Lemma NonnegativeRationals_to_NonnegativeReals_le : ∏ x y : NonnegativeRationals, (x <= y)%NRat <-> NonnegativeRationals_to_NonnegativeReals x <= NonnegativeRationals_to_NonnegativeReals y. Proof. intros x y ; split. - intros H. apply Dcuts_nlt_ge. intro H0. revert H. apply_pr2 notge_ltNonnegativeRationals. apply_pr2 NonnegativeRationals_to_NonnegativeReals_lt. exact H0. - intros H. apply notlt_geNonnegativeRationals. intros H0. revert H. apply Dcuts_gt_nle. apply NonnegativeRationals_to_NonnegativeReals_lt. exact H0. 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.