Lemma Dcuts_min_open : Dcuts_def_open Dcuts_min_val. Proof. intros r Hr. generalize (X_open _ (pr1 Hr)) (Y_open _ (pr2 Hr)). apply hinhfun2 ; intros q q'. generalize (isdecrel_ltNonnegativeRationals (pr1 q) (pr1 q')) ; apply sumofmaps ; intros H. - exists (pr1 q) ; repeat split. + exact (pr1 (pr2 q)). + apply Y_bot with (1 := pr1 (pr2 q')), lt_leNonnegativeRationals, H. + exact (pr2 (pr2 q)). - exists (pr1 q') ; repeat split. + apply X_bot with (1 := pr1 (pr2 q)), notlt_geNonnegativeRationals, H. + exact (pr1 (pr2 q')). + exact (pr2 (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.