Lemma Dcuts_max_plus : ∏ x y : Dcuts, (0 < x -> y = 0) -> Dcuts_max x y = Dcuts_plus x y. Proof. intros x y H. apply Dcuts_le_ge_eq. - intros r. apply hinhfun. intros H0. left. exact H0. - intros r. apply hinhfun. apply sumofmaps ; [intros H0 | ]. { exact H0. } intros xy ; rewrite (pr1 (pr2 xy)). apply fromempty. refine (Dcuts_zero_empty _ _). rewrite <- H. { apply (pr2 (pr2 (pr2 xy))). } apply hinhpr. exists (pr1 (pr1 xy)). split. + apply Dcuts_zero_empty. + exact (pr1 (pr2 (pr2 xy))). 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.