proposition: Proposition is_maximal_is_strongly_maximal {X : dcpo} (CX : continuous_dcpo_struct X) {x : X} (Hx : is_strongly_maximal x) : is_maximal x.
proof: Proof. intros y p. use (invmap (continuous_dcpo_struct_le_via_approximation CX y x)). intros z q. assert (H := Hx z y q). revert H. use factor_through_squash. { apply propproperty. } intro H. induction H as [ r | r ]. - exact r. - apply fromempty. exact (is_hausdorff_separated_not_le (symm_is_hausdorff_separated r) p). Qed.
imports: UniMath.MoreFoundations.All UniMath.OrderTheory.DCPOs.Core.DirectedSets UniMath.OrderTheory.DCPOs.Core.Basics UniMath.OrderTheory.DCPOs.Core.WayBelow UniMath.OrderTheory.DCPOs.Core.ScottTopology UniMath.OrderTheory.DCPOs.Core.IntrinsicApartness UniMath.OrderTheory.DCPOs.Basis.Continuous UniMath.OrderTheory.DCPOs.Basis.Basis UniMath.OrderTheory.DCPOs.Elements.Sharp