proposition: Proposition spans_double_cat_associator_laws : double_associator_laws spans_double_cat_associator_data.
proof: 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.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Core.Setcategories UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.Spans UniMath.CategoryTheory.Limits.Pullbacks UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics UniMath.Bicategories.DoubleCategories.Core.DoubleCats UniMath.Bicategories.DoubleCategories.Core.UnivalentDoubleCats UniMath.Bicategories.DoubleCategories.Core.PseudoDoubleSetCats UniMath.Bicategories.DoubleCategories.Core.CompanionsAndConjoints