Proposition is_scott_continuous_depfunction_pr (i : I) : is_scott_continuous depfunction_dcpo (D i) (λ f, f i). Proof. use make_is_scott_continuous. - exact (is_monotone_depfunction_poset_pr (λ i, D i) (λ i, D i) i). - intros E. cbn ; unfold type_prod_pointwise_lub. use antisymm_dcpo. + use dcpo_lub_is_least ; cbn. intro e. use less_than_dcpo_lub ; [ exact e | ] ; cbn. apply refl_dcpo. + use dcpo_lub_is_least ; cbn. intro e. use less_than_dcpo_lub ; [ exact e | ] ; cbn. apply refl_dcpo. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.Posets.Basics. Require Import UniMath.OrderTheory.Posets.PointedPosets. 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.Core.ScottContinuous. Require Import UniMath.OrderTheory.DCPOs.Examples.Propositions. Require Import UniMath.OrderTheory.DCPOs.Examples.SubDCPO.