proposition: Proposition is_least_upperbound_pair {X Y : hSet} (DX : dcpo_struct X) (DY : dcpo_struct Y) (I : UU) (D : I → (X × Y)%set) (x : X) (y : Y) (Hx : is_least_upperbound DX (λ i, pr1 (D i)) x) (Hy : is_least_upperbound DY (λ i, pr2 (D i)) y) : is_least_upperbound (prod_PartialOrder DX DY) D (x ,, y).
proof: Proof. pose (x' := (x ,, Hx) : lub DX _). pose (y' := (y ,, Hy) : lub DY _). split. - intros i. exact (lub_is_upperbound x' i ,, lub_is_upperbound y' i). - intros z Hz. exact (lub_is_least x' _ (λ i, pr1 (Hz i)) ,, lub_is_least y' _ (λ i, pr2 (Hz i))). 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.Core.ScottTopology UniMath.OrderTheory.DCPOs.Core.WayBelow UniMath.OrderTheory.DCPOs.Basis.Basis UniMath.OrderTheory.DCPOs.Basis.Continuous UniMath.OrderTheory.DCPOs.Elements.Maximal