Definition Action_univalence_inv_comp_eval {G:gr} {X Y:Action G} (f:ActionIso X Y) (x:X) : castAction (Action_univalence_inv f) x = f x. Proof. intros. exact (eqtohomot (maponpaths pr1weq (maponpaths underlyingIso (Action_univalence_inv_comp f))) x). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.