMath.RealNumbers.Prelim. Require Import UniMath.RealNumbers.NonnegativeRationals.