Proposition hcomp_split_globular_l_id {C : double_cat} {x y z : C} {v₁ v₂ : z -->v z} (q₁ : identity_v z = v₁) (q₂ : identity_v z = v₂) {h₁ h₂ h₃ : y -->h z} {k : x -->h y} (s₁ : square (identity_v y) v₁ h₁ h₂) (s₂ : square (identity_v y) v₂ h₂ h₃) : id_v_square k ⋆h s₁ ⋆v id_v_square k ⋆h s₂ = transportf_square (!(id_v_left _)) (idpath _) (id_v_square k ⋆h transportf_square (id_v_left _) (idpath _) (s₁ ⋆v s₂)). Proof. induction q₁, q₂ ; cbn. rewrite <- comp_h_square_comp. rewrite square_id_left_v. unfold transportb_square. rewrite transportf_hcomp_r. rewrite transportf_f_square. rewrite transportf_hcomp_l. use transportf_square_eq. rewrite transportf_hcomp_r. refine (maponpaths (λ z, z ⋆h _) _). use transportf_square_eq. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleCats. Require Import UniMath.Bicategories.DoubleCategories.DerivedLaws.TransportLaws. Require Import UniMath.Bicategories.DoubleCategories.DerivedLaws.Variations. Import TransportSquare.