Definition is_coverage (P : X → (X → hProp) → hProp) : hProp := (∀ (x y : X) (C : X → hProp), P x C ⇒ C y ⇒ y ≤ x) ∧ (∀ (x y : X) (C : X → hProp), y ≤ x ⇒ P x C ⇒ ∃ (C' : X → hProp), P y C' ∧ ∀ (z : X), C' z ⇒ ∃ (s : X), C s ∧ z ≤ s).
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.BinaryOperations. Require Import UniMath.OrderTheory.Lattice.Lattice. Require Import UniMath.OrderTheory.Lattice.Bounded. Require Import UniMath.OrderTheory.Lattice.Distributive. Require Import UniMath.OrderTheory.Lattice.Heyting. Require Import UniMath.OrderTheory.Lattice.CompleteHeyting. Require Import UniMath.OrderTheory.Lattice.Examples.FromPartialOrder.