Definition is_algebraic_dcpo_directed_set {X : dcpo} (CX : is_algebraic_dcpo X) (x : X) : directed_set X. Proof. use make_directed_set. - exact (∑ (b : X), is_compact_el b ∧ b ≪ x). - exact (λ z, pr1 z). - exact (is_algebraic_is_directed CX x). Defined.
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.