Lemma isassoc_Dcuts_max : ∏ x y z : Dcuts, Dcuts_max (Dcuts_max x y) z = Dcuts_max x (Dcuts_max y z). Proof. intros x y z. apply Dcuts_eq_is_eq ; intros r. split. - apply hinhuniv ; apply sumofmaps ; [ | intros Zr]. + apply hinhfun ; apply sumofmaps ; [ intros Xr | intros Yr]. * now left. * right ; apply hinhpr. now left. + apply hinhpr. right ; apply hinhpr. now right. - apply hinhuniv ; apply sumofmaps ; [intros Xr | ]. + apply hinhpr. left ; apply hinhpr. now left. + apply hinhfun ; apply sumofmaps ; [intros Yr | intros Zr]. * left ; apply hinhpr. now right. * now right. 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.