Proposition lub_fixpoint (D : directed_set X) (HD : ∏ (d : D), f(D d) = D d) : f (⨆ D) = ⨆ D. Proof. rewrite scott_continuous_map_on_lub. use antisymm_dcpo. - use dcpo_lub_is_least. intro i ; cbn in i ; cbn. use less_than_dcpo_lub. + exact i. + exact (eq_to_le_dcpo (HD i)). - use dcpo_lub_is_least. intro i ; cbn in i ; cbn. use less_than_dcpo_lub. + exact i. + exact (eq_to_le_dcpo (!(HD i))). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.Posets.Basics. Require Import UniMath.OrderTheory.Posets.MonotoneFunctions. Require Import UniMath.OrderTheory.Posets.PointedPosets. Require Import UniMath.OrderTheory.DCPOs.Core.DirectedSets. Require Import UniMath.OrderTheory.DCPOs.Core.Basics. Require Import UniMath.OrderTheory.DCPOs.Examples.SubDCPO. Require Import UniMath.OrderTheory.DCPOs.Core.ScottContinuous.