Definition is_ideal (P : B → hProp) : hProp := ((∃ (b : B), P b) ∧ (∀ (a₁ a₂ : B), P a₁ ⇒ P a₂ ⇒ ∃ (b : B), P b ∧ a₁ ≺ b ∧ a₂ ≺ b) ∧ (∀ (a b : B), P b ⇒ a ≺ b ⇒ P a))%logic.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.Posets. 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.Core.WayBelow. Require Import UniMath.OrderTheory.DCPOs.Basis.Continuous. Require Import UniMath.OrderTheory.DCPOs.Basis.Algebraic. Require Import UniMath.OrderTheory.DCPOs.Basis.Basis. Require Import UniMath.OrderTheory.DCPOs.Basis.CompactBasis. Require Import UniMath.OrderTheory.DCPOs.Elements.Maximal. Require Import UniMath.OrderTheory.DCPOs.Examples.Products. Require Import UniMath.OrderTheory.DCPOs.Examples.SubDCPO. Require Import UniMath.OrderTheory.DCPOs.Examples.Propositions.