Lemma to_binop_inv_inv {x y : PA} (f g : PA⟦x, y⟧) : to_binop _ _ (to_inv f) (to_inv g) = to_inv (to_binop _ _ f g). Proof. apply (to_rcan g). rewrite to_assoc. rewrite linvax. rewrite to_runax'. apply (to_rcan f). rewrite to_assoc. rewrite linvax. cbn. rewrite <- (linvax (to_binop x y f g)). apply maponpaths. apply to_commax'. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.CategoriesWithBinOps. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Core.Functors.