Lemma Dcuts_min_gt : ∏ x y z : Dcuts, z < x → z < y → z < (Dcuts_min x y). Proof. intros x y z. apply hinhfun2. intros r q. generalize (isdecrel_ltNonnegativeRationals (pr1 r) (pr1 q)) ; apply sumofmaps ; intros H. - exists (pr1 r). repeat split. + exact (pr1 (pr2 r)). + exact (pr2 (pr2 r)). + apply is_Dcuts_bot with (1 := pr2 (pr2 q)), lt_leNonnegativeRationals, H. - exists (pr1 q). repeat split. + exact (pr1 (pr2 q)). + apply is_Dcuts_bot with (1 := pr2 (pr2 r)), notlt_geNonnegativeRationals, H. + 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.