proposition: Lemma MappingCylinderIsoHomot {C1 C2 : Complex A} (f : (ComplexPreCat_Additive A)⟦C1, C2⟧) : ComplexHomot A (MappingCylinder A f) (MappingCylinder A f).
proof: Proof. intros i. unfold MappingCylinder. cbn. set (DS1 := to_BinDirectSums A (C1 (i + 1)) (C2 i)). set (DS2 := to_BinDirectSums A (C1 i) DS1). set (DS3 := to_BinDirectSums A (C1 (i - 1 + 1)) (C2 (i - 1))). set (DS4 := to_BinDirectSums A (C1 (i - 1)) DS3). use compose. - exact (C1 i). - exact (to_Pr1 DS2). - use compose. + exact DS3. + exact (transportf (λ x' : ob A, precategory_morphisms x' DS3) (maponpaths C1 (hzrminusplus i 1)) (to_In1 DS3)). + exact (to_inv (to_In2 DS4)). Defined.
imports: UniMath.Foundations.UnivalenceAxiom UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.Foundations.NaturalNumbers 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 UniMath.HomologicalAlgebra.MappingCone