Proposition is_lub_type_prod_pointwise_lub : is_least_upperbound (depfunction_poset (λ i : I, D i) (λ i : I, D i)) E type_prod_pointwise_lub. Proof. split. - intros e i ; cbn. use less_than_dcpo_lub ; [ exact e | ] ; cbn. apply refl_dcpo. - intros f Hf i. use dcpo_lub_is_least. intro e ; cbn. exact (Hf e i). 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.