Lemma chain_mor_left {C : category} {c : chain C} {i j} (Hij : i < j) (HiSj : i < S j) : chain_mor c Hij · dmor c (idpath (S j)) = chain_mor c HiSj. Proof. destruct j. - destruct (negnatlthn0 _ Hij). - simpl; destruct (natlehchoice4 i (S j) HiSj). + destruct (natlehchoice4 _ _ h). * destruct (natlehchoice4 _ _ Hij); [|destruct p, (isirreflnatlth _ h0)]. apply cancel_postcomposition, cancel_postcomposition, maponpaths, isasetbool. * destruct p; simpl. destruct (natlehchoice4 _ _ Hij); [destruct (isirreflnatlth _ h0)|]. apply cancel_postcomposition, maponpaths, isasetnat. + generalize Hij; rewrite p in Hij. intros H; destruct(isirreflnatlth (S j)). rewrite p in H; exact H. Qed.
Require Import UniMath.Foundations.PartA. Require Import UniMath.Foundations.NaturalNumbers. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.Graphs.Limits. Require Import UniMath.CategoryTheory.Limits.Initial.