Proposition is_progressive_lub {X : dcpo} (D : directed_set (monotone_map_dcpo X X)) (HD : ∏ (d : D), is_progressive (D d)) : is_progressive (⨆ D). Proof. intros x. assert (H := directed_set_el D). revert H. use factor_through_squash. { apply propproperty. } intro d. use less_than_dcpo_lub. - exact d. - exact (HD d x). Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.Posets.Basics. Require Import UniMath.OrderTheory.Posets.MonotoneFunctions. Require Import UniMath.OrderTheory.DCPOs.Core.DirectedSets. Require Import UniMath.OrderTheory.DCPOs.Core.Basics. Require Import UniMath.OrderTheory.DCPOs.Examples.Products. Require Import UniMath.OrderTheory.DCPOs.Examples.SubDCPO.