proposition: Lemma InvRotMorphismMorInvComm {C1 C2 : Complex A} (f : Morphism C1 C2) (i : hz) : (InvRotMorphismMorInv f i) · (MappingConeDiff A (to_inv (# (InvTranslationFunctor A) (MappingConePr1 A f)) · inv_from_z_iso (AddEquivUnitIso (TranslationEquiv A) C1)) i) = Diff C2 i · InvRotMorphismMorInv f (i + 1).
proof: Proof. unfold InvRotMorphismMorInv. unfold MappingConeDiff. unfold MappingConeDiff1, MappingConeDiff2, MappingConeDiff3. cbn. unfold DiffTranslationComplex. unfold InvTranslationComplex. cbn. unfold MappingConeDiff. unfold MappingConeDiff1, MappingConeDiff2, MappingConeDiff3. cbn. unfold DiffTranslationComplex. unfold InvTranslationComplex. cbn. unfold InvRotMorphismMorInv. set (DS1 := to_BinDirectSums A (C1 (i + 1 - 1 + 1)) (C2 (i + 1 - 1))). set (DS2 := to_BinDirectSums A DS1 (C1 i)). set (DS3 := to_BinDirectSums A (C1 (i + 1 + 1 - 1 + 1)) (C2 (i + 1 + 1 - 1))). set (DS4 := to_BinDirectSums A DS3 (C1 (i + 1))). set (DS5 := to_BinDirectSums A (C1 (i + 1 - 1 + 1 + 1)) (C2 (i + 1 - 1 + 1))). rewrite to_premor_linear'. rewrite to_premor_linear'. rewrite assoc. rewrite assoc. rewrite assoc. rewrite assoc. rewrite assoc. rewrite assoc. rewrite assoc. rewrite assoc. rewrite assoc. rewrite assoc. rewrite assoc. rewrite <- (assoc _ (to_In1 DS2)). rewrite <- (assoc _ (to_In1 DS2)). rewrite (to_Unel1' DS2). rewrite (to_IdIn1 DS2). rewrite id_right. rewrite ZeroArrow_comp_right. rewrite ZeroArrow_comp_left. rewrite ZeroArrow_comp_left. rewrite to_runax''. rewrite <- transport_source_precompose. rewrite <- transport_source_precompose. rewrite <- transport_source_precompose. rewrite <- PreAdditive_invrcomp. rewrite <- PreAdditive_invrcomp. rewrite <- PreAdditive_invrcomp. rewrite (to_Unel2' DS1). rewrite to_inv_zero. rewrite transport_source_ZeroArrow. rewrite ZeroArrow_comp_left. rewrite ZeroArrow_comp_left. rewrite to_runax''. rewrite <- transport_target_to_inv. rewrite <- PreAdditive_invrcomp. rewrite <- PreAdditive_invlcomp. rewrite <- PreAdditive_invlcomp. rewrite <- PreAdditive_invlcomp. rewrite <- transport_source_to_inv. rewrite <- transport_source_to_inv. rewrite inv_inv_eq. rewrite transport_source_precompose. apply cancel_postcomposition. rewrite <- transport_target_postcompose. rewrite to_premor_linear'. rewrite to_premor_linear'. rewrite assoc. rewrite assoc. rewrite assoc. rewrite assoc. rewrite <- PreAdditive_invrcomp. rewrite assoc. rewrite assoc. rewrite (to_IdIn2 DS1). rewrite id_left. rewrite (to_Unel2' DS1). rewrite ZeroArrow_comp_left. rewrite ZeroArrow_comp_left. rewrite ZeroArrow_comp_left. rewrite ZeroArrow_comp_left. rewrite to_inv_zero. rewrite to_lunax''. rewrite to_lunax''. rewrite transport_target_postcompose. rewrite transport_source_precompose. set (tmp := @transport_compose' A _ _ _ _ (Diff C2 i) (transportf (λ x' : A, A ⟦ x', DS3 ⟧) (maponpaths C2 (hzrplusminus (i + 1) 1)) (to_In2 DS3)) (maponpaths C2 (! hzrminusplus (i + 1) 1))). use (pathscomp0 _ tmp). clear tmp. set (tmp := transport_hz_section A C2 1 (Diff C2) _ _ (! hzrplusminus i 1)). assert (e : maponpaths C2 (maponpaths_2 _ (! hzrplusminus i 1) _) = maponpaths C2 (! hzrminusplus (i + 1) 1)). { apply maponpaths. apply isasethz. } rewrite <- e. clear e. rewrite tmp. clear tmp. rewrite pathsinv0inv0. apply cancel_precomposition. rewrite transport_f_f. unfold DS5, DS3. rewrite <- maponpathscomp0. set (tmp := @transport_hz_double_section A C2 (λ i0 : pr1 hz, to_BinDirectSums A (C1 (i0 + 1)) (C2 i0)) (λ i0 : hz, to_In2 (to_BinDirectSums A (C1 (i0 + 1)) (C2 i0))) _ _ (hzrminusplus (i + 1) 1 @ ! hzrplusminus (i + 1) 1)). cbn in tmp. use (pathscomp0 tmp). assert (e : (maponpaths C2 (! (hzrminusplus (i + 1) 1 @ ! hzrplusminus (i + 1) 1))) = (maponpaths C2 (hzrplusminus (i + 1) 1 @ maponpaths_2 _ (! hzrplusminus i 1) _))). { apply maponpaths. apply isasethz. } cbn in e. rewrite e. clear e. 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