proposition: Lemma InvRotMorphism_is_iso_with_inv_eq1 {C1 C2 : Complex A} (f : Morphism C1 C2) : # (ComplexHomotFunctor A) (((InvRotMorphismInv f) : (ComplexPreCat_Additive A)⟦_, _⟧) · InvRotMorphism f) = # (ComplexHomotFunctor A) (identity (C2 : (ComplexPreCat_Additive A))).
proof: Proof. apply maponpaths. unfold InvRotMorphismInv. unfold InvRotMorphism. use MorphismEq. intros i. cbn. unfold InvRotMorphismMorInv. 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 _ (to_In1 DS2)). rewrite (to_Unel1' DS2). rewrite ZeroArrow_comp_right. rewrite ZeroArrow_comp_left. rewrite to_lunax''. rewrite assoc. rewrite <- (assoc _ (to_In1 DS2)). rewrite (to_IdIn1 DS2). rewrite id_right. induction (hzrplusminus i 1). cbn. exact (to_IdIn2 DS1). Qed.
imports: UniMath.Foundations.UnivalenceAxiom UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.Foundations.NaturalNumbers UniMath.MoreFoundations.PartA UniMath.Algebra.BinaryOperations UniMath.Algebra.Monoids UniMath.NumberSystems.Integers UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.TransportMorphisms UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Coequalizers UniMath.CategoryTheory.Limits.Kernels UniMath.CategoryTheory.Limits.Cokernels UniMath.CategoryTheory.Limits.Pushouts UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.BinDirectSums UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Abelian UniMath.CategoryTheory.AbelianToAdditive UniMath.CategoryTheory.AdditiveFunctors UniMath.HomologicalAlgebra.Complexes UniMath.HomologicalAlgebra.KA UniMath.HomologicalAlgebra.TranslationFunctors