Proposition generated_sieve (P : X → hProp) : sieve. Proof. use make_sieve. - exact (λ x, ∃ (y : X), P y ∧ x ≤ y). - abstract (intros x y p ; use factor_through_squash_hProp ; intros l ; induction l as [ l [ q₁ q₂ ] ] ; refine (hinhpr (l ,, q₁ ,, _)) ; exact (istrans_posetRelation _ _ _ _ p q₂)). Defined.
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.