ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CochainComplex.mappingCone.triangleRotateShortComplex_X₂
@[simps]
noncomputable def triangleRotateShortComplex : ShortComplex (CochainComplex C ℤ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {K L : CochainComplex C ℤ} (φ : K ⟶ L)
Goal: (CochainComplex.mappingCone.triangleRotateShortComplex φ).X₂ = (CochainComplex.mappingCone.triangle φ).rotate.obj₂
-- CochainComplex.mappingCone.triangleRotateShortComplex_f
@[simps]
noncomputable def triangleRotateShortComplex : ShortComplex (CochainComplex C ℤ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {K L : CochainComplex C ℤ} (φ : K ⟶ L)
Goal: (CochainComplex.mappingCone.triangleRotateShortComplex φ).f = (CochainComplex.mappingCone.triangle φ).rotate.mor₁
-- CochainComplex.mappingCone.triangleRotateShortComplexSplitting_r
@[simps]
noncomputable def triangleRotateShortComplexSplitting (n : ℤ) :
    ((triangleRotateShortComplex φ).map (eval _ _ n)).Splitting 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {K L : CochainComplex C ℤ} (φ : K ⟶ L) (n : ℤ)
Goal: (CochainComplex.mappingCone.triangleRotateShortComplexSplitting φ n).r = (CochainComplex.mappingCone.snd φ).v n n ⋯
-- CochainComplex.mappingCone.triangleRotateShortComplexSplitting_s
@[simps]
noncomputable def triangleRotateShortComplexSplitting (n : ℤ) :
    ((triangleRotateShortComplex φ).map (eval _ _ n)).Splitting 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {K L : CochainComplex C ℤ} (φ : K ⟶ L) (n : ℤ)
Goal: (CochainComplex.mappingCone.triangleRotateShortComplexSplitting φ n).s =
  -(CochainComplex.mappingCone.inl φ).v (n + 1) n ⋯
-- CochainComplex.mappingCone.cocycleOfDegreewiseSplit_triangleRotateShortComplexSplitting_v
@[simp]
lemma cocycleOfDegreewiseSplit_triangleRotateShortComplexSplitting_v (p : ℤ) :
    (cocycleOfDegreewiseSplit _ (triangleRotateShortComplexSplitting φ)).1.v p _ rfl =
      -φ.f _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {K L : CochainComplex C ℤ} (φ : K ⟶ L) (p : ℤ)
Goal: (↑(CochainComplex.cocycleOfDegreewiseSplit (CochainComplex.mappingCone.triangleRotateShortComplex φ)
          (CochainComplex.mappingCone.triangleRotateShortComplexSplitting φ))).v
    p (p + 1) ⋯ =
  -φ.f (p + { as := 1 }.as)
-- HomotopyCategory.distinguished_iff_iso_trianglehOfDegreewiseSplit
lemma distinguished_iff_iso_trianglehOfDegreewiseSplit
    (T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))) :
    (T ∈ distTriang _) ↔ ∃ (S : ShortComplex (CochainComplex C ℤ))
      (σ : ∀ n, (S.map (HomologicalComplex.eval C _ n)).Splitting),
      Nonempty (T ≅ CochainComplex.trianglehOfDegreewiseSplit S σ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasBinaryBiproducts C] (T : CategoryTheory.Pretriangulated.Triangle (HomotopyCategory C (ComplexShape.up ℤ)))
Goal: T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles ↔
  ∃ S σ, Nonempty (T ≅ CochainComplex.trianglehOfDegreewiseSplit S σ)
-- QuasiIsoAt.quasiIso
class QuasiIsoAt (f : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i] : Prop where
  quasiIso : ShortComplex.QuasiIso ((shortComplexFunctor C c i).map f)
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c} {f : K ⟶ L} {i : ι} [K.HasHomology i] [L.HasHomology i] [self : QuasiIsoAt f i]
Goal: CategoryTheory.ShortComplex.QuasiIso ((HomologicalComplex.shortComplexFunctor C c i).map f)
-- quasiIsoAt_iff
lemma quasiIsoAt_iff (f : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i] :
    QuasiIsoAt f i ↔
      ShortComplex.QuasiIso ((shortComplexFunctor C c i).map f) 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c} (f : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i]
Goal: QuasiIsoAt f i ↔ CategoryTheory.ShortComplex.QuasiIso ((HomologicalComplex.shortComplexFunctor C c i).map f)
-- quasiIsoAt_iff'
lemma quasiIsoAt_iff' (f : K ⟶ L) (i j k : ι) (hi : c.prev j = i) (hk : c.next j = k)
    [K.HasHomology j] [L.HasHomology j] [(K.sc' i j k).HasHomology] [(L.sc' i j k).HasHomology] :
    QuasiIsoAt f j ↔
      ShortComplex.QuasiIso ((shortComplexFunctor' C c i j k).map f) 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c} (f : K ⟶ L) (i j k : ι) (hi : c.prev j = i) (hk : c.next j = k) [K.HasHomology j] [L.HasHomology j] [(K.sc' i j k).HasHomology] [(L.sc' i j k).HasHomology]
Goal: QuasiIsoAt f j ↔ CategoryTheory.ShortComplex.QuasiIso ((HomologicalComplex.shortComplexFunctor' C c i j k).map f)
-- quasiIsoAt_iff_isIso_homologyMap
lemma quasiIsoAt_iff_isIso_homologyMap (f : K ⟶ L) (i : ι)
    [K.HasHomology i] [L.HasHomology i] :
    QuasiIsoAt f i ↔ IsIso (homologyMap f i) 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c} (f : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i]
Goal: QuasiIsoAt f i ↔ CategoryTheory.IsIso (HomologicalComplex.homologyMap f i)
-- quasiIsoAt_iff_exactAt
lemma quasiIsoAt_iff_exactAt (f : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i]
    (hK : K.ExactAt i) :
    QuasiIsoAt f i ↔ L.ExactAt i 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c} (f : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i] (hK : K.ExactAt i)
Goal: QuasiIsoAt f i ↔ L.ExactAt i
-- quasiIsoAt_iff_exactAt'
lemma quasiIsoAt_iff_exactAt' (f : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i]
    (hL : L.ExactAt i) :
    QuasiIsoAt f i ↔ K.ExactAt i 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c} (f : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i] (hL : L.ExactAt i)
Goal: QuasiIsoAt f i ↔ K.ExactAt i
-- isoOfQuasiIsoAt_hom
@[simps! hom]
noncomputable def isoOfQuasiIsoAt (f : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i]
    [QuasiIsoAt f i] : K.homology i ≅ L.homology i 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c} (f : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i] [QuasiIsoAt f i]
Goal: (isoOfQuasiIsoAt f i).hom = HomologicalComplex.homologyMap f i
-- isoOfQuasiIsoAt_hom_inv_id
@[reassoc (attr := simp)]
lemma isoOfQuasiIsoAt_hom_inv_id (f : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i]
    [QuasiIsoAt f i] :
    homologyMap f i ≫ (isoOfQuasiIsoAt f i).inv = 𝟙 _ 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c} (f : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i] [QuasiIsoAt f i]
Goal: HomologicalComplex.homologyMap f i ≫ (isoOfQuasiIsoAt f i).inv = 𝟙 (K.homology i)
-- isoOfQuasiIsoAt_inv_hom_id
@[reassoc (attr := simp)]
lemma isoOfQuasiIsoAt_inv_hom_id (f : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i]
    [QuasiIsoAt f i] :
    (isoOfQuasiIsoAt f i).inv ≫ homologyMap f i = 𝟙 _ 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c} (f : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i] [QuasiIsoAt f i]
Goal: (isoOfQuasiIsoAt f i).inv ≫ HomologicalComplex.homologyMap f i = 𝟙 (L.homology i)
-- CochainComplex.quasiIsoAt₀_iff
lemma CochainComplex.quasiIsoAt₀_iff {K L : CochainComplex C ℕ} (f : K ⟶ L)
    [K.HasHomology 0] [L.HasHomology 0] [(K.sc' 0 0 1).HasHomology] [(L.sc' 0 0 1).HasHomology] :
    QuasiIsoAt f 0 ↔
      ShortComplex.QuasiIso ((HomologicalComplex.shortComplexFunctor' C _ 0 0 1).map f) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K L : CochainComplex C ℕ} (f : K ⟶ L) [HomologicalComplex.HasHomology K 0] [HomologicalComplex.HasHomology L 0] [(HomologicalComplex.sc' K 0 0 1).HasHomology] [(HomologicalComplex.sc' L 0 0 1).HasHomology]
Goal: QuasiIsoAt f 0 ↔
  CategoryTheory.ShortComplex.QuasiIso ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).map f)
-- ChainComplex.quasiIsoAt₀_iff
lemma ChainComplex.quasiIsoAt₀_iff {K L : ChainComplex C ℕ} (f : K ⟶ L)
    [K.HasHomology 0] [L.HasHomology 0] [(K.sc' 1 0 0).HasHomology] [(L.sc' 1 0 0).HasHomology] :
    QuasiIsoAt f 0 ↔
      ShortComplex.QuasiIso ((HomologicalComplex.shortComplexFunctor' C _ 1 0 0).map f) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K L : ChainComplex C ℕ} (f : K ⟶ L) [HomologicalComplex.HasHomology K 0] [HomologicalComplex.HasHomology L 0] [(HomologicalComplex.sc' K 1 0 0).HasHomology] [(HomologicalComplex.sc' L 1 0 0).HasHomology]
Goal: QuasiIsoAt f 0 ↔
  CategoryTheory.ShortComplex.QuasiIso ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.down ℕ) 1 0 0).map f)
-- QuasiIso.quasiIsoAt
class QuasiIso (f : K ⟶ L) [∀ i, K.HasHomology i] [∀ i, L.HasHomology i] : Prop where
  quasiIsoAt : ∀ i, QuasiIsoAt f i := by infer_instance
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c} {f : K ⟶ L} [∀ (i : ι), K.HasHomology i] [∀ (i : ι), L.HasHomology i] [self : QuasiIso f] (i : ι)
Goal: QuasiIsoAt f i
-- quasiIso_iff
lemma quasiIso_iff (f : K ⟶ L) [∀ i, K.HasHomology i] [∀ i, L.HasHomology i] :
    QuasiIso f ↔ ∀ i, QuasiIsoAt f i 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c} (f : K ⟶ L) [∀ (i : ι), K.HasHomology i] [∀ (i : ι), L.HasHomology i]
Goal: QuasiIso f ↔ ∀ (i : ι), QuasiIsoAt f i
-- quasiIsoAt_of_comp_left
lemma quasiIsoAt_of_comp_left (φ : K ⟶ L) (φ' : L ⟶ M) (i : ι) [K.HasHomology i]
    [L.HasHomology i] [M.HasHomology i]
    [hφ : QuasiIsoAt φ i] [hφφ' : QuasiIsoAt (φ ≫ φ') i] :
    QuasiIsoAt φ' i 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L M : HomologicalComplex C c} (φ : K ⟶ L) (φ' : L ⟶ M) (i : ι) [K.HasHomology i] [L.HasHomology i] [M.HasHomology i] [hφ : QuasiIsoAt φ i] [hφφ' : QuasiIsoAt (φ ≫ φ') i]
Goal: QuasiIsoAt φ' i
-- quasiIsoAt_iff_comp_left
lemma quasiIsoAt_iff_comp_left (φ : K ⟶ L) (φ' : L ⟶ M) (i : ι) [K.HasHomology i]
    [L.HasHomology i] [M.HasHomology i]
    [hφ : QuasiIsoAt φ i] :
    QuasiIsoAt (φ ≫ φ') i ↔ QuasiIsoAt φ' i 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L M : HomologicalComplex C c} (φ : K ⟶ L) (φ' : L ⟶ M) (i : ι) [K.HasHomology i] [L.HasHomology i] [M.HasHomology i] [hφ : QuasiIsoAt φ i]
Goal: QuasiIsoAt (φ ≫ φ') i ↔ QuasiIsoAt φ' i
-- quasiIso_iff_comp_left
lemma quasiIso_iff_comp_left (φ : K ⟶ L) (φ' : L ⟶ M) [∀ i, K.HasHomology i]
    [∀ i, L.HasHomology i] [∀ i, M.HasHomology i]
    [hφ : QuasiIso φ] :
    QuasiIso (φ ≫ φ') ↔ QuasiIso φ' 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L M : HomologicalComplex C c} (φ : K ⟶ L) (φ' : L ⟶ M) [∀ (i : ι), K.HasHomology i] [∀ (i : ι), L.HasHomology i] [∀ (i : ι), M.HasHomology i] [hφ : QuasiIso φ]
Goal: QuasiIso (φ ≫ φ') ↔ QuasiIso φ'
-- quasiIso_of_comp_left
lemma quasiIso_of_comp_left (φ : K ⟶ L) (φ' : L ⟶ M) [∀ i, K.HasHomology i]
    [∀ i, L.HasHomology i] [∀ i, M.HasHomology i]
    [hφ : QuasiIso φ] [hφφ' : QuasiIso (φ ≫ φ')] :
    QuasiIso φ' 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L M : HomologicalComplex C c} (φ : K ⟶ L) (φ' : L ⟶ M) [∀ (i : ι), K.HasHomology i] [∀ (i : ι), L.HasHomology i] [∀ (i : ι), M.HasHomology i] [hφ : QuasiIso φ] [hφφ' : QuasiIso (φ ≫ φ')]
Goal: QuasiIso φ'
-- quasiIsoAt_of_comp_right
lemma quasiIsoAt_of_comp_right (φ : K ⟶ L) (φ' : L ⟶ M) (i : ι) [K.HasHomology i]
    [L.HasHomology i] [M.HasHomology i]
    [hφ' : QuasiIsoAt φ' i] [hφφ' : QuasiIsoAt (φ ≫ φ') i] :
    QuasiIsoAt φ i 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L M : HomologicalComplex C c} (φ : K ⟶ L) (φ' : L ⟶ M) (i : ι) [K.HasHomology i] [L.HasHomology i] [M.HasHomology i] [hφ' : QuasiIsoAt φ' i] [hφφ' : QuasiIsoAt (φ ≫ φ') i]
Goal: QuasiIsoAt φ i
-- quasiIsoAt_iff_comp_right
lemma quasiIsoAt_iff_comp_right (φ : K ⟶ L) (φ' : L ⟶ M) (i : ι) [K.HasHomology i]
    [L.HasHomology i] [M.HasHomology i]
    [hφ' : QuasiIsoAt φ' i] :
    QuasiIsoAt (φ ≫ φ') i ↔ QuasiIsoAt φ i 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L M : HomologicalComplex C c} (φ : K ⟶ L) (φ' : L ⟶ M) (i : ι) [K.HasHomology i] [L.HasHomology i] [M.HasHomology i] [hφ' : QuasiIsoAt φ' i]
Goal: QuasiIsoAt (φ ≫ φ') i ↔ QuasiIsoAt φ i
-- quasiIso_iff_comp_right
lemma quasiIso_iff_comp_right (φ : K ⟶ L) (φ' : L ⟶ M) [∀ i, K.HasHomology i]
    [∀ i, L.HasHomology i] [∀ i, M.HasHomology i]
    [hφ' : QuasiIso φ'] :
    QuasiIso (φ ≫ φ') ↔ QuasiIso φ 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L M : HomologicalComplex C c} (φ : K ⟶ L) (φ' : L ⟶ M) [∀ (i : ι), K.HasHomology i] [∀ (i : ι), L.HasHomology i] [∀ (i : ι), M.HasHomology i] [hφ' : QuasiIso φ']
Goal: QuasiIso (φ ≫ φ') ↔ QuasiIso φ
-- quasiIso_of_comp_right
lemma quasiIso_of_comp_right (φ : K ⟶ L) (φ' : L ⟶ M) [∀ i, K.HasHomology i]
    [∀ i, L.HasHomology i] [∀ i, M.HasHomology i]
    [hφ : QuasiIso φ'] [hφφ' : QuasiIso (φ ≫ φ')] :
    QuasiIso φ 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L M : HomologicalComplex C c} (φ : K ⟶ L) (φ' : L ⟶ M) [∀ (i : ι), K.HasHomology i] [∀ (i : ι), L.HasHomology i] [∀ (i : ι), M.HasHomology i] [hφ : QuasiIso φ'] [hφφ' : QuasiIso (φ ≫ φ')]
Goal: QuasiIso φ
-- quasiIso_iff_of_arrow_mk_iso
lemma quasiIso_iff_of_arrow_mk_iso (φ : K ⟶ L) (φ' : K' ⟶ L') (e : Arrow.mk φ ≅ Arrow.mk φ')
    [∀ i, K.HasHomology i] [∀ i, L.HasHomology i]
    [∀ i, K'.HasHomology i] [∀ i, L'.HasHomology i] :
    QuasiIso φ ↔ QuasiIso φ' 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L K' L' : HomologicalComplex C c} (φ : K ⟶ L) (φ' : K' ⟶ L') (e : CategoryTheory.Arrow.mk φ ≅ CategoryTheory.Arrow.mk φ') [∀ (i : ι), K.HasHomology i] [∀ (i : ι), L.HasHomology i] [∀ (i : ι), K'.HasHomology i] [∀ (i : ι), L'.HasHomology i]
Goal: QuasiIso φ ↔ QuasiIso φ'
-- quasiIso_of_arrow_mk_iso
lemma quasiIso_of_arrow_mk_iso (φ : K ⟶ L) (φ' : K' ⟶ L') (e : Arrow.mk φ ≅ Arrow.mk φ')
    [∀ i, K.HasHomology i] [∀ i, L.HasHomology i]
    [∀ i, K'.HasHomology i] [∀ i, L'.HasHomology i]
    [hφ : QuasiIso φ] : QuasiIso φ' 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L K' L' : HomologicalComplex C c} (φ : K ⟶ L) (φ' : K' ⟶ L') (e : CategoryTheory.Arrow.mk φ ≅ CategoryTheory.Arrow.mk φ') [∀ (i : ι), K.HasHomology i] [∀ (i : ι), L.HasHomology i] [∀ (i : ι), K'.HasHomology i] [∀ (i : ι), L'.HasHomology i] [hφ : QuasiIso φ]
Goal: QuasiIso φ'
-- HomologicalComplex.quasiIsoAt_map_iff_of_preservesHomology
lemma quasiIsoAt_map_iff_of_preservesHomology [F.ReflectsIsomorphisms] :
    QuasiIsoAt ((F.mapHomologicalComplex c).map φ) i ↔ QuasiIsoAt φ i 
Arguments: {ι : Type u_1} {c : ComplexShape ι} {C₁ : Type u_2} {C₂ : Type u_3} [CategoryTheory.Category.{u_4, u_2} C₁] [CategoryTheory.Category.{u_5, u_3} C₂] [CategoryTheory.Preadditive C₁] [CategoryTheory.Preadditive C₂] {K L : HomologicalComplex C₁ c} (φ : K ⟶ L) (F : C₁ ⥤ C₂) [F.Additive] [F.PreservesHomology] (i : ι) [K.HasHomology i] [L.HasHomology i] [((F.mapHomologicalComplex c).obj K).HasHomology i] [((F.mapHomologicalComplex c).obj L).HasHomology i] [F.ReflectsIsomorphisms]
Goal: QuasiIsoAt ((F.mapHomologicalComplex c).map φ) i ↔ QuasiIsoAt φ i
-- HomologicalComplex.quasiIso_map_iff_of_preservesHomology
lemma quasiIso_map_iff_of_preservesHomology [F.ReflectsIsomorphisms] :
    QuasiIso ((F.mapHomologicalComplex c).map φ) ↔ QuasiIso φ 
Arguments: {ι : Type u_1} {c : ComplexShape ι} {C₁ : Type u_2} {C₂ : Type u_3} [CategoryTheory.Category.{u_4, u_2} C₁] [CategoryTheory.Category.{u_5, u_3} C₂] [CategoryTheory.Preadditive C₁] [CategoryTheory.Preadditive C₂] {K L : HomologicalComplex C₁ c} (φ : K ⟶ L) (F : C₁ ⥤ C₂) [F.Additive] [F.PreservesHomology] [∀ (i : ι), K.HasHomology i] [∀ (i : ι), L.HasHomology i] [∀ (i : ι), ((F.mapHomologicalComplex c).obj K).HasHomology i] [∀ (i : ι), ((F.mapHomologicalComplex c).obj L).HasHomology i] [F.ReflectsIsomorphisms]
Goal: QuasiIso ((F.mapHomologicalComplex c).map φ) ↔ QuasiIso φ
-- HomologicalComplex.mem_quasiIso_iff
@[simp]
lemma mem_quasiIso_iff [CategoryWithHomology C] (f : K ⟶ L) : quasiIso C c f ↔ QuasiIso f 
Arguments: {ι : Type u_1} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c} [CategoryTheory.CategoryWithHomology C] (f : K ⟶ L)
Goal: HomologicalComplex.quasiIso C c f ↔ QuasiIso f
-- homotopyEquivalences_le_quasiIso
lemma homotopyEquivalences_le_quasiIso [CategoryWithHomology C] :
    homotopyEquivalences C c ≤ quasiIso C c 
Arguments: {ι : Type u_1} (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (c : ComplexShape ι) [CategoryTheory.CategoryWithHomology C]
Goal: HomologicalComplex.homotopyEquivalences C c ≤ HomologicalComplex.quasiIso C c
-- HomologicalComplex.HasHomotopyCofiber.hasBinaryBiproduct
class HasHomotopyCofiber (φ : F ⟶ G) : Prop where
  hasBinaryBiproduct (i j : ι) (hij : c.Rel i j) : HasBinaryBiproduct (F.X j) (G.X i)
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {F G : HomologicalComplex C c} (φ : F ⟶ G) [self : HomologicalComplex.HasHomotopyCofiber φ] (i j : ι) (hij : c.Rel i j)
Goal: CategoryTheory.Limits.HasBinaryBiproduct (F.X j) (G.X i)
-- HomologicalComplex.homotopyCofiber.inrX_sndX
@[reassoc (attr := simp)]
lemma inrX_sndX (i : ι) : inrX φ i ≫ sndX φ i = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {F G : HomologicalComplex C c} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] [DecidableRel c.Rel] (i : ι)
Goal: HomologicalComplex.homotopyCofiber.inrX φ i ≫ HomologicalComplex.homotopyCofiber.sndX φ i = 𝟙 (G.X i)
-- HomologicalComplex.homotopyCofiber.sndX_inrX
@[reassoc]
lemma sndX_inrX (i : ι) (hi : ¬ c.Rel i (c.next i)) :
    sndX φ i ≫ inrX φ i = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {F G : HomologicalComplex C c} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] [DecidableRel c.Rel] (i : ι) (hi : ¬c.Rel i (c.next i))
Goal: HomologicalComplex.homotopyCofiber.sndX φ i ≫ HomologicalComplex.homotopyCofiber.inrX φ i =
  𝟙 (HomologicalComplex.homotopyCofiber.X φ i)
-- HomologicalComplex.homotopyCofiber.inlX_fstX
@[reassoc (attr := simp)]
lemma inlX_fstX (i j : ι ) (hij : c.Rel j i) :
    inlX φ i j hij ≫ fstX φ j i hij = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {F G : HomologicalComplex C c} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] [DecidableRel c.Rel] (i j : ι) (hij : c.Rel j i)
Goal: HomologicalComplex.homotopyCofiber.inlX φ i j hij ≫ HomologicalComplex.homotopyCofiber.fstX φ j i hij = 𝟙 (F.X i)
-- HomologicalComplex.homotopyCofiber.inlX_sndX
@[reassoc (attr := simp)]
lemma inlX_sndX (i j : ι) (hij : c.Rel j i) :
    inlX φ i j hij ≫ sndX φ j = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {F G : HomologicalComplex C c} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] [DecidableRel c.Rel] (i j : ι) (hij : c.Rel j i)
Goal: HomologicalComplex.homotopyCofiber.inlX φ i j hij ≫ HomologicalComplex.homotopyCofiber.sndX φ j = 0
-- HomologicalComplex.homotopyCofiber.inrX_fstX
@[reassoc (attr := simp)]
lemma inrX_fstX (i j : ι) (hij : c.Rel i j) :
    inrX φ i ≫ fstX φ i j hij = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {F G : HomologicalComplex C c} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] [DecidableRel c.Rel] (i j : ι) (hij : c.Rel i j)
Goal: HomologicalComplex.homotopyCofiber.inrX φ i ≫ HomologicalComplex.homotopyCofiber.fstX φ i j hij = 0
-- HomologicalComplex.homotopyCofiber.ext_to_X
lemma ext_to_X (i j : ι) (hij : c.Rel i j) {A : C} {f g : A ⟶ X φ i}
    (h₁ : f ≫ fstX φ i j hij = g ≫ fstX φ i j hij) (h₂ : f ≫ sndX φ i = g ≫ sndX φ i) :
    f = g 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {F G : HomologicalComplex C c} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] [DecidableRel c.Rel] (i j : ι) (hij : c.Rel i j) {A : C} {f g : A ⟶ HomologicalComplex.homotopyCofiber.X φ i} (h₁ : f ≫ HomologicalComplex.homotopyCofiber.fstX φ i j hij = g ≫ HomologicalComplex.homotopyCofiber.fstX φ i j hij) (h₂ : f ≫ HomologicalComplex.homotopyCofiber.sndX φ i = g ≫ HomologicalComplex.homotopyCofiber.sndX φ i)
Goal: f = g
-- HomologicalComplex.homotopyCofiber.ext_to_X'
lemma ext_to_X' (i : ι) (hi : ¬ c.Rel i (c.next i)) {A : C} {f g : A ⟶ X φ i}
    (h : f ≫ sndX φ i = g ≫ sndX φ i) : f = g 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {F G : HomologicalComplex C c} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] [DecidableRel c.Rel] (i : ι) (hi : ¬c.Rel i (c.next i)) {A : C} {f g : A ⟶ HomologicalComplex.homotopyCofiber.X φ i} (h : f ≫ HomologicalComplex.homotopyCofiber.sndX φ i = g ≫ HomologicalComplex.homotopyCofiber.sndX φ i)
Goal: f = g
-- HomologicalComplex.homotopyCofiber.ext_from_X
lemma ext_from_X (i j : ι) (hij : c.Rel j i) {A : C} {f g : X φ j ⟶ A}
    (h₁ : inlX φ i j hij ≫ f = inlX φ i j hij ≫ g) (h₂ : inrX φ j ≫ f = inrX φ j ≫ g) :
    f = g 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {F G : HomologicalComplex C c} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] [DecidableRel c.Rel] (i j : ι) (hij : c.Rel j i) {A : C} {f g : HomologicalComplex.homotopyCofiber.X φ j ⟶ A} (h₁ : HomologicalComplex.homotopyCofiber.inlX φ i j hij ≫ f = HomologicalComplex.homotopyCofiber.inlX φ i j hij ≫ g) (h₂ : HomologicalComplex.homotopyCofiber.inrX φ j ≫ f = HomologicalComplex.homotopyCofiber.inrX φ j ≫ g)
Goal: f = g