mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Idempotents.FunctorExtension

Goal state before:
pp: case a
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Category.{v_3, u_3} E
F G : C ⥤ Karoubi D
φ : F ⟶ G
P : Karoubi C
h : (F.map P.p).f ≫ (φ.app P.X).f = (φ.app P.X).f ≫ (G.map P.p).f
h' : (F.map P.p).f ≫ (F.map P.p).f = (F.map P.p).f
| (F.map P.p).f ≫ (F.map P.p).f ≫ (F.map P.p).f

pp: case a
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Category.{v_3, u_3} E
F G : C ⥤ Karoubi D
φ : F ⟶ G
P : Karoubi C
h : (F.map P.p).f ≫ (φ.app P.X).f = (φ.app P.X).f ≫ (G.map P.p).f
h' : (F.map P.p).f ≫ (F.map P.p).f = (F.map P.p).f
| (φ.app P.X).f

Tactic:
rw [h', h']

Goal state after:
case a
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Category.{v_3, u_3} E
F G : C ⥤ Karoubi D
φ : F ⟶ G
P : Karoubi C
h : (F.map P.p).f ≫ (φ.app P.X).f = (φ.app P.X).f ≫ (G.map P.p).f
h' : (F.map P.p).f ≫ (F.map P.p).f = (F.map P.p).f
| (F.map P.p).f

case a
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Category.{v_3, u_3} E
F G : C ⥤ Karoubi D
φ : F ⟶ G
P : Karoubi C
h : (F.map P.p).f ≫ (φ.app P.X).f = (φ.app P.X).f ≫ (G.map P.p).f
h' : (F.map P.p).f ≫ (F.map P.p).f = (F.map P.p).f
| (φ.app P.X).f
Module: Mathlib.CategoryTheory.Idempotents.FunctorExtension

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Category.{v_3, u_3} E
G : Karoubi C ⥤ Karoubi D
P Q : Karoubi C
f : P ⟶ Q
⊢ (((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) ⋙ functorExtension₁ C D).obj G).map f ≫
      { f := (G.map Q.decompId_p).f, comm := ⋯ } =
    { f := (G.map P.decompId_p).f, comm := ⋯ } ≫ ((𝟭 (Karoubi C ⥤ Karoubi D)).obj G).map f

Tactic:
by
              simpa only [hom_ext_iff, G.map_comp]
                using! (G.congr_map (decompId_p_naturality f)).symm

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

Goal state before:
pp: case h.inr
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hR : IsNoetherian R R
J' : SelfLERadical J
j1 j2 : CostructuredArrow (idealPowersToSelfLERadical J) J'
h : unop j2.left ≤ unop j1.left
⊢ Zag j1 j2

Tactic:
left; exact ⟨CostructuredArrow.homMk (homOfLE h).op rfl⟩

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 : C ⥤ D
inst✝¹ : F.Additive
X : C
inst✝ : Projective X
⊢ IsIso
    (((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj ((ChainComplex.single₀ C).obj X)).descOpcycles (F.map (𝟙 X)) 1
      fromLeftDerivedZero'._proof_1 ⋯)

Tactic:
ChainComplex.isIso_descOpcycles_iff

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 : C ⥤ D
inst✝¹ : F.Additive
X : C
inst✝ : Projective X
⊢ { X₁ := ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj ((ChainComplex.single₀ C).obj X)).X 1,
        X₂ := ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj ((ChainComplex.single₀ C).obj X)).X 0, X₃ := F.obj X,
        f := ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj ((ChainComplex.single₀ C).obj X)).d 1 0,
        g := F.map (𝟙 X), zero := ⋯ }.Exact ∧
    Epi (F.map (𝟙 X))
Module: Mathlib.Algebra.Homology.ShortComplex.Linear

Goal state before:
pp: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{v_1, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ (a • φ₁).τ₂ = S₁.g ≫ (a • h.h₂) + (a • h.h₁) ≫ S₂.f + (a • φ₂).τ₂

Tactic:
by
    dsimp
    rw [h.comm₂]
    simp only [smul_add, Linear.comp_smul, Linear.smul_comp]

Goal state after:
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
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₁₂ : i₀ ⟶ i₂
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
f₃₄ : i₂ ⟶ i₄
h₃₄ : f₃ ≫ f₄ = f₃₄
f₄₅ : i₃ ⟶ i₅
h₄₅ : f₄ ≫ f₅ = f₄₅
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ (X.H n₁).map (twoδ₂Toδ₁ f₃ f₄ (f₃ ≫ f₄) ⋯) ≫
      X.δ (f₁ ≫ f₂) (f₃ ≫ f₄) n₁ n₂ hn₂ ≫ X.toCycles f₁ f₂ (f₁ ≫ f₂) ⋯ n₂ ≫ X.πE f₁ f₂ f₃ n₁ n₂ n₃ hn₂ hn₃ =
    0

Tactic:
by
      rw [X.δ_naturality_assoc (f₁ ≫ f₂) f₃ (f₁ ≫ f₂) (f₃ ≫ f₄)
        (𝟙 _) (twoδ₂Toδ₁ f₃ f₄ _ rfl) n₁ n₂ rfl hn₂, Functor.map_id, id_comp,
        δ_toCycles_assoc .., δToCycles_πE ..]

Goal state after:
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
⊢ { X₁ := X.cycles f₁₂ f₃ n₀, X₂ := X.cycles f₂ f₃ n₀, X₃ := X.opcycles f₁ f₂ n₁,
      f := X.cyclesMap f₁₂ f₃ f₂ f₃ (threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀, g := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁, zero := ⋯ }.Exact

Tactic:
rw [ShortComplex.exact_iff_exact_up_to_refinements]

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⦄
    (x₂ :
      A ⟶
        { X₁ := X.cycles f₁₂ f₃ n₀, X₂ := X.cycles f₂ f₃ n₀, X₃ := X.opcycles f₁ f₂ n₁,
            f := X.cyclesMap f₁₂ f₃ f₂ f₃ (threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀, g := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁,
            zero := ⋯ }.X₂),
    x₂ ≫
          { X₁ := X.cycles f₁₂ f₃ n₀, X₂ := X.cycles f₂ f₃ n₀, X₃ := X.opcycles f₁ f₂ n₁,
              f := X.cyclesMap f₁₂ f₃ f₂ f₃ (threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀, g := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁,
              zero := ⋯ }.g =
        0 →
      ∃ A' π,
        ∃ (_ : Epi π),
          ∃ x₁,
            π ≫ x₂ =
              x₁ ≫
                { X₁ := X.cycles f₁₂ f₃ n₀, X₂ := X.cycles f₂ f₃ n₀, X₃ := X.opcycles f₁ f₂ n₁,
                    f := X.cyclesMap f₁₂ f₃ f₂ f₃ (threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀, g := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁,
                    zero := ⋯ }.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
f₃₄ : i₂ ⟶ i₄
h₃₄ : f₃ ≫ f₄ = f₃₄
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ (X.dCokernelSequence f₁ f₂ f₃ f₄ f₅ f₃₄ h₃₄ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).Exact

Tactic:
rw [ShortComplex.exact_iff_exact_up_to_refinements]

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.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
⊢ { 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 := ⋯ }.Exact

Tactic:
rw [ShortComplex.exact_iff_exact_up_to_refinements]

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⦄
    (x₂ :
      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),
    x₂ ≫
          { 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 →
      ∃ A' π,
        ∃ (_ : Epi π),
          ∃ x₁,
            π ≫ x₂ =
              x₁ ≫
                { 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 := ⋯ }.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
⊢ (X.dKernelSequence f₁ f₂ f₃ f₄ f₅ f₂₃ h₂₃ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).Exact

Tactic:
rw [ShortComplex.exact_iff_exact_up_to_refinements]

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: case refine_2
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 n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hi : IsLimit (KernelFork.ofι (X.kernelSequenceCycles f₁ f₂ n₁ n₂ hn₂).f ⋯) := ⋯.fIsKernel
this : hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯).f ⋯) = X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯
⊢ (Cocone.precompose
          (parallelPair.ext
              (Iso.refl
                ((parallelPair (hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯)) 0).obj
                  WalkingParallelPair.zero))
              (Iso.refl
                ((parallelPair (hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯)) 0).obj
                  WalkingParallelPair.one))
              ⋯ ⋯).inv).obj
      (CokernelCofork.ofπ (cokernel.π (X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯)) ⋯) ≅
    Cofork.ofπ (cokernel.π (X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯)) ⋯

Tactic:
· exact Cofork.ext (Iso.refl _)

Goal state after:
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
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hp : IsColimit (CokernelCofork.ofπ (X.cokernelSequenceOpcycles f₂ f₃ n₀ n₁ hn₁).g ⋯) := ⋯.gIsCokernel
⊢ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).RightHomologyData

Tactic:
have : hp.desc (CokernelCofork.ofπ _ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).zero) =
      X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂ :=
    Cofork.IsColimit.hom_ext hp (by simpa using! hp.fac _ .one)

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 n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hp : IsColimit (CokernelCofork.ofπ (X.cokernelSequenceOpcycles f₂ f₃ n₀ n₁ hn₁).g ⋯) := ⋯.gIsCokernel
this : hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯) = X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂
⊢ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).RightHomologyData
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
⊢ (X.cokernelSequenceE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).Exact

Tactic:
rw [ShortComplex.exact_iff_exact_up_to_refinements]

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.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
⊢ (X.kernelSequenceE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).Exact

Tactic:
rw [ShortComplex.exact_iff_exact_up_to_refinements]

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.HasSpectralSequence

Goal state before:
pp: 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
c : ℤ → ComplexShape κ
r₀ : ℤ
⊢ ∀ (r r' : ℤ) (pq pq' : ℤ × ℤ),
    (ComplexShape.up' (r, 1 - r)).Rel pq pq' →
      autoParam (r + 1 = r') SpectralSequenceDataCore._auto_41 →
        autoParam (2 ≤ r) SpectralSequenceDataCore._auto_43 → WithBotTop.coe (pq.2 - r' + 2) = WithBotTop.coe pq'.2

Tactic:
rintro r r' hr pq rfl _ _

Goal state after:
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
c : ℤ → ComplexShape κ
r₀ r r' : ℤ
hr : ℤ × ℤ
hrr'✝ : autoParam (r + 1 = r') SpectralSequenceDataCore._auto_41
hr✝ : autoParam (2 ≤ r) SpectralSequenceDataCore._auto_43
⊢ WithBotTop.coe (hr.2 - r' + 2) = WithBotTop.coe (hr + (r, 1 - r)).2
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: 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
c : ℤ → ComplexShape κ
r₀ : ℤ
⊢ ∀ (r r' : ℤ) (pq pq' : ℤ × ℤ),
    (ComplexShape.up' (r, 1 - r)).Rel pq pq' →
      autoParam (r + 1 = r') SpectralSequenceDataCore._auto_47 →
        autoParam (2 ≤ r) SpectralSequenceDataCore._auto_49 →
          WithBotTop.coe (pq'.2 + r' - 1) = WithBotTop.coe (pq.2 + 1)

Tactic:
rintro r r' hr pq rfl _ _

Goal state after:
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
c : ℤ → ComplexShape κ
r₀ r r' : ℤ
hr : ℤ × ℤ
hrr'✝ : autoParam (r + 1 = r') SpectralSequenceDataCore._auto_47
hr✝ : autoParam (2 ≤ r) SpectralSequenceDataCore._auto_49
⊢ WithBotTop.coe ((hr + (r, 1 - r)).2 + r' - 1) = WithBotTop.coe (hr.2 + 1)
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
⊢ (X.cokernelSequenceOpcyclesE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).Exact

Tactic:
rw [ShortComplex.exact_iff_exact_up_to_refinements]

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
⊢ (X.kernelSequenceCyclesE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).Exact

Tactic:
rw [ShortComplex.exact_iff_exact_up_to_refinements]

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: L : Type v
M : Type w
inst✝² : LieRing L
inst✝¹ : AddCommGroup M
inst✝ : LieRingModule L M
x : L
m : M
⊢ ⁅x, -m⁆ = -⁅x, m⁆

Tactic:
rw [← sub_eq_zero, sub_neg_eq_add, ← lie_add]

Goal state after:
L : Type v
M : Type w
inst✝² : LieRing L
inst✝¹ : AddCommGroup M
inst✝ : LieRingModule L M
x : L
m : M
⊢ ⁅x, -m + m⁆ = 0
Module: Mathlib.RingTheory.Artinian.Module

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
f : Module.End R M
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → range (f ^ n) = range (f ^ m)
m : ℕ
hm : n ≤ m
⊢ (f ^ m).ker ⊔ (f ^ m).range = ⊤

Tactic:
simp_rw [← hn _ hm, Submodule.eq_top_iff', Submodule.mem_sup]

Goal state after:
R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
f : Module.End R M
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → range (f ^ n) = range (f ^ m)
m : ℕ
hm : n ≤ m
⊢ ∀ (x : M), ∃ y ∈ (f ^ m).ker, ∃ z ∈ range (f ^ n), y + z = x
Module: Mathlib.RingTheory.Artinian.Module

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
inst✝ : IsArtinianRing R
a : R
ha : a ∈ R⁰
⊢ IsRegular a

Tactic:
isRegular_iff_mem_nonZeroDivisors

Goal state after:
R : Type u_1
inst✝¹ : Ring R
inst✝ : IsArtinianRing R
a : R
ha : a ∈ R⁰
⊢ a ∈ R⁰
Module: Mathlib.RingTheory.Artinian.Module

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
inst✝ : IsArtinianRing Rᵐᵒᵖ
a : R
ha : a ∈ R⁰
⊢ IsRegular a

Tactic:
isRegular_iff_mem_nonZeroDivisors

Goal state after:
R : Type u_1
inst✝¹ : Ring R
inst✝ : IsArtinianRing Rᵐᵒᵖ
a : R
ha : a ∈ R⁰
⊢ a ∈ R⁰
Module: Mathlib.RingTheory.Artinian.Module

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
inst✝ : IsArtinianRing R
a : R
⊢ IsRegular a ↔ a ∈ R⁰

Tactic:
isRegular_iff_mem_nonZeroDivisors

Goal state after:
R : Type u_1
inst✝¹ : Ring R
inst✝ : IsArtinianRing R
a : R
⊢ a ∈ R⁰ ↔ a ∈ R⁰
Module: Mathlib.RingTheory.Artinian.Module

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
inst✝ : IsArtinianRing Rᵐᵒᵖ
a : R
⊢ IsRegular a ↔ a ∈ R⁰

Tactic:
isRegular_iff_mem_nonZeroDivisors

Goal state after:
R : Type u_1
inst✝¹ : Ring R
inst✝ : IsArtinianRing Rᵐᵒᵖ
a : R
⊢ a ∈ R⁰ ↔ a ∈ R⁰
Module: Mathlib.RingTheory.Artinian.Module

Goal state before:
pp: case h
R : Type u_1
inst✝¹ : Ring R
inst✝ : IsArtinianRing R
Jac : Ideal R := Ring.jacobson R
n : ℕ
hn✝ : ∀ (m : ℕ), n ≤ m → { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } n = { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } m
hn : Jac * Jac ^ n = Jac ^ n
ne✝ : ¬Ring.jacobson R ^ n = 0
N : Ideal R
eq : Jac * N = N
ne : N ≠ ⊥
min : ∀ x ∈ {N | Jac * N = N ∧ N ≠ ⊥}, ¬x < N
this✝ : Jac ^ n * N = N
I : R → Submodule R R := fun x ↦ Submodule.map (LinearMap.toSpanSingleton R R x) (Jac ^ n)
hI : ∀ (x : R), I x ≤ Ideal.span {x}
x : R
hx : Ideal.span {x} ≤ N ∧ I x ≠ ⊥
this : I x = N
⊢ False

Tactic:
have : Ideal.span {x} = N := le_antisymm hx.1 (this.symm.trans_le <| hI x)

Goal state after:
case h
R : Type u_1
inst✝¹ : Ring R
inst✝ : IsArtinianRing R
Jac : Ideal R := Ring.jacobson R
n : ℕ
hn✝ : ∀ (m : ℕ), n ≤ m → { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } n = { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } m
hn : Jac * Jac ^ n = Jac ^ n
ne✝ : ¬Ring.jacobson R ^ n = 0
N : Ideal R
eq : Jac * N = N
ne : N ≠ ⊥
min : ∀ x ∈ {N | Jac * N = N ∧ N ≠ ⊥}, ¬x < N
this✝¹ : Jac ^ n * N = N
I : R → Submodule R R := fun x ↦ Submodule.map (LinearMap.toSpanSingleton R R x) (Jac ^ n)
hI : ∀ (x : R), I x ≤ Ideal.span {x}
x : R
hx : Ideal.span {x} ≤ N ∧ I x ≠ ⊥
this✝ : I x = N
this : Ideal.span {x} = N
⊢ False
Module: Mathlib.Algebra.Lie.Ideal

Goal state before:
pp: R : Type u
L : Type v
L' : Type w₂
M : Type w
M' : Type w₁
inst✝¹² : CommRing R
inst✝¹¹ : LieRing L
inst✝¹⁰ : LieRing L'
inst✝⁹ : LieAlgebra R L'
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : AddCommGroup M'
inst✝⁴ : Module R M'
inst✝³ : LieRingModule L M'
inst✝² : LieAlgebra R L
inst✝¹ : LieModule R L M
inst✝ : LieModule R L M'
f : L →ₗ⁅R⁆ L'
I I₂ : LieIdeal R L
J : LieIdeal R L'
x y : L
h : y ∈ (Submodule.comap (↑f) (toLieSubalgebra R L' J).toSubmodule).carrier
this : ⁅f x, f y⁆ ∈ J
⊢ ⁅x, y⁆ ∈ (Submodule.comap (↑f) (toLieSubalgebra R L' J).toSubmodule).carrier

Tactic:
simp only [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup,
          Submodule.mem_toAddSubmonoid, Submodule.mem_comap, LieHom.coe_toLinearMap, LieHom.map_lie,
          LieSubalgebra.mem_toSubmodule]

Goal state after:
R : Type u
L : Type v
L' : Type w₂
M : Type w
M' : Type w₁
inst✝¹² : CommRing R
inst✝¹¹ : LieRing L
inst✝¹⁰ : LieRing L'
inst✝⁹ : LieAlgebra R L'
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : AddCommGroup M'
inst✝⁴ : Module R M'
inst✝³ : LieRingModule L M'
inst✝² : LieAlgebra R L
inst✝¹ : LieModule R L M
inst✝ : LieModule R L M'
f : L →ₗ⁅R⁆ L'
I I₂ : LieIdeal R L
J : LieIdeal R L'
x y : L
h : y ∈ (Submodule.comap (↑f) (toLieSubalgebra R L' J).toSubmodule).carrier
this : ⁅f x, f y⁆ ∈ J
⊢ ⁅f x, f y⁆ ∈ toLieSubalgebra R L' J