Proposition isaprop_is_hor_gregarious_equivalence (H : vertically_saturated B) (HB_2_1 : locally_univalent_verity_double_bicat B) {x y : B} (h : x --> y) : isaprop (is_hor_gregarious_equivalence h). Proof. use invproofirrelevance. intros φ₁ φ₂. use (path_is_hor_gregarious_equivalence H HB_2_1). - use (v_sq_isotoid_2_1 H HB_2_1). use make_invertible_vertical_square. + use make_invertible_vertical_square_data. * use (square_between_companions φ₁). apply (pr12 φ₂). * use (square_between_companions φ₂). apply (pr12 φ₁). + split. * apply comp_square_between_companions. * apply comp_square_between_companions. - rewrite v_sq_idtoiso_isotoid_2_1 ; cbn. apply square_between_companions_unit. - rewrite v_sq_idtoiso_isotoid_2_1 ; cbn. apply square_between_companions_counit. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Discrete. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.VerityDoubleBicat. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.DerivedLaws. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.UnderlyingCats. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.CellsAndSquares. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.LocalUnivalence. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.CompanionPairs. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.CompanionPairUnique. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.CompanionPairAdjEquiv.