Proposition is_scott_continuous_eval (X Y : dcpo) : is_scott_continuous (X × dcpo_funspace X Y) Y (λ xf, (pr12 xf) (pr1 xf)). Proof. use is_scott_continuous_coordinates. - intros x. use make_is_scott_continuous. + intros f g p ; cbn in *. apply p. + intros D. cbn ; unfold pointwise_lub ; cbn. use dcpo_lub_eq_pointwise. intros f ; cbn. apply idpath. - intros f. use make_is_scott_continuous. + intros x₁ x₂ p ; cbn in *. exact (is_monotone_scott_continuous_map f p). + intros D. exact (scott_continuous_map_on_lub f D). 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.