Lemma two_type_bicat_laws : prebicat_laws two_type_bicat_data. Proof. repeat (use tpair) ; try (intros; apply idpath). - intros ; cbn in *. apply pathscomp0rid. - intros ; cbn in *. apply path_assoc. - intros ; cbn in *. symmetry. apply maponpathscomp0. - intros ; cbn in *. rewrite maponpathscomp0. apply idpath. - intros x y p₁ p₂ q ; cbn in *. induction q ; cbn. apply idpath. - intros x y p₁ p₂ q ; cbn in *. induction q ; cbn. exact (!(pathscomp0rid _)). - intros w x y z p₁ p₂ p₃ p₄ q ; cbn in *. induction q ; cbn. exact (!(pathscomp0rid _)). - intros w x y z p₁ p₂ p₃ p₄ q ; cbn in *. induction q ; cbn. exact (!(pathscomp0rid _)). - intros w x y z p₁ p₂ p₃ p₄ q ; cbn in *. induction q ; cbn. exact ((pathscomp0rid _)). - intros w x y z p₁ p₂ p₃ p₄ q ; cbn in *. induction q ; cbn. exact ((pathscomp0rid _)). - intros x y p ; cbn in *. apply pathsinv0r. - intros x y p ; cbn in *. apply pathsinv0l. - intros w x y z p₁ p₂ p₃ ; cbn in *. apply pathsinv0r. - intros w x y z p₁ p₂ p₃ ; cbn in *. apply pathsinv0l. - intros x y z p q ; cbn in *. induction p ; cbn. apply idpath. - intros v w x y z p₁ p₂ p₃ p₄ ; cbn in *. induction p₁, p₂, p₃, p₄ ; cbn. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.AdjointUnique.