Proposition double_triangle_alt_inv {C : double_cat} {x y z : C} (h : x -->h y) (k : y -->h z) : (id_v_square h ⋆h linvunitor_h k) ⋆v lassociator_h _ _ _ = transportb_square (id_v_left _) (id_v_left _) (rinvunitor_h _ ⋆h id_v_square _). Proof. refine (!(square_id_right_v' _) @ _ @ square_id_right_v' _). apply maponpaths. rewrite <- rassociator_lassociator_h'. rewrite !transportf_square_postwhisker. apply maponpaths. refine (square_assoc_v _ _ _ @ _ @ !(square_assoc_v _ _ _)). apply maponpaths. apply maponpaths_2. rewrite <- !square_assoc_v'. rewrite lassociator_rassociator_h. unfold transportb_square. rewrite transportf_square_postwhisker. rewrite transportf_f_square. rewrite square_id_right_v. unfold transportb_square. rewrite transportf_f_square. refine (!(square_id_right_v' _) @ _ @ square_id_right_v' _). apply maponpaths. rewrite <- comp_h_square_id. rewrite <- lunitor_linvunitor_h'. etrans. { apply maponpaths. use hcomp_split_globular_l_id' ; apply idpath. } refine (!_). etrans. { apply maponpaths. use hcomp_split_globular_l_id' ; apply idpath. } rewrite !transportf_square_postwhisker. apply maponpaths. refine (square_assoc_v _ _ _ @ _ @ !(square_assoc_v _ _ _)). apply maponpaths. apply maponpaths_2. refine (!(square_assoc_v' _ _ _) @ _). rewrite double_triangle_alt. unfold transportb_square. rewrite transportf_square_postwhisker. rewrite !transportf_square_prewhisker. rewrite !transportf_f_square. rewrite (hcomp_split_globular_l_id (idpath _) (idpath _)). rewrite (hcomp_split_globular_r_id (idpath _) (idpath _)). rewrite !transportf_f_square. rewrite rinvunitor_runitor_h. rewrite linvunitor_lunitor_h. unfold transportb_square. rewrite !transportf_f_square. rewrite transportf_hcomp_l. rewrite transportf_f_square. rewrite transportf_hcomp_r. rewrite !transportf_square_id. rewrite transportf_hcomp_r. rewrite transportf_f_square. rewrite transportf_hcomp_l. rewrite !transportf_square_id. 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.