mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.GuitartExact.Opposite

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
inst✝¹ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
inst✝ : w.GuitartExact
X₃ : C₃ᵒᵖ
X₂ : C₂ᵒᵖ
g : B.op.obj X₃ ⟶ R.op.obj X₂
⊢ IsConnected (w.op.StructuredArrowRightwards g)

Tactic:
rw [← isConnected_op_iff_isConnected,
    isConnected_iff_of_equivalence (w.structuredArrowRightwardsOpEquivalence g)]

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
inst✝¹ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
inst✝ : w.GuitartExact
X₃ : C₃ᵒᵖ
X₂ : C₂ᵒᵖ
g : B.op.obj X₃ ⟶ R.op.obj X₂
⊢ IsConnected (w.CostructuredArrowDownwards g.unop)
Module: Mathlib.CategoryTheory.GuitartExact.Basic

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
X₂ : C₂
X₃ : C₃
g : R.obj X₂ ⟶ B.obj X₃
f₁ f₂ : w.StructuredArrowRightwards g
φ : f₁ ⟶ f₂
⊢ StructuredArrow.Hom.right
      ((w.structuredArrowDownwards X₂).map (StructuredArrow.homMk (StructuredArrow.Hom.right φ).left ⋯) ≫
        (CostructuredArrow.mk (StructuredArrow.homMk (StructuredArrow.right f₂).hom ⋯)).hom) =
    StructuredArrow.Hom.right (CostructuredArrow.mk (StructuredArrow.homMk (StructuredArrow.right f₁).hom ⋯)).hom

Tactic:
exact CostructuredArrow.w φ.right

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

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
inst✝¹ : Category.{v₁, u₁} C₁
inst✝ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
F : W₁.Localization ⥤ W₂.Localization := Φ.localizedFunctor W₁.Q W₂.Q
e : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ F := CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)
⊢ Φ.IsLeftDerivabilityStructure ↔ Φ.op.IsRightDerivabilityStructure

Tactic:
let e' : Φ.functor.op ⋙ W₂.Q.op ≅ W₁.Q.op ⋙ F.op := NatIso.op e.symm

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
inst✝¹ : Category.{v₁, u₁} C₁
inst✝ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
F : W₁.Localization ⥤ W₂.Localization := Φ.localizedFunctor W₁.Q W₂.Q
e : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ F := CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)
e' : Φ.functor.op ⋙ W₂.Q.op ≅ W₁.Q.op ⋙ F.op := NatIso.op e.symm
⊢ Φ.IsLeftDerivabilityStructure ↔ Φ.op.IsRightDerivabilityStructure
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.Basic

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
inst✝¹ : Category.{v₁, u₁} C₁
inst✝ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
F : W₁.Localization ⥤ W₂.Localization := Φ.localizedFunctor W₁.Q W₂.Q
e : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ F := CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)
⊢ Φ.IsRightDerivabilityStructure ↔ Φ.op.IsLeftDerivabilityStructure

Tactic:
let e' : Φ.functor.op ⋙ W₂.Q.op ≅ W₁.Q.op ⋙ F.op := NatIso.op e.symm

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
inst✝¹ : Category.{v₁, u₁} C₁
inst✝ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
F : W₁.Localization ⥤ W₂.Localization := Φ.localizedFunctor W₁.Q W₂.Q
e : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ F := CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)
e' : Φ.functor.op ⋙ W₂.Q.op ≅ W₁.Q.op ⋙ F.op := NatIso.op e.symm
⊢ Φ.IsRightDerivabilityStructure ↔ Φ.op.IsLeftDerivabilityStructure
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝² : EnoughInjectives C
n : ℤ
inst✝¹ : K.IsStrictlyGE (n + 1)
inst✝ : L.IsStrictlyGE n
this : K.IsStrictlyGE n
⊢ ∃ K', ∃ (_ : K'.IsStrictlyGE n), ∃ ι π, Mono ι ∧ QuasiIso ι ∧ degreewiseEpiWithInjectiveKernel π ∧ ι ≫ π = f

Tactic:
obtain ⟨L', _, i, p, _, hp, _, rfl⟩ := cm5b f n

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
K L : CochainComplex C ℤ
inst✝² : EnoughInjectives C
n : ℤ
inst✝¹ : K.IsStrictlyGE (n + 1)
inst✝ : L.IsStrictlyGE n
this : K.IsStrictlyGE n
L' : CochainComplex C ℤ
w✝² : L'.IsStrictlyGE n
i : K ⟶ L'
p : L' ⟶ L
w✝¹ : Mono i
hp : degreewiseEpiWithInjectiveKernel p
w✝ : QuasiIso p
⊢ ∃ K', ∃ (_ : K'.IsStrictlyGE n), ∃ ι π, Mono ι ∧ QuasiIso ι ∧ degreewiseEpiWithInjectiveKernel π ∧ ι ≫ π = i ≫ p
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.OfLocalizedEquivalences

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝⁹ : Category.{v_1, u_1} C₁
inst✝⁸ : Category.{v_2, u_2} C₂
inst✝⁷ : Category.{v_3, u_3} D₁
inst✝⁶ : Category.{v_4, u_4} D₂
W₁ : MorphismProperty C₁
W₁' : MorphismProperty D₁
W₂ : MorphismProperty C₂
W₂' : MorphismProperty D₂
T : LocalizerMorphism W₁ W₂
L : LocalizerMorphism W₁ W₁'
R : LocalizerMorphism W₂ W₂'
B : LocalizerMorphism W₁' W₂'
inst✝⁵ : W₂'.RespectsIso
inst✝⁴ : L.IsLocalizedEquivalence
inst✝³ : R.IsLocalizedEquivalence
inst✝² : R.functor.EssSurj
inst✝¹ : T.IsLeftDerivabilityStructure
iso : T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor
inst✝ : TwoSquare.GuitartExact iso.inv
Y₂ : D₂
⊢ Nonempty (B.LeftResolution Y₂)

Tactic:
obtain ⟨X₂, ⟨e₂⟩⟩ := Functor.EssSurj.mem_essImage R.functor Y₂

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝⁹ : Category.{v_1, u_1} C₁
inst✝⁸ : Category.{v_2, u_2} C₂
inst✝⁷ : Category.{v_3, u_3} D₁
inst✝⁶ : Category.{v_4, u_4} D₂
W₁ : MorphismProperty C₁
W₁' : MorphismProperty D₁
W₂ : MorphismProperty C₂
W₂' : MorphismProperty D₂
T : LocalizerMorphism W₁ W₂
L : LocalizerMorphism W₁ W₁'
R : LocalizerMorphism W₂ W₂'
B : LocalizerMorphism W₁' W₂'
inst✝⁵ : W₂'.RespectsIso
inst✝⁴ : L.IsLocalizedEquivalence
inst✝³ : R.IsLocalizedEquivalence
inst✝² : R.functor.EssSurj
inst✝¹ : T.IsLeftDerivabilityStructure
iso : T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor
inst✝ : TwoSquare.GuitartExact iso.inv
Y₂ : D₂
X₂ : C₂
e₂ : R.functor.obj X₂ ≅ Y₂
⊢ Nonempty (B.LeftResolution Y₂)
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.OfLocalizedEquivalences

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D₁
inst✝⁸ : Category.{v_4, u_4} D₂
W₁ : MorphismProperty C₁
W₁' : MorphismProperty D₁
W₂ : MorphismProperty C₂
W₂' : MorphismProperty D₂
T : LocalizerMorphism W₁ W₂
L : LocalizerMorphism W₁ W₁'
R : LocalizerMorphism W₂ W₂'
B : LocalizerMorphism W₁' W₂'
inst✝⁷ : W₂'.RespectsIso
inst✝⁶ : L.IsLocalizedEquivalence
inst✝⁵ : R.IsLocalizedEquivalence
inst✝⁴ : R.functor.EssSurj
inst✝³ : L.functor.EssSurj
inst✝² : R.functor.Full
inst✝¹ : R.IsInduced
iso : T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor
inst✝ : TwoSquare.GuitartExact iso.hom
iso' : T.op.functor ⋙ R.op.functor ≅ L.op.functor ⋙ B.op.functor := NatIso.op iso.symm
this : TwoSquare.GuitartExact iso'.inv
⊢ T.op.IsLeftDerivabilityStructure ↔ B.op.IsLeftDerivabilityStructure

Tactic:
exact isLeftDerivabilityStructure_iff_of_isLocalizedEquivalence iso'

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.KProjective

Goal state before:
pp: case refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
K : CochainComplex C ℤ
x✝ : K.IsKProjective
L : HomologicalComplex C (ComplexShape.up ℤ)
hL : L.Acyclic
f : K ⟶ L
⊢ (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map 0 = 0

Tactic:
Functor.map_zero

Goal state after:
case refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
K : CochainComplex C ℤ
x✝ : K.IsKProjective
L : HomologicalComplex C (ComplexShape.up ℤ)
hL : L.Acyclic
f : K ⟶ L
⊢ 0 = 0
Module: Mathlib.Algebra.Homology.HomotopyCategory.KProjective

Goal state before:
pp: case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
K : CochainComplex C ℤ
hK : (HomotopyCategory.subcategoryAcyclic C).leftOrthogonal ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K)
L : CochainComplex C ℤ
f : K ⟶ L
hL : HomotopyCategory.subcategoryAcyclic C ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj L)
⊢ 0 = (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map 0

Tactic:
Functor.map_zero

Goal state after:
case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
K : CochainComplex C ℤ
hK : (HomotopyCategory.subcategoryAcyclic C).leftOrthogonal ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K)
L : CochainComplex C ℤ
f : K ⟶ L
hL : HomotopyCategory.subcategoryAcyclic C ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj L)
⊢ 0 = 0
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q : ℤ
f : X ⟶ K.X q
n : ℤ
h : p + n = q
q' : ℤ
hq' : q + 1 = q'
hf : f ≫ K.d q q' = 0
L : CochainComplex C ℤ
g : K ⟶ L
⊢ fromSingleMk (f ≫ g.f q) h q' hq' ⋯ = (fromSingleMk f h q' hq' hf).postcomp g

Tactic:
ext : 1
  exact (Cochain.fromSingleEquiv h).injective (by simp [Cochain.fromSingleMk_postcomp])

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

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

Tactic:
by simp [Cochain.toSingleMk_precomp]

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

Goal state before:
pp: case zero
R : Type u
inst✝ : CommRing R
J : Ideal R
k : ℕᵒᵖ
⊢ J ≤ (J ^ 0).radical

Tactic:
simp [Ideal.radical_top, pow_zero, Ideal.one_eq_top, le_top]

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.LeftDerived

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{v_1, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
β : (projectiveResolution X).complex ⟶ P.complex
hβ : (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β = P.iso.hom
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((G.mapHomologicalComplex (ComplexShape.down ℕ)).map β) =
    (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map β) ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)

Tactic:
simp only [← Functor.map_comp, NatTrans.mapHomologicalComplex_naturality]

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{v_1, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
β : (projectiveResolution X).complex ⟶ P.complex
hβ : (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β = P.iso.hom
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
      ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as ≫
        (G.mapHomologicalComplex (ComplexShape.down ℕ)).map β) =
    (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
      ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app (projectiveResolution X).complex ≫
        (G.mapHomologicalComplex (ComplexShape.down ℕ)).map β)
Module: Mathlib.Algebra.Homology.SpectralObject.Homology

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₃₄ : i₂ ⟶ i₄
h₃₄ : f₃ ≫ f₄ = f₃₄
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
A : C
⊢ ∀ (x₂ : A ⟶ (X.dCokernelSequence f₁ f₂ f₃ f₄ f₅ f₃₄ h₃₄ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).X₂),
    x₂ ≫ (X.dCokernelSequence f₁ f₂ f₃ f₄ f₅ f₃₄ h₃₄ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).g = 0 →
      ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.dCokernelSequence f₁ f₂ f₃ f₄ f₅ f₃₄ h₃₄ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).f

Tactic:
x₂

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₃₄ : i₂ ⟶ i₄
h₃₄ : f₃ ≫ f₄ = f₃₄
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
A : C
x₂ : A ⟶ (X.dCokernelSequence f₁ f₂ f₃ f₄ f₅ f₃₄ h₃₄ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).X₂
⊢ x₂ ≫ (X.dCokernelSequence f₁ f₂ f₃ f₄ f₅ f₃₄ h₃₄ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).g = 0 →
    ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.dCokernelSequence f₁ f₂ f₃ f₄ f₅ f₃₄ h₃₄ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).f
Module: Mathlib.Algebra.Homology.SpectralObject.Homology

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
A : C
⊢ ∀ (x₂ : A ⟶ (X.dKernelSequence f₁ f₂ f₃ f₄ f₅ f₂₃ h₂₃ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).X₂),
    x₂ ≫ (X.dKernelSequence f₁ f₂ f₃ f₄ f₅ f₂₃ h₂₃ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).g = 0 →
      ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.dKernelSequence f₁ f₂ f₃ f₄ f₅ f₂₃ h₂₃ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).f

Tactic:
x₂

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
A : C
x₂ : A ⟶ (X.dKernelSequence f₁ f₂ f₃ f₄ f₅ f₂₃ h₂₃ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).X₂
⊢ x₂ ≫ (X.dKernelSequence f₁ f₂ f₃ f₄ f₅ f₂₃ h₂₃ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).g = 0 →
    ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.dKernelSequence f₁ f₂ f₃ f₄ f₅ f₂₃ h₂₃ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).f
Module: Mathlib.Algebra.Homology.SpectralObject.Page

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
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
⊢ ∀ (x₂ : A ⟶ (X.cokernelSequenceE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).X₂),
    x₂ ≫ (X.cokernelSequenceE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).g = 0 →
      ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.cokernelSequenceE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).f

Tactic:
x₂

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
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
x₂ : A ⟶ (X.cokernelSequenceE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).X₂
⊢ x₂ ≫ (X.cokernelSequenceE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).g = 0 →
    ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.cokernelSequenceE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).f
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: ι : Type u_2
κ : Type u_3
inst✝ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq' : κ
i₀' i₀ : ι
hi₀' : i₀' = data.i₀ r' pq' ⋯
hi₀ : i₀ = data.i₀ r pq' ⋯
⊢ i₀' ≤ i₀

Tactic:
rw [hi₀', hi₀]

Goal state after:
ι : Type u_2
κ : Type u_3
inst✝ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq' : κ
i₀' i₀ : ι
hi₀' : i₀' = data.i₀ r' pq' ⋯
hi₀ : i₀ = data.i₀ r pq' ⋯
⊢ data.i₀ r' pq' ⋯ ≤ data.i₀ r pq' ⋯
Module: Mathlib.Algebra.Homology.SpectralObject.Page

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
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
⊢ ∀ (x₂ : A ⟶ (X.kernelSequenceE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).X₂),
    x₂ ≫ (X.kernelSequenceE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).g = 0 →
      ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.kernelSequenceE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).f

Tactic:
x₂

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
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
x₂ : A ⟶ (X.kernelSequenceE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).X₂
⊢ x₂ ≫ (X.kernelSequenceE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).g = 0 →
    ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.kernelSequenceE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).f
Module: Mathlib.Algebra.Homology.SpectralObject.Page

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
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₁₂ : i₀ ⟶ i₂
h₁₂ : f₁ ≫ f₂ = f₁₂
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
⊢ ∀ (x₂ : A ⟶ (X.cokernelSequenceOpcyclesE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).X₂),
    x₂ ≫ (X.cokernelSequenceOpcyclesE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).g = 0 →
      ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.cokernelSequenceOpcyclesE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).f

Tactic:
x₂

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
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₁₂ : i₀ ⟶ i₂
h₁₂ : f₁ ≫ f₂ = f₁₂
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
x₂ : A ⟶ (X.cokernelSequenceOpcyclesE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).X₂
⊢ x₂ ≫ (X.cokernelSequenceOpcyclesE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).g = 0 →
    ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.cokernelSequenceOpcyclesE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).f
Module: Mathlib.Algebra.Homology.SpectralObject.Page

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
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
⊢ ∀ (x₂ : A ⟶ (X.kernelSequenceCyclesE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).X₂),
    x₂ ≫ (X.kernelSequenceCyclesE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).g = 0 →
      ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.kernelSequenceCyclesE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).f

Tactic:
x₂

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
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
x₂ : A ⟶ (X.kernelSequenceCyclesE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).X₂
⊢ x₂ ≫ (X.kernelSequenceCyclesE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).g = 0 →
    ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₂ = x₁ ≫ (X.kernelSequenceCyclesE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).f
Module: Mathlib.Algebra.Lie.Basic

Goal state before:
pp: R : Type u
L : Type v
M : Type w
N : Type w₁
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : LieRingModule L N
inst✝ : LieModule R L N
t : R
x✝ y z : L
m✝ n✝ : M
x : L
f : M →ₗ[R] N
m n : M
⊢ ⁅x, f (m + n)⁆ - f ⁅x, m + n⁆ = ⁅x, f m⁆ - f ⁅x, m⁆ + (⁅x, f n⁆ - f ⁅x, n⁆)

Tactic:
by
        simp only [lie_add, map_add]
        abel

Goal state after:
Module: Mathlib.Algebra.Lie.Basic

Goal state before:
pp: R : Type u
L₁ : Type v
L₂ : Type w
inst✝⁴ : CommRing R
inst✝³ : LieRing L₁
inst✝² : LieAlgebra R L₁
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L₁ →ₗ⁅R⁆ L₂
h₁ : ∀ (x y : L₁), f (x + y) = f x + f y
h₂ :
  ∀ (m : R) (x : L₁),
    { toFun := ⇑f, map_add' := h₁ }.toFun (m • x) = (RingHom.id R) m • { toFun := ⇑f, map_add' := h₁ }.toFun x
h₃ :
  ∀ {x y : L₁},
    { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ }.toFun ⁅x, y⁆ =
      ⁅{ toFun := ⇑f, map_add' := h₁, map_smul' := h₂ }.toFun x,
        { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ }.toFun y⁆
⊢ { toFun := ⇑f, map_add' := h₁, map_smul' := h₂, map_lie' := h₃ } = f

Tactic:
ext
  rfl

Goal state after:
Module: Mathlib.RingTheory.Artinian.Module

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
r : R
n m : ℕ
h : n ≤ m
x : M
x✝ : x ∈ (fun n ↦ (r ^ n • LinearMap.id).range) m
y : M
hy : r ^ m • y = x
⊢ r ^ (n + (m - n)) • y = x

Tactic:
← hy,

Goal state after:
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
r : R
n m : ℕ
h : n ≤ m
x : M
x✝ : x ∈ (fun n ↦ (r ^ n • LinearMap.id).range) m
y : M
hy : r ^ m • y = x
⊢ r ^ (n + (m - n)) • y = r ^ m • y
Module: Mathlib.Algebra.Lie.OfAssociative

Goal state before:
pp: case succ
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x y : L
z : M
n : ℕ
ih : (φ x ^ n) ⁅y, z⁆ = ∑ ij ∈ antidiagonal n, n.choose ij.1 • ⁅((ad R L) x ^ ij.1) y, (φ x ^ ij.2) z⁆
i j : ℕ
hij : (i, j) ∈ antidiagonal n
⊢ n.choose (i, j).1 • ⁅((ad R L) x) (((ad R L) x ^ (i, j).1) y), (φ x ^ (i, j).2) z⁆ =
    n.choose (i, j).2 • ⁅((ad R L) x) (((ad R L) x ^ (i, j).1) y), (φ x ^ (i, j).2) z⁆

Tactic:
rw [mem_antidiagonal] at hij

Goal state after:
case succ
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x y : L
z : M
n : ℕ
ih : (φ x ^ n) ⁅y, z⁆ = ∑ ij ∈ antidiagonal n, n.choose ij.1 • ⁅((ad R L) x ^ ij.1) y, (φ x ^ ij.2) z⁆
i j : ℕ
hij : (i, j).1 + (i, j).2 = n
⊢ n.choose (i, j).1 • ⁅((ad R L) x) (((ad R L) x ^ (i, j).1) y), (φ x ^ (i, j).2) z⁆ =
    n.choose (i, j).2 • ⁅((ad R L) x) (((ad R L) x ^ (i, j).1) y), (φ x ^ (i, j).2) z⁆
Module: Mathlib.Algebra.Lie.Submodule

Goal state before:
pp: R : Type u
L : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : LieRingModule L M
N N' : LieSubmodule R L M
s : Set M
hs : s = ↑N
a✝ b✝ : M
x : a✝ ∈ s
y : b✝ ∈ s
⊢ a✝ + b✝ ∈ s

Tactic:
hs

Goal state after:
R : Type u
L : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : LieRingModule L M
N N' : LieSubmodule R L M
s : Set M
hs : s = ↑N
a✝ b✝ : M
x : a✝ ∈ ↑N
y : b✝ ∈ ↑N
⊢ a✝ + b✝ ∈ ↑N