Definition lattice_SW2NE {X : hSet} (b : interlaced_prebilattice X) : lattice (setquotinset (rel_same_SW2NE b)). Proof. set (iscc := iscomp_consensus_rel_same_SW2NE b). set (iscg := iscomp_gullibility_rel_same_SW2NE b). set (cc := setquotfun2 (eqrel_same_SW2NE b) (eqrel_same_SW2NE b) (consensus b) iscc). set (gg := setquotfun2 (eqrel_same_SW2NE b) (eqrel_same_SW2NE b) (gullibility b) iscg). exists cc, gg. do 4 (try split). - use (isassocsetquotfun2 (consensus b) iscc (isassoc_Lmin (klattice b))). - use (iscommsetquotfun2 (consensus b) iscc (iscomm_consensus b)). - use (isassocsetquotfun2 (gullibility b) iscg (isassoc_Lmax (klattice b))). - use (iscommsetquotfun2 (gullibility b) iscg (iscomm_gullibility b)). - use (setquotuniv2prop _ (λ x y, @eqset (setquotinset _) (cc x (gg x y)) x)). intros x y. cbn. rewrite (consensus_gullibility_absorb b). reflexivity. - use (setquotuniv2prop _ (λ x y, @eqset (setquotinset _) (gg x (cc x y)) x)). intros x y. cbn. rewrite (gullibility_consensus_absorb b). reflexivity. Defined.
Require Import UniMath.OrderTheory.Prebilattice.Prebilattice. Require Import UniMath.OrderTheory.Prebilattice.Interlaced. Require Import UniMath.OrderTheory.Prebilattice.PrebilatticeDisplayedCat. Require Import UniMath.MoreFoundations.Equivalences. Require Import UniMath.Foundations.Sets. Require Import UniMath.Algebra.BinaryOperations. Require Import UniMath.OrderTheory.Lattice.Lattice. Require Import UniMath.OrderTheory.Lattice.Bounded. Require Import UniMath.CategoryTheory.Core.Prelude.