Proposition lower_set_is_scott_closed (x : X) : is_scott_closed (λ y, y ⊑ x). Proof. split. - intros y₁ y₂ p q. exact (trans_dcpo q p). - intros D H. use dcpo_lub_is_least. exact H. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.DCPOs.Core.DirectedSets. Require Import UniMath.OrderTheory.DCPOs.Core.Basics. Require Import UniMath.OrderTheory.DCPOs.Core.WayBelow. Require Import UniMath.OrderTheory.DCPOs.Basis.Continuous.