Lemma rel_same_SW2NE_meet_consensus {X : hSet} (b : interlaced_prebilattice X) (x y : X): rel_same_SW2NE b (meet b x y) (consensus b x y). Proof. use (isantisymm_Lle (tlattice b)). - use Lmax_le_case. -- rewrite (iscomm_consensus _ (meet _ _ _) _ ) . use interlaced_leqt_leqk_leqtconsensus. exists (consensus b x y). split. --- use (bottom_interlacing_consensus_t (Lmin_le_l _ _ _) (Lmin_le_r _ _ _)). --- use isrefl_Lle. -- use interlaced_leqk_leqt_leqtconsensus. exists (meet b x y). split. --- use (bottom_interlacing_meet_k (Lmin_le_l _ _ _) (Lmin_le_r _ _ _)). --- use isrefl_Lle. - use (top_interlacing_consensus_t (Lmax_le_l _ _ _) (Lmax_le_r _ _ _)). 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.