Lemma MappingCone_Diff2_Diff1 {C1 C2 : Complex A} (f : Morphism C1 C2) (i : hz) : let DS1 := to_BinDirectSums A ((TranslationComplex A C1) i) (C2 i) in let DS2 := to_BinDirectSums A ((TranslationComplex A C1) (i + 1)) (C2 (i + 1)) in let DS3 := to_BinDirectSums A ((TranslationComplex A C1) (i + 1 + 1)) (C2 (i + 1 + 1)) in (MappingConeDiff2 f i) · (MappingConeDiff1 f (i + 1)) = ZeroArrow (Additive.to_Zero A) _ _. Proof. intros DS1 DS2 DS3. unfold MappingConeDiff2. unfold MappingConeDiff1. fold DS1. fold DS2. fold DS3. rewrite assoc. rewrite assoc. rewrite assoc. rewrite <- (assoc _ _ (to_Pr1 DS2)). rewrite (to_Unel2 DS2). rewrite (PreAdditive_unel_zero _ (Additive.to_Zero A)). rewrite ZeroArrow_comp_right. rewrite ZeroArrow_comp_left. apply ZeroArrow_comp_left. 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.