proposition: Lemma RotMorphismCommHomot2 {C1 C2 : Complex A} (f : Morphism C1 C2) (i : hz) : MappingConeDiff A f i · RotMorphismCommHomot f (i + 1) = RotMorphismCommMor2 f i.
proof: Proof. unfold RotMorphismCommHomot. unfold MappingConeDiff. cbn. unfold RotMorphismCommMor2. 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 (DS3 := to_BinDirectSums A (C1 (i + 1 + 1)) (C2 (i + 1))). set (DS9 := to_BinDirectSums A (C1 (i + 1 - 1 + 1)) (C2 (i + 1 - 1))). set (DS10 := to_BinDirectSums A (C2 (i + 1 - 1 + 1)) DS9). unfold DiffTranslationComplex. rewrite to_postmor_linear'. rewrite assoc. rewrite assoc. rewrite <- (assoc _ _ (to_Pr2 DS3)). rewrite (to_Unel1' DS3). rewrite ZeroArrow_comp_right. rewrite ZeroArrow_comp_left. rewrite to_lunax''. rewrite to_postmor_linear'. rewrite assoc. rewrite assoc. rewrite <- (assoc _ _ (to_Pr2 DS3)). rewrite (to_IdIn2 DS3). rewrite id_right. rewrite assoc. rewrite assoc. rewrite <- (assoc _ _ (to_Pr2 DS3)). rewrite (to_IdIn2 DS3). rewrite id_right. use maponpaths_12. - rewrite assoc. apply idpath. - rewrite assoc. 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