`{Univalence} (X : pType) `{IsConnected 0 X} : NatEquiv (opyon (Pi1 X)) (opyon X o B). Proof. nrefine (natequiv_compose (G := opyon (Pi1 (pTr 1 X))) _ _). 2: exact (natequiv_opyon_equiv (A:=Group) (grp_iso_inverse (grp_iso_pi_Tr 0 X))). refine (natequiv_compose _ (natequiv_grp_homo_pmap_bg _)). refine (natequiv_compose (G := opyon (pTr 1 X) o B) _ _); revgoals. { refine (natequiv_prewhisker _ _). refine (natequiv_opyon_equiv _^-1$). rapply pequiv_pclassifyingspace_pi1. } snrapply Build_NatEquiv. 1: intro; exact pequiv_ptr_rec. rapply is1natural_prewhisker. Defined.
Require Import Basics Types. Require Import Pointed WildCat. Require Import Cubical.DPath Cubical.PathSquare Cubical.DPathSquare. Require Import Algebra.AbGroups. Require Import Homotopy.HSpace.Core. Require Import TruncType. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.HomotopyGroup. Require Import Homotopy.WhiteheadsPrinciple. Module Import ClassifyingSpaceNotation.