proposition: Proposition is_scott_continuous_depfunction_map {W : dcpo} (fs : ∏ (i : I), scott_continuous_map W (D i)) : is_scott_continuous W depfunction_dcpo (λ w i, fs i w).
proof: Proof. use make_is_scott_continuous. - exact (is_monotone_depfunction_poset_pair fs (λ i, pr12 (fs i))). - intros E. use funextsec ; intro i. cbn ; unfold type_prod_pointwise_lub. rewrite scott_continuous_map_on_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.
imports: UniMath.MoreFoundations.All UniMath.OrderTheory.Posets.Basics UniMath.OrderTheory.Posets.PointedPosets UniMath.OrderTheory.Posets.MonotoneFunctions UniMath.OrderTheory.DCPOs.Core.DirectedSets UniMath.OrderTheory.DCPOs.Core.Basics UniMath.OrderTheory.DCPOs.Core.ScottContinuous UniMath.OrderTheory.DCPOs.Examples.Propositions UniMath.OrderTheory.DCPOs.Examples.SubDCPO