Proposition is_scott_continuous_pointwise_lub : is_scott_continuous X Y pointwise_lub. Proof. use make_is_scott_continuous. - exact is_monotone_pointwise_lub. - intros D' ; unfold pointwise_lub. use antisymm_dcpo. + use dcpo_lub_is_least. cbn ; intro i. rewrite (scott_continuous_map_on_lub (D i)). use dcpo_lub_is_least. cbn ; intro j. use less_than_dcpo_lub. * exact j. * cbn. use less_than_dcpo_lub. ** exact i. ** cbn. apply refl_dcpo. + use dcpo_lub_is_least. cbn ; intro i. use dcpo_lub_is_least. cbn ; intro j. use less_than_dcpo_lub. * exact j. * cbn. rewrite (scott_continuous_map_on_lub (D j)). use less_than_dcpo_lub. ** exact i. ** cbn. apply refl_dcpo. 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.