proposition: Lemma RotMorphismIsoEq2''_2 {C1 C2 : Complex A} (f : Morphism C1 C2) (i : hz) : let DS1 := to_BinDirectSums A (C1 (i + 1)) (C2 i) in let DS2 := to_BinDirectSums A (C2 (i + 1)) DS1 in let DS3 := to_BinDirectSums A (C1 (i + 1 + 1)) (C2 (i + 1)) in let DS4 := to_BinDirectSums A (C2 (i + 1 + 1)) DS3 in let DS9 := to_BinDirectSums A (C1 (i + 1 - 1 + 1)) (C2 (i + 1 - 1)) in let DS10 := to_BinDirectSums A (C2 (i + 1 - 1 + 1)) DS9 in transportf (precategory_morphisms DS2) (@maponpaths hz A (λ i0 : pr1 hz, to_BinDirectSums A (C2 (i0 + 1)) (to_BinDirectSums A (C1 (i0 + 1)) (C2 i0))) _ _ (hzrplusminus i 1)) (to_Pr1 DS2 · to_In2 DS3 · (to_Pr2 DS3 · transportf (λ x' : A, A ⟦ x', DS10 ⟧) (maponpaths C2 (hzrminusplus (i + 1) 1)) (to_In1 DS10))) = to_Pr1 DS2 · to_In1 DS2.
proof: Proof. intros DS1 DS2 DS3 DS4 DS9 DS10. rewrite assoc. rewrite <- (assoc _ (to_In2 DS3)). rewrite (to_IdIn2 DS3). rewrite id_right. rewrite transport_target_postcompose. apply cancel_precomposition. set (tmp := transport_hz_double_section_source_target A (λ i0 : hz, C2 (i0 + 1)) (λ i0 : hz, (to_BinDirectSums A (C2 (i0 + 1)) (to_BinDirectSums A (C1 (i0 + 1)) (C2 i0)))) (λ i0 : hz, to_In1 (to_BinDirectSums A (C2 (i0 + 1)) (to_BinDirectSums A (C1 (i0 + 1)) (C2 i0)))) _ _ (hzrplusminus i 1)). cbn beta in tmp. use (pathscomp0 _ (! tmp)). clear tmp. apply maponpaths. assert (e2 : maponpaths (λ i0 : hz, C2 (i0 + 1)) (hzrplusminus i 1) = maponpaths C2 (hzrminusplus (i + 1) 1)). { assert (e3 : maponpaths (λ i0 : hz, C2 (i0 + 1)) (hzrplusminus i 1) = maponpaths C2 (maponpaths (λ i0 : hz, i0 + 1) (hzrplusminus i 1))). { induction (hzrplusminus i 1). apply idpath. } rewrite e3. apply maponpaths. apply isasethz. } rewrite e2. 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