proposition: Lemma RotMorphismCommHomot1 {C1 C2 : Complex A} (f : Morphism C1 C2) (i : hz) : RotMorphismCommHomot f i · MappingConeDiff A (MappingConeIn2 A f) (i - 1) = RotMorphismCommMor1 f i.
proof: Proof. unfold RotMorphismCommHomot. unfold MappingConeDiff. cbn. unfold RotMorphismCommMor1. cbn. unfold MappingConeDiff1, MappingConeDiff2, MappingConeDiff3. cbn. set (DS1 := to_BinDirectSums A (C1 (i + 1)) (C2 i)). set (DS2 := to_BinDirectSums A (C2 (i + 1)) DS1). set (DS5 := to_BinDirectSums A (C1 (i - 1 + 1 + 1)) (C2 (i - 1 + 1))). set (DS6 := to_BinDirectSums A (C2 (i - 1 + 1 + 1)) DS5). set (DS7 := to_BinDirectSums A (C1 (i - 1 + 1)) (C2 (i - 1))). set (DS8 := to_BinDirectSums A (C2 (i - 1 + 1)) DS7). unfold DiffTranslationComplex. rewrite to_premor_linear'. rewrite to_premor_linear'. use maponpaths_12. - rewrite <- assoc. apply cancel_precomposition. rewrite <- transport_source_precompose. rewrite assoc. rewrite (to_IdIn1 DS8). rewrite id_left. induction (hzrminusplus i 1). apply idpath. - rewrite <- assoc. rewrite <- transport_source_precompose. rewrite assoc. rewrite (to_IdIn1 DS8). rewrite id_left. rewrite <- assoc. rewrite <- transport_source_precompose. rewrite assoc. rewrite (to_Unel1' DS8). rewrite ZeroArrow_comp_left. rewrite transport_source_ZeroArrow. rewrite ZeroArrow_comp_right. rewrite to_runax''. rewrite <- transport_source_precompose. apply idpath. 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