ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.FreeMonoidalCategory.Hom.inductionOn
theorem Hom.inductionOn {motive : {X Y : F C} → (X ⟶ Y) → Prop} {X Y : F C} (t : X ⟶ Y)
    (id : (X : F C) → motive (𝟙 X))
    (α_hom : (X Y Z : F C) → motive (α_ X Y Z).hom)
    (α_inv : (X Y Z : F C) → motive (α_ X Y Z).inv)
    (l_hom : (X : F C) → motive (λ_ X).hom)
    (l_inv : (X : F C) → motive (λ_ X).inv)
    (ρ_hom : (X : F C) → motive (ρ_ X).hom)
    (ρ_inv : (X : F C) → motive (ρ_ X).inv)
    (comp : {X Y Z : F C} → (f : X ⟶ Y) → (g : Y ⟶ Z) → motive f → motive g → motive (f ≫ g))
    (whiskerLeft : (X : F C) → {Y Z : F C} → (f : Y ⟶ Z) → motive f → motive (X ◁ f))
    (whiskerRight : {X Y : F C} → (f : X ⟶ Y) → (Z : F C) → motive f → motive (f ▷ Z)) :
    motive t 
Arguments: {C : Type u} {motive : {X Y : CategoryTheory.FreeMonoidalCategory C} → (X ⟶ Y) → Prop} {X Y : CategoryTheory.FreeMonoidalCategory C} (t : X ⟶ Y) (id : ∀ (X : CategoryTheory.FreeMonoidalCategory C), motive (CategoryTheory.CategoryStruct.id X)) (α_hom :
  ∀ (X Y Z : CategoryTheory.FreeMonoidalCategory C), motive (CategoryTheory.MonoidalCategory.associator X Y Z).hom) (α_inv :
  ∀ (X Y Z : CategoryTheory.FreeMonoidalCategory C), motive (CategoryTheory.MonoidalCategory.associator X Y Z).inv) (l_hom : ∀ (X : CategoryTheory.FreeMonoidalCategory C), motive (CategoryTheory.MonoidalCategory.leftUnitor X).hom) (l_inv : ∀ (X : CategoryTheory.FreeMonoidalCategory C), motive (CategoryTheory.MonoidalCategory.leftUnitor X).inv) (ρ_hom : ∀ (X : CategoryTheory.FreeMonoidalCategory C), motive (CategoryTheory.MonoidalCategory.rightUnitor X).hom) (ρ_inv : ∀ (X : CategoryTheory.FreeMonoidalCategory C), motive (CategoryTheory.MonoidalCategory.rightUnitor X).inv) (comp :
  ∀ {X Y Z : CategoryTheory.FreeMonoidalCategory C} (f : X ⟶ Y) (g : Y ⟶ Z),
    motive f → motive g → motive (CategoryTheory.CategoryStruct.comp f g)) (whiskerLeft :
  ∀ (X : CategoryTheory.FreeMonoidalCategory C) {Y Z : CategoryTheory.FreeMonoidalCategory C} (f : Y ⟶ Z),
    motive f → motive (CategoryTheory.MonoidalCategory.whiskerLeft X f)) (whiskerRight :
  ∀ {X Y : CategoryTheory.FreeMonoidalCategory C} (f : X ⟶ Y) (Z : CategoryTheory.FreeMonoidalCategory C),
    motive f → motive (CategoryTheory.MonoidalCategory.whiskerRight f Z))
Goal: motive t
-- CategoryTheory.Functor.IsHomological.exact
class IsHomological extends F.PreservesZeroMorphisms : Prop where
  exact (T : Triangle C) (hT : T ∈ distTriang C) :
    ((shortComplexOfDistTriangle T hT).map F).Exact
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_5, u_3} A] [CategoryTheory.Abelian A] {F : CategoryTheory.Functor C A} [self : F.IsHomological] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: ((CategoryTheory.Pretriangulated.shortComplexOfDistTriangle T hT).map F).Exact
-- CategoryTheory.Functor.map_distinguished_exact
lemma map_distinguished_exact [F.IsHomological] (T : Triangle C) (hT : T ∈ distTriang C) :
    ((shortComplexOfDistTriangle T hT).map F).Exact 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_5, u_3} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor C A) [F.IsHomological] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: ((CategoryTheory.Pretriangulated.shortComplexOfDistTriangle T hT).map F).Exact
-- CategoryTheory.Functor.IsHomological.mk'
lemma IsHomological.mk' [F.PreservesZeroMorphisms]
    (hF : ∀ (T : Pretriangulated.Triangle C) (hT : T ∈ distTriang C),
      ∃ (T' : Pretriangulated.Triangle C) (e : T ≅ T'),
      ((shortComplexOfDistTriangle T' (isomorphic_distinguished _ hT _ e.symm)).map F).Exact) :
    F.IsHomological 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_5, u_3} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor C A) [F.PreservesZeroMorphisms] (hF :
  ∀ (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles),
    ∃ T' e, ((CategoryTheory.Pretriangulated.shortComplexOfDistTriangle T' ⋯).map F).Exact)
Goal: F.IsHomological
-- CategoryTheory.Functor.IsHomological.of_iso
lemma IsHomological.of_iso {F₁ F₂ : C ⥤ A} [F₁.IsHomological] (e : F₁ ≅ F₂) :
    F₂.IsHomological 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_5, u_3} A] [CategoryTheory.Abelian A] {F₁ F₂ : CategoryTheory.Functor C A} [F₁.IsHomological] (e : F₁ ≅ F₂)
Goal: F₂.IsHomological
-- CategoryTheory.Functor.mem_homologicalKernel_iff
lemma mem_homologicalKernel_iff [F.IsHomological] [F.ShiftSequence ℤ] (X : C) :
    F.homologicalKernel.P X ↔ ∀ (n : ℤ), IsZero ((F.shift n).obj X) 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_5, u_3} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor C A) [F.IsHomological] [F.ShiftSequence ℤ] (X : C)
Goal: F.homologicalKernel.P X ↔ ∀ (n : ℤ), CategoryTheory.Limits.IsZero ((F.shift n).obj X)
-- CategoryTheory.Functor.isHomological_of_localization
lemma isHomological_of_localization (L : C ⥤ D)
    [L.CommShift ℤ] [L.IsTriangulated] [L.mapArrow.EssSurj] (F : D ⥤ A)
    (G : C ⥤ A) (e : L ⋙ F ≅ G) [G.IsHomological] :
    F.IsHomological 
Arguments: {C : Type u_1} {D : Type u_2} {A : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Limits.HasZeroObject D] [CategoryTheory.HasShift D ℤ] [CategoryTheory.Preadditive D] [∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive] [CategoryTheory.Pretriangulated D] [CategoryTheory.Category.{u_6, u_3} A] [CategoryTheory.Abelian A] (L : CategoryTheory.Functor C D) [L.CommShift ℤ] [L.IsTriangulated] [L.mapArrow.EssSurj] (F : CategoryTheory.Functor D A) (G : CategoryTheory.Functor C A) (e : L.comp F ≅ G) [G.IsHomological]
Goal: F.IsHomological
-- CategoryTheory.Functor.homologySequenceδ_naturality
@[reassoc]
lemma homologySequenceδ_naturality :
    (F.shift n₀).map φ.hom₃ ≫ F.homologySequenceδ T' n₀ n₁ h =
      F.homologySequenceδ T n₀ n₁ h ≫ (F.shift n₁).map φ.hom₁ 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Category.{u_4, u_3} A] (F : CategoryTheory.Functor C A) [F.ShiftSequence ℤ] {T T' : CategoryTheory.Pretriangulated.Triangle C} (φ : T ⟶ T') (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁)
Goal: CategoryTheory.CategoryStruct.comp ((F.shift n₀).map φ.hom₃) (F.homologySequenceδ T' n₀ n₁ h) =
  CategoryTheory.CategoryStruct.comp (F.homologySequenceδ T n₀ n₁ h) ((F.shift n₁).map φ.hom₁)
-- CategoryTheory.Functor.comp_homologySequenceδ
@[reassoc]
lemma comp_homologySequenceδ :
    (F.shift n₀).map T.mor₂ ≫ F.homologySequenceδ T n₀ n₁ h = 0 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_4, u_3} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor C A) [F.IsHomological] [F.ShiftSequence ℤ] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁)
Goal: CategoryTheory.CategoryStruct.comp ((F.shift n₀).map T.mor₂) (F.homologySequenceδ T n₀ n₁ h) = 0
-- CategoryTheory.Functor.homologySequenceδ_comp
@[reassoc]
lemma homologySequenceδ_comp :
    F.homologySequenceδ T n₀ n₁ h ≫ (F.shift n₁).map T.mor₁ = 0 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_4, u_3} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor C A) [F.IsHomological] [F.ShiftSequence ℤ] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁)
Goal: CategoryTheory.CategoryStruct.comp (F.homologySequenceδ T n₀ n₁ h) ((F.shift n₁).map T.mor₁) = 0
-- CategoryTheory.Functor.homologySequence_comp
@[reassoc]
lemma homologySequence_comp  :
    (F.shift n₀).map T.mor₁ ≫ (F.shift n₀).map T.mor₂ = 0 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_4, u_3} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor C A) [F.IsHomological] [F.ShiftSequence ℤ] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (n₀ : ℤ)
Goal: CategoryTheory.CategoryStruct.comp ((F.shift n₀).map T.mor₁) ((F.shift n₀).map T.mor₂) = 0
-- CategoryTheory.Functor.homologySequence_exact₂
lemma homologySequence_exact₂ :
    (ShortComplex.mk _ _ (F.homologySequence_comp T hT n₀)).Exact 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_4, u_3} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor C A) [F.IsHomological] [F.ShiftSequence ℤ] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (n₀ : ℤ)
Goal: (CategoryTheory.ShortComplex.mk ((F.shift n₀).map T.mor₁) ((F.shift n₀).map T.mor₂) ⋯).Exact
-- CategoryTheory.Functor.homologySequence_exact₃
lemma homologySequence_exact₃ :
    (ShortComplex.mk _ _ (F.comp_homologySequenceδ T hT _ _ h)).Exact 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_4, u_3} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor C A) [F.IsHomological] [F.ShiftSequence ℤ] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁)
Goal: (CategoryTheory.ShortComplex.mk ((F.shift n₀).map T.mor₂) (F.homologySequenceδ T n₀ n₁ h) ⋯).Exact
-- CategoryTheory.Functor.homologySequence_exact₁
lemma homologySequence_exact₁ :
    (ShortComplex.mk _ _ (F.homologySequenceδ_comp T hT _ _ h)).Exact 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_4, u_3} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor C A) [F.IsHomological] [F.ShiftSequence ℤ] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁)
Goal: (CategoryTheory.ShortComplex.mk (F.homologySequenceδ T n₀ n₁ h) ((F.shift n₁).map T.mor₁) ⋯).Exact
-- CategoryTheory.Functor.homologySequence_epi_shift_map_mor₁_iff
lemma homologySequence_epi_shift_map_mor₁_iff :
    Epi ((F.shift n₀).map T.mor₁) ↔ (F.shift n₀).map T.mor₂ = 0 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_4, u_3} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor C A) [F.IsHomological] [F.ShiftSequence ℤ] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (n₀ : ℤ)
Goal: CategoryTheory.Epi ((F.shift n₀).map T.mor₁) ↔ (F.shift n₀).map T.mor₂ = 0
-- CategoryTheory.Functor.homologySequence_mono_shift_map_mor₁_iff
lemma homologySequence_mono_shift_map_mor₁_iff :
    Mono ((F.shift n₁).map T.mor₁) ↔ F.homologySequenceδ T n₀ n₁ h = 0 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_4, u_3} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor C A) [F.IsHomological] [F.ShiftSequence ℤ] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁)
Goal: CategoryTheory.Mono ((F.shift n₁).map T.mor₁) ↔ F.homologySequenceδ T n₀ n₁ h = 0
-- CategoryTheory.Functor.homologySequence_epi_shift_map_mor₂_iff
lemma homologySequence_epi_shift_map_mor₂_iff :
    Epi ((F.shift n₀).map T.mor₂) ↔ F.homologySequenceδ T n₀ n₁ h = 0 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_4, u_3} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor C A) [F.IsHomological] [F.ShiftSequence ℤ] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁)
Goal: CategoryTheory.Epi ((F.shift n₀).map T.mor₂) ↔ F.homologySequenceδ T n₀ n₁ h = 0
-- CategoryTheory.Functor.homologySequence_mono_shift_map_mor₂_iff
lemma homologySequence_mono_shift_map_mor₂_iff :
    Mono ((F.shift n₀).map T.mor₂) ↔ (F.shift n₀).map T.mor₁ = 0 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_4, u_3} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor C A) [F.IsHomological] [F.ShiftSequence ℤ] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (n₀ : ℤ)
Goal: CategoryTheory.Mono ((F.shift n₀).map T.mor₂) ↔ (F.shift n₀).map T.mor₁ = 0
-- CategoryTheory.Functor.mem_homologicalKernel_W_iff
lemma mem_homologicalKernel_W_iff {X Y : C} (f : X ⟶ Y) :
    F.homologicalKernel.W f ↔ ∀ (n : ℤ), IsIso ((F.shift n).map f) 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_5, u_3} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor C A) [F.IsHomological] [F.ShiftSequence ℤ] {X Y : C} (f : X ⟶ Y)
Goal: F.homologicalKernel.W f ↔ ∀ (n : ℤ), CategoryTheory.IsIso ((F.shift n).map f)
-- CategoryTheory.Functor.homologySequenceComposableArrows₅_exact
lemma homologySequenceComposableArrows₅_exact :
    (F.homologySequenceComposableArrows₅ T n₀ n₁ h).Exact 
Arguments: {C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Category.{u_4, u_3} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor C A) [F.IsHomological] [F.ShiftSequence ℤ] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁)
Goal: (F.homologySequenceComposableArrows₅ T n₀ n₁ h).Exact
-- CategoryTheory.Tor_obj
@[simps]
def Tor (n : ℕ) : C ⥤ C ⥤ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Abelian C] [CategoryTheory.MonoidalPreadditive C] [CategoryTheory.HasProjectiveResolutions C] (n : ℕ) (X : C)
Goal: (CategoryTheory.Tor C n).obj X = ((CategoryTheory.MonoidalCategory.tensoringLeft C).obj X).leftDerived n
-- CategoryTheory.Tor_map
@[simps]
def Tor (n : ℕ) : C ⥤ C ⥤ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Abelian C] [CategoryTheory.MonoidalPreadditive C] [CategoryTheory.HasProjectiveResolutions C] (n : ℕ)
Goal: ∀ {X Y : C} (f : X ⟶ Y),
  (CategoryTheory.Tor C n).map f =
    CategoryTheory.NatTrans.leftDerived ((CategoryTheory.MonoidalCategory.tensoringLeft C).map f) n
-- CategoryTheory.Tor'_obj_obj
@[simps! obj_obj]
def Tor' (n : ℕ) : C ⥤ C ⥤ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Abelian C] [CategoryTheory.MonoidalPreadditive C] [CategoryTheory.HasProjectiveResolutions C] (n : ℕ) (k j : C)
Goal: ((CategoryTheory.Tor' C n).obj k).obj j =
  (((CategoryTheory.MonoidalCategory.tensoringRight C).obj j).leftDerived n).obj k
-- CategoryTheory.Tor'_map_app'
@[simp]
lemma Tor'_map_app' (n : ℕ) {X Y : C} (f : X ⟶ Y) (Z : C) :
    ((Tor' C n).map f).app Z = (Functor.leftDerived ((tensoringRight C).obj Z) n).map f 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Abelian C] [CategoryTheory.MonoidalPreadditive C] [CategoryTheory.HasProjectiveResolutions C] (n : ℕ) {X Y : C} (f : X ⟶ Y) (Z : C)
Goal: ((CategoryTheory.Tor' C n).map f).app Z =
  (((CategoryTheory.MonoidalCategory.tensoringRight C).obj Z).leftDerived n).map f
-- CategoryTheory.Tor'_obj_map
@[simp]
lemma Tor'_obj_map (n : ℕ) {X Y : C} (Z : C) (f : X ⟶ Y) :
    ((Tor' C n).obj Z).map f = (NatTrans.leftDerived ((tensoringRight C).map f) n).app Z 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Abelian C] [CategoryTheory.MonoidalPreadditive C] [CategoryTheory.HasProjectiveResolutions C] (n : ℕ) {X Y : C} (Z : C) (f : X ⟶ Y)
Goal: ((CategoryTheory.Tor' C n).obj Z).map f =
  (CategoryTheory.NatTrans.leftDerived ((CategoryTheory.MonoidalCategory.tensoringRight C).map f) n).app Z
-- CategoryTheory.isZero_Tor_succ_of_projective
lemma isZero_Tor_succ_of_projective (X Y : C) [Projective Y] (n : ℕ) :
    IsZero (((Tor C (n + 1)).obj X).obj Y) 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Abelian C] [CategoryTheory.MonoidalPreadditive C] [CategoryTheory.HasProjectiveResolutions C] (X Y : C) [CategoryTheory.Projective Y] (n : ℕ)
Goal: CategoryTheory.Limits.IsZero (((CategoryTheory.Tor C (n + 1)).obj X).obj Y)
-- CategoryTheory.isZero_Tor'_succ_of_projective
lemma isZero_Tor'_succ_of_projective (X Y : C) [Projective X] (n : ℕ) :
    IsZero (((Tor' C (n + 1)).obj X).obj Y) 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Abelian C] [CategoryTheory.MonoidalPreadditive C] [CategoryTheory.HasProjectiveResolutions C] (X Y : C) [CategoryTheory.Projective X] (n : ℕ)
Goal: CategoryTheory.Limits.IsZero (((CategoryTheory.Tor' C (n + 1)).obj X).obj Y)
-- CategoryTheory.Limits.limitFunctorial_map
@[simp]
theorem limitFunctorial_map {F G : J ⥤ C} (α : F ⟶ G) :
    map (fun F : J ⥤ C => limit F) α = Limits.lim.map α 
Arguments: {J : Type w} [CategoryTheory.SmallCategory J] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimitsOfShape J C] {F G : J ⥤ C} (α : F ⟶ G)
Goal: CategoryTheory.map (fun F => CategoryTheory.Limits.limit F) α = CategoryTheory.Limits.lim.map α
-- CategoryTheory.Limits.limitLaxMonoidal_ε
@[simps]
instance limitLaxMonoidal : LaxMonoidal fun F : J ⥤ C => limit F 
Arguments: {J : Type w} [CategoryTheory.SmallCategory J] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimitsOfShape J C] [CategoryTheory.MonoidalCategory C]
Goal: CategoryTheory.LaxMonoidal.ε =
  CategoryTheory.Limits.limit.lift ((CategoryTheory.Functor.const J).obj (𝟙_ C))
    { pt := 𝟙_ C, π := { app := fun j => 𝟙 (((CategoryTheory.Functor.const J).obj (𝟙_ C)).obj j), naturality := ⋯ } }
-- CategoryTheory.Limits.limitLaxMonoidal_μ
@[simps]
instance limitLaxMonoidal : LaxMonoidal fun F : J ⥤ C => limit F 
Arguments: {J : Type w} [CategoryTheory.SmallCategory J] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimitsOfShape J C] [CategoryTheory.MonoidalCategory C] (F G : J ⥤ C)
Goal: CategoryTheory.LaxMonoidal.μ F G =
  CategoryTheory.Limits.limit.lift (F ⊗ G)
    { pt := CategoryTheory.Limits.limit F ⊗ CategoryTheory.Limits.limit G,
      π := { app := fun j => CategoryTheory.Limits.limit.π F j ⊗ CategoryTheory.Limits.limit.π G j, naturality := ⋯ } }
-- CategoryTheory.Limits.limLax_obj
@[simp]
theorem limLax_obj (F : J ⥤ C) : limLax.obj F = limit F 
Arguments: {J : Type w} [CategoryTheory.SmallCategory J] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimitsOfShape J C] [CategoryTheory.MonoidalCategory C] (F : J ⥤ C)
Goal: CategoryTheory.Limits.limLax.obj F = CategoryTheory.Limits.limit F
-- CategoryTheory.Limits.limLax_obj'
theorem limLax_obj' (F : J ⥤ C) : limLax.obj F = lim.obj F 
Arguments: {J : Type w} [CategoryTheory.SmallCategory J] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimitsOfShape J C] [CategoryTheory.MonoidalCategory C] (F : J ⥤ C)
Goal: CategoryTheory.Limits.limLax.obj F = CategoryTheory.Limits.lim.obj F
-- CategoryTheory.Limits.limLax_map
@[simp]
theorem limLax_map {F G : J ⥤ C} (α : F ⟶ G) : limLax.map α = lim.map α 
Arguments: {J : Type w} [CategoryTheory.SmallCategory J] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimitsOfShape J C] [CategoryTheory.MonoidalCategory C] {F G : J ⥤ C} (α : F ⟶ G)
Goal: CategoryTheory.Limits.limLax.map α = CategoryTheory.Limits.lim.map α
-- CategoryTheory.Limits.limLax_ε
@[simp]
theorem limLax_ε :
    (@limLax J _ C _ _ _).ε =
      limit.lift _
        { pt := _
          π := { app := fun j => 𝟙 _ } } 
Arguments: {J : Type w} [CategoryTheory.SmallCategory J] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimitsOfShape J C] [CategoryTheory.MonoidalCategory C]
Goal: CategoryTheory.Limits.limLax.ε =
  CategoryTheory.Limits.limit.lift ((CategoryTheory.Functor.const J).obj (𝟙_ C))
    { pt := 𝟙_ C, π := { app := fun j => 𝟙 (((CategoryTheory.Functor.const J).obj (𝟙_ C)).obj j), naturality := ⋯ } }