proposition: Lemma ComplexEq_transport_target {C1 C2 C2' : Complex A} (H : ∏ (i : hz), C2 i = C2' i) (H1 : ∏ (i : hz), transportf (λ x : A, C2' i --> x) (H (i + 1)) (transportf (λ x : A, x --> C2 (i + 1)) (H i) (Diff C2 i)) = Diff C2' i) (f : Morphism C1 C2) (i : hz) : (transportf (λ x : Complex A, Morphism C1 x) (ComplexEq A C2 C2' H H1) f) i = transportf (λ x : A, C1 i --> x) (H i) (f i).
proof: Proof. assert (e : (transportf (λ x : Complex A, Morphism C1 x) (ComplexEq A C2 C2' H H1) f) i = (transportf (λ x : Complex A, C1 i --> x i) (ComplexEq A C2 C2' H H1) (f i))). { induction (ComplexEq A C2 C2' H H1). apply idpath. } use (pathscomp0 e). clear e. rewrite <- (@transport_target_funextfun hz _ C2 C2' H i (C1 i) (f i)). unfold ComplexEq. unfold Complex_Funclass. set (e := (funextfun (pr1 (pr1 C2)) (pr1 (pr1 C2')) (λ x0 : pr1 hz, H x0))). cbn. cbn in e. fold e. set (tmp := @transportf_total2_base _ _ (λ x0 : _ , A ⟦ pr1 (pr1 C1) i, (pr1 x0) i ⟧) C2 C2' (total2_paths_f e (ComplexEq' A C2 C2' H H1)) (ComplexEq'' A C2 C2' H H1) (f i)). cbn beta in tmp. use (pathscomp0 tmp). clear tmp. use transportf_total2_base. 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