Definition reverse_graph_zig_zag_of_length {g : graph} {n : nat} {x y : vertex g} (gs : graph_zig_zag_of_length n x y) : graph_zig_zag_of_length n y x. Proof. revert x y gs. induction n as [ | n IHn ]. - intros x y gs. exact (pathsinv0 gs). - intros x y gs. induction gs as [ z gs ]. induction gs as [ g' gs ]. induction g' as [ g' | g' ]. + exact (post_cons_right_graph_zig_zag_of_length (IHn _ _ gs) g'). + exact (post_cons_left_graph_zig_zag_of_length (IHn _ _ gs) g'). Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Adjunctions.Core.