Proposition spans_double_cat_associator_laws : double_associator_laws spans_double_cat_associator_data. Proof. intros w₁ w₂ x₁ x₂ y₁ y₂ z₁ z₂ h₁ h₂ j₁ j₂ k₁ k₂ vw vx vy vz sq₁ sq₂ sq₃. use span_sqr_eq. rewrite transportb_disp_mor2_span ; cbn. use (MorphismsIntoPullbackEqual (isPullback_Pullback (PC _ _ _ _ _))) ; cbn. - rewrite !assoc'. unfold span_associator_mor, mor_of_comp_span_mor ; cbn. rewrite !PullbackArrow_PullbackPr1. rewrite !assoc. rewrite PullbackArrow_PullbackPr1. unfold mor_of_comp_span_mor. use (MorphismsIntoPullbackEqual (isPullback_Pullback (PC _ _ _ _ _))) ; cbn. + rewrite !assoc'. rewrite !PullbackArrow_PullbackPr1. rewrite !assoc. rewrite PullbackArrow_PullbackPr1. apply idpath. + rewrite !assoc'. rewrite !PullbackArrow_PullbackPr2. rewrite !assoc. rewrite !PullbackArrow_PullbackPr2. rewrite !assoc'. rewrite !PullbackArrow_PullbackPr1. apply idpath. - rewrite !assoc'. unfold span_associator_mor, mor_of_comp_span_mor ; cbn. rewrite !PullbackArrow_PullbackPr2. rewrite !assoc. rewrite !PullbackArrow_PullbackPr2. rewrite !assoc'. apply maponpaths. unfold mor_of_comp_span_mor. rewrite !PullbackArrow_PullbackPr2. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.Spans. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.UnivalentDoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.PseudoDoubleSetCats. Require Import UniMath.Bicategories.DoubleCategories.Core.CompanionsAndConjoints.