Proposition lub_post_fixpoint {X : dcpo} (f : monotone_function X X) (D : directed_set X) (HD : ∏ (d : D), D d ⊑ f (D d)) : ⨆ D ⊑ f (⨆ D). Proof. use dcpo_lub_is_least. intro d. refine (trans_dcpo (HD d) _). apply f. use less_than_dcpo_lub. - exact d. - apply refl_dcpo. 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.