Proposition scott_continuous_lub_map_is_least_upperbound : is_least_upperbound (scott_continuous_map_PartialOrder X Y) D scott_continuous_lub_map. Proof. split. - intros i x. use less_than_dcpo_lub. + exact i. + apply refl_dcpo. - intros f Hf x ; cbn. unfold pointwise_lub. use dcpo_lub_is_least. intro i ; cbn. exact (Hf i x). 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.Core.ScottContinuous. Require Import UniMath.OrderTheory.DCPOs.Core.FubiniTheorem. Require Import UniMath.OrderTheory.DCPOs.Core.CoordinateContinuity. Require Import UniMath.OrderTheory.DCPOs.Examples.BinaryProducts.