Lemma Dcuts_locatedness : ∏ X : Dcuts, ∏ p q : NonnegativeRationals, (p < q)%NRat -> p ∈ X ∨ ¬ (q ∈ X). Proof. intros X p q Hlt. apply ispositive_minusNonnegativeRationals in Hlt. generalize (is_Dcuts_corr X _ Hlt). apply_pr2_in ispositive_minusNonnegativeRationals Hlt. apply hinhuniv ; apply sumofmaps ; [ intros Xr | ]. - apply hinhpr ; right. intro H ; apply Xr. apply is_Dcuts_bot with (1 := H). now apply minusNonnegativeRationals_le. - intros r. generalize (isdecrel_leNonnegativeRationals p (pr1 r)) ; apply sumofmaps ; [ intros Hle | intros Hnle]. + apply hinhpr ; left. now apply is_Dcuts_bot with (1 := pr1 (pr2 r)). + apply notge_ltNonnegativeRationals in Hnle. apply hinhpr ; right. intro H ; apply (pr2 (pr2 r)). apply is_Dcuts_bot with (1 := H). apply_pr2 (plusNonnegativeRationals_lecompat_r p). rewrite isassoc_plusNonnegativeRationals, minusNonnegativeRationals_plus_r, iscomm_plusNonnegativeRationals. * apply plusNonnegativeRationals_lecompat_l. now apply lt_leNonnegativeRationals. * now apply lt_leNonnegativeRationals. 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.