Proposition is_glb_closed_sieve_type_intersection (I : UU) (f : I → set_of_closed_sieves) : is_greatest_lowerbound_lattice bounded_lattice_set_of_closed_sieves f (closed_sieve_type_intersection I f). Proof. split. - intros i. use eq_closed_sieve. use eq_sieve. + intros x H j ; cbn in *. exact (pr1 H j). + intros x p ; cbn in *. refine (p ,, _). exact (p i). - intros S H ; cbn. use eq_closed_sieve. use eq_sieve. + intros x p ; cbn in *. exact (pr1 p). + intros x p ; cbn in *. refine (p ,, _). intro i. exact (pr2 (eq_sieve_r (maponpaths pr1 (H i)) x p)). Qed.
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.