Lemma assoc_step_5 (p₁ : (identity_v x ·v v₁) ·v identity_v y = v₁) (p₃ : (identity_v x ·v v₃) ·v identity_v y = v₃) (p₄ : identity_v x ·v (v₄ ·v identity_v y) = v₄) : (transportf_square p₁ p₃ (linvunitor_h (identity_h x) ⋆v s₁ ⋆h s₂ ⋆v lunitor_h (identity_h y))) ⋆h s₃ = transportf_square (assocl_v _ _ _ @ p₁) p₄ ((linvunitor_h (identity_h x) ⋆v ((s₁ ⋆h s₂) ⋆v lunitor_h (identity_h y))) ⋆h (id_v_square _ ⋆v (s₃ ⋆v id_v_square _))). Proof. rewrite square_id_left_v, square_id_right_v. unfold transportb_square. rewrite transportf_f_square. rewrite square_assoc_v. unfold transportb_square. rewrite !transportf_hcomp_l. rewrite !transportf_f_square. refine (!_). rewrite transportf_hcomp_r. rewrite transportf_f_square. use transportf_square_eq. use eq_hcomp_square. - refine (!_ @ !(id_v_right _)). apply id_v_left. - use transportf_square_eq. apply idpath. - rewrite transportf_f_square. rewrite transportf_square_id. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.Core.TwoCategories. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.Comma. 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.DerivedLaws. Import TransportSquare.