proposition: Proposition comp_ul_dr_corner {B : verity_double_bicat} {x y z : B} {v₁ : x -|-> y} {v₂ : y -|-> z} {h₁ : x --> y} {h₂ : y --> z} (s₁ : square_double_bicat h₁ (id_h y) v₁ (id_v y)) (s₂ : square_double_bicat h₂ (id_h z) v₂ (id_v z)) (s₃ : square_double_bicat (id_h x) h₁ (id_v x) v₁) (s₄ : square_double_bicat (id_h y) h₂ (id_v y) v₂) : linvunitor _ ◃s (runitor _ ▹s comp_diag_ul_corner s₃ s₄ ⋆v comp_diag_dr_corner s₁ s₂) = (linvunitor _ • (_ ◃ (_ ◃ linvunitor _)) • lassociator _ _ _) ◃s ((rassociator _ _ _ • (_ ◃ (lunitor _ • runitor _))) ▹s ((s₃ ⋆v s₁) ⋆v (s₄ ⋆v s₂))).
proof: Proof. unfold comp_diag_ul_corner, comp_diag_dr_corner. rewrite r_rwhisker_v_comp_square. rewrite <- rwhisker_square_comp. rewrite !vassocl. rewrite lwhisker_square_comp. apply maponpaths. rewrite <- lwhisker_vcomp. rewrite !vassocr. rewrite lunitor_lwhisker. rewrite l_lwhisker_v_comp_square. rewrite rwhisker_lwhisker_square. rewrite <- uwhisker_vcomp_square. rewrite <- dwhisker_vcomp_square. etrans. { rewrite runitor_h_comp_square'. rewrite dwhisker_uwhisker_square. rewrite <- !uwhisker_vcomp_square. rewrite dwhisker_uwhisker_square. rewrite <- uwhisker_square_comp. rewrite lunitor_h_comp_square'. rewrite <- !dwhisker_vcomp_square. rewrite <- dwhisker_square_comp. rewrite lassociator_v_comp_square'. rewrite <- lwhisker_dwhisker_square. rewrite <- lwhisker_uwhisker_square. rewrite rwhisker_lwhisker_square. rewrite <- lwhisker_square_comp. rewrite <- rwhisker_dwhisker_square. rewrite <- rwhisker_uwhisker_square. rewrite <- rwhisker_square_comp. etrans. { do 4 apply maponpaths. apply maponpaths. apply lassociator_v_comp_square''. } rewrite r_lwhisker_v_comp_square. rewrite r_rwhisker_v_comp_square. rewrite <- lwhisker_dwhisker_square. rewrite <- lwhisker_uwhisker_square. rewrite rwhisker_lwhisker_square. rewrite <- lwhisker_square_comp. rewrite <- rwhisker_dwhisker_square. rewrite <- rwhisker_uwhisker_square. rewrite <- rwhisker_square_comp. rewrite <- double_bicat_interchange. rewrite lunitor_v_comp_square'. rewrite rwhisker_lwhisker_square. rewrite <- lwhisker_hcomp_square. rewrite l_lwhisker_v_comp_square. rewrite r_lwhisker_v_comp_square. rewrite <- lwhisker_dwhisker_square. rewrite <- lwhisker_uwhisker_square. rewrite rwhisker_lwhisker_square. rewrite <- lwhisker_square_comp. rewrite runitor_v_comp_square'. rewrite <- rwhisker_hcomp_square. rewrite l_rwhisker_v_comp_square. rewrite r_rwhisker_v_comp_square. rewrite <- rwhisker_dwhisker_square. rewrite <- rwhisker_uwhisker_square. rewrite <- rwhisker_square_comp. rewrite <- ud_whisker_vcomp_square. etrans. { do 4 apply maponpaths. apply maponpaths_2. refine (!_). apply runitor_h_comp_square''. } etrans. { do 4 apply maponpaths. rewrite ud_whisker_vcomp_square. rewrite <- dwhisker_square_comp. rewrite runitor_rinvunitor. rewrite dwhisker_square_id. rewrite <- uwhisker_vcomp_square. apply maponpaths. rewrite lassociator_v_comp_square''. do 2 apply maponpaths. rewrite <- double_bicat_interchange. rewrite r_rwhisker_v_comp_square. rewrite <- lrwhisker_hcomp_square. apply maponpaths_2. apply maponpaths. rewrite r_lwhisker_v_comp_square. rewrite <- lwhisker_square_comp. rewrite runitor_lunitor_identity. rewrite lwhisker_vcomp. rewrite linvunitor_lunitor. rewrite lwhisker_id2. apply lwhisker_square_id. } rewrite dwhisker_uwhisker_square. rewrite <- uwhisker_square_comp. rewrite linvunitor_lunitor. rewrite dwhisker_square_id. rewrite !vassocl. rewrite runitor_rinvunitor. rewrite id2_right. rewrite <- lwhisker_uwhisker_square. rewrite rwhisker_lwhisker_square. rewrite <- lwhisker_square_comp. rewrite lwhisker_vcomp. rewrite lunitor_triangle. rewrite lunitor_lwhisker. rewrite runitor_lunitor_identity. rewrite rwhisker_vcomp. rewrite linvunitor_lunitor. rewrite id2_rwhisker. rewrite id2_left. etrans. { apply maponpaths. apply maponpaths_2. etrans. { do 2 apply maponpaths. rewrite !vassocr. rewrite <- lwhisker_lwhisker. rewrite !vassocl. apply idpath. } etrans. { apply maponpaths. rewrite !vassocr. rewrite lwhisker_vcomp. rewrite lunitor_lwhisker. apply idpath. } rewrite !vassocr. rewrite lwhisker_vcomp. rewrite rwhisker_vcomp. rewrite rinvunitor_runitor. rewrite id2_rwhisker. rewrite lwhisker_id2. rewrite id2_left. apply idpath. } apply idpath. } rewrite lwhisker_lwhisker. rewrite lwhisker_square_comp. apply maponpaths. rewrite <- runitor_triangle. rewrite !vassocr. rewrite lassociator_rassociator. rewrite id2_left. rewrite <- rwhisker_lwhisker_square. rewrite rwhisker_square_comp. apply maponpaths. etrans. { do 2 apply maponpaths. apply maponpaths_2. etrans. { apply maponpaths_2. refine (!_). apply runitor_v_comp_square''. } rewrite rwhisker_lwhisker_square. rewrite <- lwhisker_hcomp_square. apply maponpaths. rewrite <- lrwhisker_hcomp_square. rewrite <- id_h_square_bicat_id. apply idpath. } pose (double_bicat_interchange B (s₃ ⋆v s₁) (id_h_square_bicat (id_v y)) (id_h_square_bicat _) s₄) as p. rewrite runitor_h_comp_square' in p. rewrite lunitor_h_comp_square' in p. rewrite <- dwhisker_vcomp_square in p. rewrite dwhisker_uwhisker_square in p. rewrite <- uwhisker_vcomp_square in p. rewrite <- ud_whisker_vcomp_square in p. rewrite lunitor_runitor_identity in p. rewrite <- uwhisker_square_comp in p. rewrite rinvunitor_runitor in p. rewrite uwhisker_square_id in p. rewrite lassociator_v_comp_square''. rewrite !rwhisker_lwhisker_square. rewrite <- lwhisker_square_comp. rewrite <- rwhisker_square_comp. pose (maponpaths (λ z, rinvunitor _ ▵s (lunitor _ ▿s z)) p) as p'. cbn -[ver_bicat_of_ver_bicat_sq_bicat] in p'. rewrite <- dwhisker_square_comp in p'. rewrite linvunitor_lunitor in p'. rewrite dwhisker_square_id in p'. rewrite <- uwhisker_square_comp in p'. rewrite rinvunitor_runitor in p'. rewrite uwhisker_square_id in p'. refine (!_). etrans. { do 2 apply maponpaths. apply maponpaths_2. exact (!p'). } rewrite <- uwhisker_vcomp_square. rewrite rwhisker_uwhisker_square. rewrite lwhisker_uwhisker_square. rewrite lunitor_V_id_is_left_unit_V_id. apply maponpaths. rewrite <- ud_whisker_vcomp_square. rewrite <- id_h_square_bicat_comp. rewrite id_h_square_bicat_id. etrans. { do 2 apply maponpaths. apply maponpaths_2. apply maponpaths. apply maponpaths_2. apply runitor_v_comp_square'. } rewrite l_rwhisker_v_comp_square. rewrite l_lwhisker_v_comp_square. rewrite <- rwhisker_hcomp_square. rewrite l_rwhisker_v_comp_square. rewrite <- rwhisker_square_comp. etrans. { apply maponpaths. apply maponpaths_2. rewrite !vassocr. rewrite rwhisker_rwhisker_alt. rewrite !vassocl. rewrite rwhisker_vcomp. rewrite rinvunitor_runitor. rewrite id2_rwhisker. apply id2_right. } rewrite <- rwhisker_lwhisker_square. apply maponpaths. etrans. { apply maponpaths_2. rewrite lwhisker_hcomp. rewrite triangle_l_inv. rewrite <- rwhisker_hcomp. apply idpath. } rewrite <- l_lwhisker_v_comp_square. rewrite !lwhisker_hcomp_square. apply maponpaths_2. rewrite <- lunitor_lwhisker. rewrite <- runitor_triangle. rewrite runitor_lunitor_identity. apply idpath. Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence UniMath.CategoryTheory.TwoSidedDisplayedCats.Discrete UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.BicategoryLaws UniMath.Bicategories.Core.Unitors UniMath.Bicategories.DoubleCategories.DoubleBicat.VerityDoubleBicat