proposition: Lemma RotMorphismIsoEq2''_1 {C1 C2 : Complex A} (f : Morphism C1 C2) (i : hz) : transportf (precategory_morphisms (to_BinDirectSums A (C2 (i + 1)) (to_BinDirectSums A (C1 (i + 1)) (C2 i)))) (@maponpaths hz A (λ i0 : pr1 hz, to_BinDirectSums A (C2 (i0 + 1)) (to_BinDirectSums A (C1 (i0 + 1)) (C2 i0))) _ _ (hzrminusplus i 1)) (RotMorphismIsoHomot f i · MappingConeDiff A (MappingConeIn2 A f) (i - 1)) = to_binop (to_BinDirectSums A (C2 (i + 1)) (to_BinDirectSums A (C1 (i + 1)) (C2 i))) (to_BinDirectSums A (C2 (i + 1)) (to_BinDirectSums A (C1 (i + 1)) (C2 i))) (to_Pr2 (to_BinDirectSums A (C2 (i + 1)) (to_BinDirectSums A (C1 (i + 1)) (C2 i))) · to_Pr2 (to_BinDirectSums A (C1 (i + 1)) (C2 i)) · to_inv (Diff C2 i · to_In1 (to_BinDirectSums A (C2 (i + 1)) (to_BinDirectSums A (C1 (i + 1)) (C2 i))))) (to_Pr2 (to_BinDirectSums A (C2 (i + 1)) (to_BinDirectSums A (C1 (i + 1)) (C2 i))) · to_Pr2 (to_BinDirectSums A (C1 (i + 1)) (C2 i)) · (to_In2 (to_BinDirectSums A (C1 (i + 1)) (C2 i)) · to_In2 (to_BinDirectSums A (C2 (i + 1)) (to_BinDirectSums A (C1 (i + 1)) (C2 i))))).
proof: Proof. unfold RotMorphismIsoHomot. unfold MappingConeDiff. 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 (DS4 := to_BinDirectSums A (C2 (i + 1 + 1)) DS3). 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). 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_premor_linear'. rewrite assoc. rewrite transport_target_postcompose. rewrite <- assoc. apply cancel_precomposition. rewrite <- transport_source_precompose. rewrite <- transport_target_postcompose. rewrite to_premor_linear'. rewrite to_premor_linear'. rewrite assoc. rewrite (assoc (to_In1 DS8)). rewrite (assoc (to_In1 DS8)). rewrite (to_IdIn1 DS8). rewrite (to_Unel1' DS8). rewrite id_left. rewrite ZeroArrow_comp_left. rewrite to_runax''. rewrite id_left. rewrite <- transport_target_to_binop. rewrite <- transport_source_to_binop. use maponpaths_12. + rewrite PreAdditive_invlcomp. unfold DS2, DS1, DS6, DS5. set (tmp := transport_hz_double_section_source_target A C2 (λ i0 : hz, (to_BinDirectSums A (C2 (i0 + 1)) (to_BinDirectSums A (C1 (i0 + 1)) (C2 i0)))) (λ i0 : hz, to_inv (Diff C2 i0) · (to_In1 (to_BinDirectSums A (C2 (i0 + 1)) (to_BinDirectSums A (C1 (i0 + 1)) (C2 i0))))) _ _ (hzrminusplus i 1)). cbn beta in tmp. use (pathscomp0 _ (! tmp)). clear tmp. rewrite transport_source_target_comm. apply idpath. + unfold DS2, DS1, DS6, DS5. set (tmp := transport_hz_double_section_source_target A C2 (λ i0 : hz, (to_BinDirectSums A (C2 (i0 + 1)) (to_BinDirectSums A (C1 (i0 + 1)) (C2 i0)))) (λ i0 : hz, to_In2 (to_BinDirectSums A (C1 (i0 + 1)) (C2 i0)) · to_In2 (to_BinDirectSums A (C2 (i0 + 1)) (to_BinDirectSums A (C1 (i0 + 1)) (C2 i0)))) _ _ (hzrminusplus i 1)). cbn beta in tmp. use (pathscomp0 _ (! tmp)). clear tmp. rewrite transport_source_target_comm. 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