Lemma InvRotMorphismComm2 {C1 C2 : Complex A} (f : Morphism C1 C2 ) : ((MappingConeIn2 A (to_inv (# (InvTranslationFunctor A) (MappingConePr1 A f)) · inv_from_z_iso (AddEquivUnitIso (TranslationEquiv A) C1)) : (ComplexPreCat_Additive A)⟦_, _⟧)) · InvRotMorphism f = f. Proof. use MorphismEq. intros i. cbn. unfold InvRotMorphismMor. set (DS1 := to_BinDirectSums A (C1 (i + 1 - 1 + 1)) (C2 (i + 1 - 1))). set (DS2 := to_BinDirectSums A DS1 (C1 i)). rewrite to_premor_linear'. rewrite assoc. rewrite assoc. rewrite (to_IdIn2 DS2). rewrite id_left. rewrite (to_Unel2' DS2). rewrite ZeroArrow_comp_left. 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.