mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.ModelCategory.PathObject

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
A : C
inst✝⁴ : CategoryWithWeakEquivalences C
P : PathObject A
inst✝³ : HasBinaryProducts C
inst✝² : CategoryWithFibrations C
inst✝¹ : P.IsGood
inst✝ : (fibrations C).RespectsIso
hp : fibrations C P.p
⊢ Fibration P.symm.p

Tactic:
rw [P.symm_p, fibration_iff]

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
A : C
inst✝⁴ : CategoryWithWeakEquivalences C
P : PathObject A
inst✝³ : HasBinaryProducts C
inst✝² : CategoryWithFibrations C
inst✝¹ : P.IsGood
inst✝ : (fibrations C).RespectsIso
hp : fibrations C P.p
⊢ fibrations C (P.p ≫ (prod.braiding A A).hom)
Module: Mathlib.AlgebraicTopology.ModelCategory.Basic

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : CategoryWithFibrations C
inst✝⁶ : CategoryWithCofibrations C
inst✝⁵ : CategoryWithWeakEquivalences C
inst✝⁴ : HasFiniteLimits C
inst✝³ : HasFiniteColimits C
inst✝² : (weakEquivalences C).HasTwoOutOfThreeProperty
inst✝¹ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
inst✝ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)
⊢ (fibrations C).IsStableUnderRetracts

Tactic:
← rlp_eq_of_wfs (trivialCofibrations C) (fibrations C)

Goal state after:
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : CategoryWithFibrations C
inst✝⁶ : CategoryWithCofibrations C
inst✝⁵ : CategoryWithWeakEquivalences C
inst✝⁴ : HasFiniteLimits C
inst✝³ : HasFiniteColimits C
inst✝² : (weakEquivalences C).HasTwoOutOfThreeProperty
inst✝¹ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
inst✝ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)
⊢ (trivialCofibrations C).rlp.IsStableUnderRetracts
Module: Mathlib.Algebra.Homology.HomotopyCategory.Plus

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Preadditive C
inst✝⁴ : Preadditive D
A : Type u_3
inst✝³ : Category.{v_3, u_3} A
inst✝² : Abelian A
inst✝¹ : HasZeroObject C
inst✝ : HasBinaryBiproducts C
n : ℤ
X : C
⊢ CochainComplex.plus C ((CochainComplex.singleFunctor C n).obj X)

Tactic:
exact ⟨n, inferInstance⟩

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLEGT

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
Y X : C
f₁ f₂ : Y ⟶ (t.truncLE n).obj X
h : f₁ ≫ (t.truncLEι n).app X = f₂ ≫ (t.truncLEι n).app X
inst✝ : t.IsLE Y n
⊢ f₁ = f₂

Tactic:
by
  have : t.IsLE Y (n + 1 - 1) := by simpa
  rw [← cancel_mono ((t.truncLEIsoTruncLT n (n + 1) rfl).hom.app _)]
  exact t.to_truncLT_obj_ext (by simpa)

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: C : Type u_1
inst✝⁸ : Category.{v_1, u_1} C
A B : ObjectProperty C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : A.IsTriangulated
inst✝¹ : B.IsTriangulated
inst✝ : B.IsClosedUnderIsomorphisms
⊢ A.IsVerdierLeftLocalizing B ↔ ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A Y → B.trW s → ∃ Z s' a, A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s'

Tactic:
rw [← isVerdierRightLocalizing_op_iff, isVerdierRightLocalizing_iff]

Goal state after:
C : Type u_1
inst✝⁸ : Category.{v_1, u_1} C
A B : ObjectProperty C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : A.IsTriangulated
inst✝¹ : B.IsTriangulated
inst✝ : B.IsClosedUnderIsomorphisms
⊢ (∀ ⦃X Y : Cᵒᵖ⦄ (s : X ⟶ Y), A.op X → B.op.trW s → ∃ Z s' b, A.op Z ∧ (A.op ⊓ B.op).trW s' ∧ s ≫ b = s') ↔
    ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A Y → B.trW s → ∃ Z s' a, A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s'
Module: Mathlib.Algebra.Homology.DerivedCategory.Plus

Goal state before:
pp: case refine_1
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
K : CochainComplex C ℤ
hK : HomologicalComplex.Acyclic K
L : CochainComplex C ℤ
n : ℤ
hn : L.IsStrictlyGE n
φ : K ⟶ L
⊢ CochainComplex.plus C (K.truncGE n)

Tactic:
exact ⟨n, inferInstance⟩

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n₀ : ℤ
X : C
x✝ : t.IsLE X n₀
e : contractibleTriangle X ≅ (t.triangleLTGE (n₀ + 1)).obj X
he : e.hom.hom₂ = (Iso.refl X).hom
⊢ e.inv.hom₂ = 𝟙 X

Tactic:
rw [← cancel_mono e.hom.hom₂, ← comp_hom₂, e.inv_hom_id, he]

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n₀ : ℤ
X : C
x✝ : t.IsLE X n₀
e : contractibleTriangle X ≅ (t.triangleLTGE (n₀ + 1)).obj X
he : e.hom.hom₂ = (Iso.refl X).hom
⊢ (𝟙 ((t.triangleLTGE (n₀ + 1)).obj X)).hom₂ = 𝟙 X ≫ (Iso.refl X).hom
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
X Y : C
f₁ f₂ : (t.truncGE n).obj X ⟶ Y
h : (t.truncGEπ n).app X ≫ f₁ = (t.truncGEπ n).app X ≫ f₂
inst✝ : t.IsGE Y n
f : (t.truncGE n).obj X ⟶ Y
hf : (t.truncGEπ n).app X ≫ f = 0
g : (shiftFunctor C 1).obj ((t.triangleLTGE n).obj X).obj₁ ⟶ Y
hg : f = ((t.triangleLTGE n).obj X).mor₃ ≫ g
hg' : g = 0
⊢ ((t.triangleLTGE n).obj X).mor₃ ≫ 0 = 0

Tactic:
comp_zero

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
X Y : C
f₁ f₂ : (t.truncGE n).obj X ⟶ Y
h : (t.truncGEπ n).app X ≫ f₁ = (t.truncGEπ n).app X ≫ f₂
inst✝ : t.IsGE Y n
f : (t.truncGE n).obj X ⟶ Y
hf : (t.truncGEπ n).app X ≫ f = 0
g : (shiftFunctor C 1).obj ((t.triangleLTGE n).obj X).obj₁ ⟶ Y
hg : f = ((t.triangleLTGE n).obj X).mor₃ ≫ g
hg' : g = 0
⊢ 0 = 0
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X' Y' X Y : C
f' : X' ⟶ Y'
f : X ⟶ Y
r : RetractArrow f' f
left✝ : Epi f
hf : Injective (kernel f)
⊢ epiWithInjectiveKernel f'

Tactic:
have : Epi f' :=
      (MorphismProperty.epimorphisms C).of_retract r (.infer_property _)

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X' Y' X Y : C
f' : X' ⟶ Y'
f : X ⟶ Y
r : RetractArrow f' f
left✝ : Epi f
hf : Injective (kernel f)
this : Epi f'
⊢ epiWithInjectiveKernel f'
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
T : Triangle C
hT : T ∈ distinguishedTriangles
n : ℤ
h₁ : t.IsLE T.obj₁ n
h₃ : t.IsLE T.obj₃ n
Y : C
f : T.obj₂ ⟶ Y
hY : t.IsGE Y (n + 1)
f' : T.obj₃ ⟶ Y
hf' : f = T.mor₂ ≫ f'
⊢ T.mor₂ ≫ 0 = 0

Tactic:
comp_zero

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
T : Triangle C
hT : T ∈ distinguishedTriangles
n : ℤ
h₁ : t.IsLE T.obj₁ n
h₃ : t.IsLE T.obj₃ n
Y : C
f : T.obj₂ ⟶ Y
hY : t.IsGE Y (n + 1)
f' : T.obj₃ ⟶ Y
hf' : f = T.mor₂ ≫ f'
⊢ 0 = 0
Module: Mathlib.Algebra.Homology.ModelCategory.Lifting

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
A B X Y : CochainComplex C ℤ
t : A ⟶ X
i : A ⟶ B
p : X ⟶ Y
b : B ⟶ Y
sq : CommSq t i p b
hsq : (n : ℤ) → ⋯.LiftStruct
Q : CochainComplex C ℤ
π : B ⟶ Q
hπ : i ≫ π = 0
hQ : IsColimit (CokernelCofork.ofπ π hπ)
K : CochainComplex C ℤ
ι : K ⟶ X
hι : ι ≫ p = 0
hK : IsLimit (KernelFork.ofι ι hι)
this✝¹ : Epi π
this✝ : Mono ι
n : ℤ
this :
  (↑(cocycle₁' sq hsq)).v n (n + 1) ⋯ ≫ X.d (n + 1) (n + 2) +
      B.d n (n + 1) ≫ (↑(cocycle₁' sq hsq)).v (n + 1) (n + 2) ⋯ =
    0
⊢ (cochain₁ sq hsq hQ hK).v n (n + 1) ⋯ ≫ K.d (n + 1) (n + 2) +
      Q.d n (n + 1) ≫ (cochain₁ sq hsq hQ hK).v (n + 1) (n + 2) ⋯ =
    0

Tactic:
rwa [← cancel_mono (ι.f (n + 2)), ← cancel_epi (π.f n),
      Preadditive.add_comp, Category.assoc, Category.assoc, Preadditive.comp_add,
      HomologicalComplex.Hom.comm_assoc,
      π_f_cochain₁_v_ι_f, zero_comp, comp_zero, ← ι.comm,
      π_f_cochain₁_v_ι_f_assoc]

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n₀✝ n₁ : ℤ
hn₁ : n₀✝ + 1 = n₁
hf : ∀ i ≤ n₀✝, QuasiIsoAt f i
n₀ : ℤ := n₁ - 1
A : C
x₁ : A ⟶ K.X n₁
x✝ : x₁ ≫ K.d n₁ (n₁ + 1) = 0
y₀ : A ⟶ L.X n₀
h₁ : x₁ ≫ (i K n₁).f n₁ = 0
h₂ : x₁ ≫ f.f n₁ = y₀ ≫ L.d n₀ n₁
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁_1, π ≫ x₁ = x₁_1 ≫ K.d n₀ n₁

Tactic:
replace h₁ : x₁ ≫ K.pOpcycles n₁ = 0 := by
    rw [← cancel_mono (Injective.ι _)]
    simpa [i, ← cancel_mono (singleObjXSelf (.up ℤ) n₁ _).hom] using h₁

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n₀✝ n₁ : ℤ
hn₁ : n₀✝ + 1 = n₁
hf : ∀ i ≤ n₀✝, QuasiIsoAt f i
n₀ : ℤ := n₁ - 1
A : C
x₁ : A ⟶ K.X n₁
x✝ : x₁ ≫ K.d n₁ (n₁ + 1) = 0
y₀ : A ⟶ L.X n₀
h₂ : x₁ ≫ f.f n₁ = y₀ ≫ L.d n₀ n₁
h₁ : x₁ ≫ pOpcycles K n₁ = 0
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁_1, π ≫ x₁ = x₁_1 ≫ K.d n₀ n₁
Module: Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : ModelCategory C
X Y : C
f : X ⟶ Y
⊢ WeakEquivalence (f ≫ iResolutionObj Y) ↔ WeakEquivalence f

Tactic:
weakEquivalence_postcomp_iff

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : ModelCategory C
X Y : C
f : X ⟶ Y
⊢ WeakEquivalence f ↔ WeakEquivalence f
Module: Mathlib.CategoryTheory.Localization.Resolution

Goal state before:
pp: case mpr
C₁ : Type u_1
C₂ : Type u_2
D₂ : Type u_4
H : Type u_5
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_4, u_4} D₂
inst✝² : Category.{v_5, u_5} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
inst✝ : Φ.HasRightResolutions
F G : D₂ ⥤ H
α : F ⟶ G
⊢ (∀ (X₁ : C₁), IsIso (α.app (L₂.obj (Φ.functor.obj X₁)))) → IsIso α

Tactic:
· intro hα
    have : ∀ (X₂ : D₂), IsIso (α.app X₂) := fun X₂ => by
      have := Φ.essSurj_of_hasRightResolutions L₂
      rw [← NatTrans.isIso_app_iff_of_iso α ((Φ.functor ⋙ L₂).objObjPreimageIso X₂)]
      apply hα
    exact NatIso.isIso_of_isIso_app α

Goal state after:
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
inst✝⁶ : Category.{v_1, u_1} C₁
inst✝⁵ : Category.{v_2, u_2} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
inst✝⁴ : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution X₂)
inst✝³ : Φ.arrow.HasRightResolutions
inst✝² : W₂.ContainsIdentities
D : Type u_3
inst✝¹ : Category.{v_3, u_3} D
L : C₂ ⥤ D
inst✝ : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
R R' : Φ.RightResolution X₂
φ : R ⟶ R'
⊢ L.map R'.w ≫ (isoOfHom L W₂ R'.w ⋯).inv ≫ y = y

Tactic:
isoOfHom_hom_inv_id_assoc

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
inst✝⁶ : Category.{v_1, u_1} C₁
inst✝⁵ : Category.{v_2, u_2} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
inst✝⁴ : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution X₂)
inst✝³ : Φ.arrow.HasRightResolutions
inst✝² : W₂.ContainsIdentities
D : Type u_3
inst✝¹ : Category.{v_3, u_3} D
L : C₂ ⥤ D
inst✝ : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
R R' : Φ.RightResolution X₂
φ : R ⟶ R'
⊢ y = y
Module: Mathlib.Algebra.Homology.DerivedCategory.DerivabilityStructureInjectives

Goal state before:
pp: C : Type u_1
H : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} H
⊢ (L C).IsLocalizedEquivalence

Tactic:
by
  have :
      ((L C).functor ⋙ 𝟭 (Plus (InjectiveObject C))).IsLocalization
        ((CochainComplex.Plus.quasiIso C).inverseImage
          (InjectiveObject.ι C).mapCochainComplexPlus) :=
    inferInstanceAs ((HomotopyCategory.Plus.quotient (InjectiveObject C)).IsLocalization _)
  exact LocalizerMorphism.IsLocalizedEquivalence.of_isLocalization_of_isLocalization (L C) (𝟭 _)

Goal state after:
Module: Mathlib.Algebra.Homology.Double

Goal state before:
pp: case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
X₀ X₁ : C
f : X₀ ⟶ X₁
ι : Type u_2
c : ComplexShape ι
i j k : ι
hij : c.Rel i j
hjk : c.Rel j k
hi₀₁ : c.Rel i j
⊢ (if hk : i = i ∧ j = j ∧ ¬i = j then eqToHom ⋯ ≫ f ≫ eqToHom ⋯ else 0) ≫ 0 = 0

Tactic:
comp_zero

Goal state after:
case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
X₀ X₁ : C
f : X₀ ⟶ X₁
ι : Type u_2
c : ComplexShape ι
i j k : ι
hij : c.Rel i j
hjk : c.Rel j k
hi₀₁ : c.Rel i j
⊢ 0 = 0
Module: Mathlib.Algebra.Homology.EulerCharacteristic

Goal state before:
pp: case h
R : Type u_1
inst✝¹ : Ring R
ι : Type u_2
c : ComplexShape ι
inst✝ : c.EulerCharSigns
X : GradedObject ι (ModuleCat R)
indices : Finset ι
h_support : finrankSupport X ⊆ ↑indices
⊢ (Function.support fun i ↦ ↑(c.χ i) * ↑(Module.finrank R ↑(X i))) ⊆ ↑indices

Tactic:
exact (support_eulerChar_summand c X).symm ▸ h_support

Goal state after:
Module: Mathlib.CategoryTheory.DifferentialObject

Goal state before:
pp: S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X✝ Y✝ Z✝ : DifferentialObject S C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ autoParam
    ((CategoryTheory.shiftFunctor C n).map (f.f ≫ g.f) =
      (CategoryTheory.shiftFunctor C n).map f.f ≫ (CategoryTheory.shiftFunctor C n).map g.f)
    ext._auto_1

Tactic:
rw [Functor.map_comp]

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: case neg
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
n m p : ℤ
hnm : n + 1 = m
hmp : ¬m + 1 = p
⊢ h.d n m ≫ 0 = 0

Tactic:
comp_zero

Goal state after:
case neg
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
n m p : ℤ
hnm : n + 1 = m
hmp : ¬m + 1 = p
⊢ 0 = 0
Module: Mathlib.Algebra.Homology.Functor

Goal state before:
pp: T : Type u_1
inst✝² : Category.{v_1, u_1} T
V : Type u_2
inst✝¹ : Category.{v_2, u_2} V
inst✝ : HasZeroMorphisms V
ι : Type u_3
c : ComplexShape ι
C : HomologicalComplex (T ⥤ V) c
X✝ Y✝ Z✝ : T
h₁ : X✝ ⟶ Y✝
h₂ : Y✝ ⟶ Z✝
i : ι
⊢ (C.X i).map (h₁ ≫ h₂) = (C.X i).map h₁ ≫ (C.X i).map h₂

Tactic:
rw [Functor.map_comp]

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Projective.Resolution

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
Z : C
P : ProjectiveResolution Z
⊢ P.π.f 1 ≫ 0 = 0

Tactic:
comp_zero

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
Z : C
P : ProjectiveResolution Z
⊢ 0 = 0
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
q n : ℤ
α : Cocycle K ((singleFunctor C q).obj X) n
p : ℤ
h : p + n = q
p' : ℤ
hp' : p' + 1 = p
f : K.X p ⟶ X
hf : Cochain.toSingleMk f h = ↑α
hα : Cochain.toSingleMk (K.d p' p ≫ f) ⋯ = 0
⊢ toSingleMk f h p' hp' ⋯ = α

Tactic:
· ext : 1; assumption

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Projective.Resolution

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughProjectives C
Z : C
n : ℕ
⊢ HomologicalComplex.ExactAt (ofComplex Z) (n + 1)

Tactic:
rw [HomologicalComplex.exactAt_iff' _ (n + 1 + 1) (n + 1) n (by simp) (by simp)]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughProjectives C
Z : C
n : ℕ
⊢ (HomologicalComplex.sc' (ofComplex Z) (n + 1 + 1) (n + 1) n).Exact
Module: Mathlib.CategoryTheory.Abelian.Projective.Resolution

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughProjectives C
Z : C
⊢ Projective.π (kernel (Projective.π Z)) ≫ 0 = 0

Tactic:
comp_zero

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughProjectives C
Z : C
⊢ 0 = 0
Module: Mathlib.CategoryTheory.GradedObject.Unitor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D : Type u_4
I₁ : Type u_5
I₂ : Type u_6
I₃ : Type u_7
J : Type u_8
inst✝¹⁵ : Category.{v_1, u_1} C₁
inst✝¹⁴ : Category.{v_2, u_2} C₂
inst✝¹³ : Category.{v_3, u_3} C₃
inst✝¹² : Category.{v_4, u_4} D
inst✝¹¹ : Zero I₂
inst✝¹⁰ : DecidableEq I₂
inst✝⁹ : HasInitial C₂
F₁ : C₁ ⥤ C₂ ⥤ C₁
F₂ : C₂ ⥤ C₃ ⥤ C₃
G : C₁ ⥤ C₃ ⥤ D
associator : bifunctorComp₁₂ F₁ G ≅ bifunctorComp₂₃ G F₂
X₂ : C₂
e₁ : F₁.flip.obj X₂ ≅ 𝟭 C₁
e₂ : F₂.obj X₂ ≅ 𝟭 C₃
inst✝⁸ : ∀ (X₁ : C₁), PreservesColimit (Functor.empty C₂) (F₁.obj X₁)
inst✝⁷ : ∀ (X₃ : C₃), PreservesColimit (Functor.empty C₂) (F₂.flip.obj X₃)
r : I₁ × I₂ × I₃ → J
π : I₁ × I₃ → J
τ : TriangleIndexData r π
X₁ : GradedObject I₁ C₁
X₃ : GradedObject I₃ C₃
inst✝⁶ : (((mapBifunctor F₁ I₁ I₂).obj X₁).obj ((single₀ I₂).obj X₂)).HasMap τ.p₁₂
inst✝⁵ : (((mapBifunctor G I₁ I₃).obj (mapBifunctorMapObj F₁ τ.p₁₂ X₁ ((single₀ I₂).obj X₂))).obj X₃).HasMap π
inst✝⁴ : (((mapBifunctor F₂ I₂ I₃).obj ((single₀ I₂).obj X₂)).obj X₃).HasMap τ.p₂₃
inst✝³ : (((mapBifunctor G I₁ I₃).obj X₁).obj (mapBifunctorMapObj F₂ τ.p₂₃ ((single₀ I₂).obj X₂) X₃)).HasMap π
inst✝² : HasGoodTrifunctor₁₂Obj F₁ G τ.ρ₁₂ X₁ ((single₀ I₂).obj X₂) X₃
inst✝¹ : HasGoodTrifunctor₂₃Obj G F₂ τ.ρ₂₃ X₁ ((single₀ I₂).obj X₂) X₃
inst✝ : (((mapBifunctor G I₁ I₃).obj X₁).obj X₃).HasMap π
triangle :
  ∀ (X₁ : C₁) (X₃ : C₃),
    ((associator.hom.app X₁).app X₂).app X₃ ≫ (G.obj X₁).map (e₂.hom.app X₃) = (G.map (e₁.hom.app X₁)).app X₃
j : J
i₁ : I₁
i₃ : I₃
hj : π (i₁, i₃) = j
⊢ r (i₁, 0, i₃) = j

Tactic:
by rw [τ.r_zero, hj]

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Cycles

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k : ι
f : i ⟶ j
g : j ⟶ k
i' j' k' : ι
f' : i' ⟶ j'
g' : j' ⟶ k'
i'' j'' k'' : ι
f'' : i'' ⟶ j''
g'' : j'' ⟶ k''
α : mk₂ f g ⟶ mk₂ f' g'
n : ℤ
⊢ (X.H (n - 1)).map (homMk₁ (α.app 1) (α.app 2) ⋯) ≫ 0 = 0

Tactic:
comp_zero

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k : ι
f : i ⟶ j
g : j ⟶ k
i' j' k' : ι
f' : i' ⟶ j'
g' : j' ⟶ k'
i'' j'' k'' : ι
f'' : i'' ⟶ j''
g'' : j'' ⟶ k''
α : mk₂ f g ⟶ mk₂ f' g'
n : ℤ
⊢ 0 = 0
Module: Mathlib.Algebra.Homology.SpectralObject.Differentials

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ X.iCycles f₂ f₃ n₀ ≫ 0 = 0

Tactic:
comp_zero

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ 0 = 0
Module: Mathlib.Algebra.Homology.SpectralObject.Differentials

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
A✝ : C
z₀ :
  A✝ ⟶
    { X₁ := X.cycles f₂ f₃ n₀, X₂ := X.opcycles f₁ f₂ n₁, X₃ := X.opcycles f₁ f₂₃ n₁, f := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁,
        g := X.opcyclesMap f₁ f₂ f₁ f₂₃ (threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) n₁, zero := ⋯ }.X₂
hz₀ :
  z₀ ≫
      { X₁ := X.cycles f₂ f₃ n₀, X₂ := X.opcycles f₁ f₂ n₁, X₃ := X.opcycles f₁ f₂₃ n₁, f := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁,
          g := X.opcyclesMap f₁ f₂ f₁ f₂₃ (threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) n₁, zero := ⋯ }.g =
    0
A₁ : C
π₁ : A₁ ⟶ A✝
w✝ : Epi π₁
z₁ : A₁ ⟶ (X.H n₁).obj (mk₁ f₁)
hz₁ : π₁ ≫ z₀ = z₁ ≫ X.pOpcycles f₁ f₂ n₁
H : X.pOpcycles f₁ f₂ n₁ ≫ X.opcyclesMap f₁ f₂ f₁ f₂₃ (threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) n₁ = X.pOpcycles f₁ f₂₃ n₁
⊢ π₁ ≫ 0 = 0

Tactic:
comp_zero

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
A✝ : C
z₀ :
  A✝ ⟶
    { X₁ := X.cycles f₂ f₃ n₀, X₂ := X.opcycles f₁ f₂ n₁, X₃ := X.opcycles f₁ f₂₃ n₁, f := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁,
        g := X.opcyclesMap f₁ f₂ f₁ f₂₃ (threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) n₁, zero := ⋯ }.X₂
hz₀ :
  z₀ ≫
      { X₁ := X.cycles f₂ f₃ n₀, X₂ := X.opcycles f₁ f₂ n₁, X₃ := X.opcycles f₁ f₂₃ n₁, f := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁,
          g := X.opcyclesMap f₁ f₂ f₁ f₂₃ (threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) n₁, zero := ⋯ }.g =
    0
A₁ : C
π₁ : A₁ ⟶ A✝
w✝ : Epi π₁
z₁ : A₁ ⟶ (X.H n₁).obj (mk₁ f₁)
hz₁ : π₁ ≫ z₀ = z₁ ≫ X.pOpcycles f₁ f₂ n₁
H : X.pOpcycles f₁ f₂ n₁ ≫ X.opcyclesMap f₁ f₂ f₁ f₂₃ (threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) n₁ = X.pOpcycles f₁ f₂₃ n₁
⊢ 0 = 0
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: case pos
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : (c r).Rel pq' pq''
⊢ (pageXIso X data r ⋯ pq (data.i₀ r pq ⋯) (data.i₁ pq) (data.i₂ pq) (data.i₃ r pq ⋯) ⋯ ⋯ ⋯ ⋯ (data.deg pq - 1)
          (data.deg pq) (data.deg pq + 1) ⋯ ⋯ ⋯).hom ≫
      0 =
    0

Tactic:
comp_zero

Goal state after:
case pos
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : (c r).Rel pq' pq''
⊢ 0 = 0
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: case neg
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : ¬(c r).Rel pq' pq''
⊢ (if hpq : (c r).Rel pq pq' then
        X.d (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (data.deg pq - 1) (data.deg pq)
            (data.deg pq + 1) (data.deg pq + 2) ⋯ ⋯ ⋯ ≫
          (pageXIso X data r hr pq' (data.i₀ r pq' ⋯) (data.i₁ pq') (data.i₀ r pq ⋯) (data.i₁ pq) ⋯ ⋯ ⋯ ⋯ (data.deg pq)
              (data.deg pq + 1) (data.deg pq + 2) ⋯ ⋯ ⋯).inv
      else 0) ≫
      0 =
    0

Tactic:
comp_zero

Goal state after:
case neg
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : ¬(c r).Rel pq' pq''
⊢ 0 = 0