Definition fubini_monotone_function_l {X Y Z : dcpo} (f : monotone_function (X × Y) Z) (D : directed_set (X × Y)) : monotone_function X Z. Proof. refine ((λ x, ⨆_{π₂ {{ D }}} (f ·l x)) ,, _). abstract (intros x₁ x₂ p ; use dcpo_lub_is_least ; cbn ; intro i ; use less_than_dcpo_lub ; [ exact i | ] ; cbn ; apply f ; split ; [ exact p | ] ; apply refl_PartialOrder). Defined.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.Posets.Basics. 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.BinaryProducts.