Lemma KAOctaComm3 {x y z : Complex A} (f1 : Morphism x y) (f2 : Morphism y z) : MorphismComp (KAOctaMor3 f1 f2) (MappingConePr1 A (KAOctaMor1 f1 f2)) = MorphismComp (MappingConePr1 A f2) (# (TranslationFunctor A) (MappingConeIn2 A f1)). Proof. use MorphismEq. intros i. cbn. set (DS1 := to_BinDirectSums A (y (i + 1)) (z i)). set (DS2 := to_BinDirectSums A (x (i + 1 + 1)) (y (i + 1))). set (DS3 := to_BinDirectSums A (x (i + 1)) (z i)). set (DS4 := to_BinDirectSums A DS2 DS3). rewrite to_postmor_linear'. rewrite <- (assoc _ _ (to_Pr1 DS4)). rewrite <- (assoc _ _ (to_Pr1 DS4)). rewrite (to_IdIn1 DS4). rewrite id_right. rewrite (to_Unel2' DS4). rewrite ZeroArrow_comp_right. rewrite to_runax''. apply idpath. Qed.
Require Import UniMath.Foundations.UnivalenceAxiom. Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.Foundations.NaturalNumbers. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.Algebra.BinaryOperations. Require Import UniMath.Algebra.Monoids. Require Import UniMath.NumberSystems.Integers. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.TransportMorphisms. Require Import UniMath.CategoryTheory.Limits.Zero. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Limits.Kernels. Require Import UniMath.CategoryTheory.Limits.Cokernels. Require Import UniMath.CategoryTheory.Limits.Pushouts. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.BinDirectSums. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.CategoriesWithBinOps. Require Import UniMath.CategoryTheory.PrecategoriesWithAbgrops. Require Import UniMath.CategoryTheory.PreAdditive. Require Import UniMath.CategoryTheory.Additive. Require Import UniMath.CategoryTheory.Abelian. Require Import UniMath.CategoryTheory.AbelianToAdditive. Require Import UniMath.CategoryTheory.AdditiveFunctors. Require Import UniMath.HomologicalAlgebra.Complexes. Require Import UniMath.HomologicalAlgebra.KA. Require Import UniMath.HomologicalAlgebra.TranslationFunctors.