proposition: Lemma InvTranslationFunctorH_isAdditiveFunctor : isAdditiveFunctor InvTranslationFunctorH.
proof: Proof. use make_isAdditiveFunctor'. - intros C1 C2. use (pathscomp0 _ (@AdditiveFunctorZeroArrow (ComplexPreCat_Additive A) (ComplexHomot_Additive A) (ComplexHomotFunctor A) (InvTranslationFunctorH C1) (InvTranslationFunctorH C2))). set (tmp := (@AdditiveFunctorZeroArrow (ComplexPreCat_Additive A) (ComplexPreCat_Additive A) InvTranslationFunctor_Additive C1 C2)). apply (maponpaths (# (ComplexHomotFunctor A))) in tmp. use (pathscomp0 _ tmp). clear tmp. use InvTranslationFunctorHImEq. use AdditiveFunctorZeroArrow. - intros C1 C2 f g. use (squash_to_prop (ComplexHomotFunctor_issurj A f)). { apply to_has_homsets. } intros f'. use (squash_to_prop (ComplexHomotFunctor_issurj A g)). { apply to_has_homsets. } intros g'. rewrite (InvTranslationFunctorH_Mor_Im f f'). rewrite (InvTranslationFunctorH_Mor_Im g g'). use (pathscomp0 _ (AdditiveFunctorLinear (ComplexHomotFunctor A) (# InvTranslationFunctor (hfiberpr1 _ _ f')) (# InvTranslationFunctor (hfiberpr1 _ _ g')))). set (tmp := AdditiveFunctorLinear InvTranslationFunctor_Additive (hfiberpr1 _ _ f') (hfiberpr1 _ _ g')). apply (maponpaths (# (ComplexHomotFunctor A))) in tmp. use (pathscomp0 _ tmp). clear tmp. use InvTranslationFunctorHImEq. rewrite AdditiveFunctorLinear. rewrite (hfiberpr2 _ _ f'). rewrite (hfiberpr2 _ _ g'). 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.Core.Univalence 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