mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Functor.Derived.RightDerived

Goal state before:
pp: C : Type u_1
D : Type u_3
H : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_3, u_3} D
inst✝¹ : Category.{v_5, u_2} H
RF RF' : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : F ⟶ L ⋙ RF
α' : F ⟶ L ⋙ RF'
W : MorphismProperty C
inst✝ : L.IsLocalization W
e : RF ≅ RF'
comm : α ≫ L.whiskerLeft e.hom = α'
⊢ RF.IsRightDerivedFunctor α W ↔ RF'.IsRightDerivedFunctor α' W

Tactic:
by
  simp only [isRightDerivedFunctor_iff_isLeftKanExtension]
  exact isLeftKanExtension_iff_of_iso e _ _ comm

Goal state after:
Module: Mathlib.CategoryTheory.Localization.StructuredArrow

Goal state before:
pp: case cons
C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X : C
P : StructuredArrow (W.Q.obj X) W.Q → Prop
hP₀ : P (StructuredArrow.mk (𝟙 (W.Q.obj X)))
hP₁ :
  ∀ ⦃Y₁ Y₂ : C⦄ (f : Y₁ ⟶ Y₂) (φ : W.Q.obj X ⟶ W.Q.obj Y₁),
    P (StructuredArrow.mk φ) → P (StructuredArrow.mk (φ ≫ W.Q.map f))
hP₂ :
  ∀ ⦃Y₁ Y₂ : C⦄ (w : Y₁ ⟶ Y₂) (hw : W w) (φ : W.Q.obj X ⟶ W.Q.obj Y₂),
    P (StructuredArrow.mk φ) → P (StructuredArrow.mk (φ ≫ (isoOfHom W.Q W w hw).inv))
g✝ : StructuredArrow (W.Q.obj X) W.Q
X₀ : Paths (LocQuiver W) := { obj := X }
Y₀ b✝ c✝ : Paths (LocQuiver W)
f : Quiver.Path X₀ b✝
g : b✝ ⟶ c✝
hf : P (StructuredArrow.mk ((Quotient.functor (relations W)).map f))
⊢ P (StructuredArrow.mk ((Quotient.functor (relations W)).map (f.cons g)))

Tactic:
obtain (g | ⟨w, hw⟩) := g
      · exact hP₁ g _ hf
      · simpa only [← Construction.wInv_eq_isoOfHom_inv w hw] using! hP₂ w hw _ hf

Goal state after:
Module: Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived

Goal state before:
pp: C : Type u₁
D : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} H
F' : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : F ⟶ L ⋙ F'
W : MorphismProperty C
G : D ⥤ H
e : F ≅ L ⋙ G
inst✝ : L.IsLocalization W
Y : C
s : Cocone (CostructuredArrow.proj L (L.obj Y) ⋙ F)
j : CostructuredArrow L (L.obj Y)
⊢ ((LeftExtension.mk G e.hom).coconeAt (L.obj Y)).ι.app j ≫ e.inv.app Y ≫ s.ι.app (CostructuredArrow.mk (𝟙 (L.obj Y))) =
    s.ι.app j

Tactic:
by
    refine Localization.induction_costructuredArrow L W _ (by simp)
      (fun X₁ X₂ f φ hφ ↦ ?_) (fun X₁ X₂ w hw φ hφ ↦ ?_) j
    · have eq := s.ι.naturality
        (CostructuredArrow.homMk f : CostructuredArrow.mk (L.map f ≫ φ) ⟶ CostructuredArrow.mk φ)
      dsimp at eq hφ ⊢
      rw [comp_id] at eq
      rw [assoc] at hφ
      rw [assoc, map_comp_assoc, ← eq, ← hφ, NatTrans.naturality_assoc, comp_map]
    · have : IsIso (F.map w) := by
        have := Localization.inverts L W w hw
        rw [← NatIso.naturality_2 e w]
        dsimp
        infer_instance
      have eq := s.ι.naturality
        (CostructuredArrow.homMk w : CostructuredArrow.mk φ ⟶ CostructuredArrow.mk
          ((Localization.isoOfHom L W w hw).inv ≫ φ))
      dsimp at eq hφ ⊢
      rw [comp_id] at eq
      rw [assoc] at hφ
      rw [map_comp, assoc, assoc, ← cancel_epi (F.map w), eq, ← hφ,
        NatTrans.naturality_assoc, comp_map, ← G.map_comp_assoc]
      simp

Goal state after:
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 : W₁'.RespectsIso
inst✝⁴ : W₂'.RespectsIso
inst✝³ : L.IsInduced
inst✝² : L.functor.IsEquivalence
inst✝¹ : R.IsInduced
inst✝ : R.functor.IsEquivalence
iso : T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor
e : T.op.functor ⋙ R.op.functor ≅ L.op.functor ⋙ B.op.functor := NatIso.op iso.symm
⊢ T.IsRightDerivabilityStructure ↔ B.IsRightDerivabilityStructure

Tactic:
simp only [isRightDerivabilityStructure_iff_op,
    isLeftDerivabilityStructure_iff_of_equivalences e]

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Map

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : Abelian C
D : Type u'
inst✝⁶ : Category.{v', u'} D
inst✝⁵ : Abelian D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : PreservesFiniteLimits F
inst✝² : PreservesFiniteColimits F
inst✝¹ : HasExt C
inst✝ : HasExt D
S : ShortComplex C
hS : S.ShortExact
⊢ (mapExactFunctor F hS.extClass).hom = ⋯.extClass.hom

Tactic:
rw [Ext.mapExactFunctor_hom, hS.extClass_hom]

Goal state after:
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : Abelian C
D : Type u'
inst✝⁶ : Category.{v', u'} D
inst✝⁵ : Abelian D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : PreservesFiniteLimits F
inst✝² : PreservesFiniteColimits F
inst✝¹ : HasExt C
inst✝ : HasExt D
S : ShortComplex C
hS : S.ShortExact
⊢ (F.mapDerivedCategorySingleFunctor 0).inv.app S.X₃ ≫
      ShiftedHom.map hS.singleδ F.mapDerivedCategory ≫
        (shiftFunctor (DerivedCategory D) ↑1).map ((F.mapDerivedCategorySingleFunctor 0).hom.app S.X₁) =
    ⋯.extClass.hom
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
m₁ m₂ m₃ : S
X : DifferentialObject S C
⊢ autoParam
    (((shiftFunctorAdd C (m₁ + m₂) m₃).hom.app X ≫ (shiftFunctor C m₃).map ((shiftFunctorAdd C m₁ m₂).hom.app X)).f =
      (eqToHom ⋯ ≫
          (shiftFunctorAdd C m₁ (m₂ + m₃)).hom.app X ≫ (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X)).f)
    ext._auto_1

Tactic:
convert! shiftFunctorAdd_assoc_hom_app m₁ m₂ m₃ X.obj

Goal state after:
case e'_3
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
m₁ m₂ m₃ : S
X : DifferentialObject S C
e_1✝ :
  (((shiftFunctor C (m₁ + m₂ + m₃)).obj X).obj ⟶
      ((shiftFunctor C m₃).obj ((shiftFunctor C m₁ ⋙ shiftFunctor C m₂).obj X)).obj) =
    ((CategoryTheory.shiftFunctor C (m₁ + m₂ + m₃)).obj X.obj ⟶
      (CategoryTheory.shiftFunctor C m₃).obj
        ((CategoryTheory.shiftFunctor C m₁ ⋙ CategoryTheory.shiftFunctor C m₂).obj X.obj))
⊢ (eqToHom ⋯ ≫
        (shiftFunctorAdd C m₁ (m₂ + m₃)).hom.app X ≫ (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X)).f =
    (shiftFunctorAdd' C m₁ (m₂ + m₃) (m₁ + m₂ + m₃) ⋯).hom.app X.obj ≫
      (CategoryTheory.shiftFunctorAdd C m₂ m₃).hom.app ((CategoryTheory.shiftFunctor C m₁).obj X.obj)
Module: Mathlib.Algebra.Homology.ExactSequenceFour

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
n : ℕ
S : ComposableArrows C (n + 3)
hS : S.IsComplex
k : ℕ
hk : k ≤ n
cc : CokernelCofork (S.map' k (k + 1) ⋯ ⋯)
kf : KernelFork (S.map' (k + 2) (k + 3) ⋯ ⋯)
hcc : IsColimit cc
hkf : IsLimit kf
hS' : (S.sc hS (k + 1) ⋯).Exact
this✝¹ : HasZeroObject C
this✝ : (S.sc hS (k + 1) ⋯).HasHomology
h : (S.sc hS (k + 1) ⋯).LeftHomologyData := hS'.leftHomologyDataOfIsLimitKernelFork kf hkf
this : Epi h.f'
⊢ Cofork.π cc ≫ hS.cokerToKer' k hk cc kf hcc hkf = h.f'

Tactic:
rw [← cancel_mono h.i, h.f'_i, ShortComplex.Exact.leftHomologyDataOfIsLimitKernelFork_i,
      assoc, IsComplex.cokerToKer'_fac]

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
⊢ δ n (n + 1) (Cochain.toSingleMk f h) = 0

Tactic:
rw [Cochain.δ_toSingleMk _ _ _ p' (by lia), hf]

Goal state after:
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
⊢ (n + 1).negOnePow • Cochain.toSingleMk 0 (CochainComplex.HomComplex.Cocycle.toSingleMk._proof_1 p q n h p' hp') = 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 : K.X p ⟶ X
n : ℤ
h : p + n = q
p' : ℤ
hp' : p' + 1 = p
hf : K.d p' p ≫ f = 0
X' : C
g : X ⟶ X'
⊢ (Cochain.toSingleEquiv h) ↑(toSingleMk (f ≫ g) h p' hp' ⋯) =
    (Cochain.toSingleEquiv h) ↑((toSingleMk f h p' hp' hf).postcomp ((singleFunctor C q).map g))

Tactic:
simp [Cochain.toSingleMk_postcomp]

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
⊢ { 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))

Tactic:
refine ⟨ShortComplex.Splitting.exact ?_, inferInstance⟩

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 := ⋯ }.Splitting
Module: Mathlib.CategoryTheory.GradedObject.Unitor

Goal state before:
pp: case neg
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Zero I
inst✝² : DecidableEq I
inst✝¹ : HasInitial C
F : D ⥤ C ⥤ D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst✝ : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
j : J
s : Cofan ((((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y)).mapObjFun p j)
j' : J
i : I
h : (j', i) ∈ p ⁻¹' {j}
hi : ¬i = 0
⊢ Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j) ⟨(j', i), h⟩ ≫
      e.inv.app (X j) ≫ (F.obj (X j)).map (singleObjApplyIso 0 Y).inv ≫ s.inj ⟨(j, 0), ⋯⟩ =
    s.inj ⟨(j', i), h⟩

Tactic:
apply IsInitial.hom_ext
        exact mapBifunctorObjObjSingle₀IsInitial _ _ _ _ hi

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

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_3, u_2} ι
inst✝¹ : Abelian C
X : SpectralObject C ι
n : ℤ
i₀ i₁ : ι
f : i₀ ⟶ i₁
inst✝ : IsIso f
φ : mk₁ f ⟶ mk₁ (inv f) := twoδ₂Toδ₁ f (inv f) (𝟙 i₀) ⋯ ≫ twoδ₁Toδ₀ f (inv f) (𝟙 i₀) ⋯
this : IsIso φ
⊢ 𝟙 ((X.H n).obj (mk₁ f)) = 0

Tactic:
← cancel_mono ((X.H n).map φ),

Goal state after:
C : Type u_1
ι : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_3, u_2} ι
inst✝¹ : Abelian C
X : SpectralObject C ι
n : ℤ
i₀ i₁ : ι
f : i₀ ⟶ i₁
inst✝ : IsIso f
φ : mk₁ f ⟶ mk₁ (inv f) := twoδ₂Toδ₁ f (inv f) (𝟙 i₀) ⋯ ≫ twoδ₁Toδ₀ f (inv f) (𝟙 i₀) ⋯
this : IsIso φ
⊢ 𝟙 ((X.H n).obj (mk₁ f)) ≫ (X.H n).map φ = 0 ≫ (X.H n).map φ
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
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ X.map f₁ f₂ f₃ f₁ f₂ f₃ (𝟙 (mk₃ f₁ f₂ f₃)) n₀ n₁ n₂ hn₁ hn₂ = 𝟙 (X.E f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂)

Tactic:
dsimp only [map]

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
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ ShortComplex.homologyMap (X.shortComplexMap f₁ f₂ f₃ f₁ f₂ f₃ (𝟙 (mk₃ f₁ f₂ f₃)) n₀ n₁ n₂ ⋯ ⋯) =
    𝟙 (X.E f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂)
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
i' j' k' l' : ι
f₁' : i' ⟶ j'
f₂' : j' ⟶ k'
f₃' : k' ⟶ l'
i'' j'' k'' l'' : ι
f₁'' : i'' ⟶ j''
f₂'' : j'' ⟶ k''
f₃'' : k'' ⟶ l''
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
β : mk₃ f₁' f₂' f₃' ⟶ mk₃ f₁'' f₂'' f₃''
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ X.map f₁ f₂ f₃ f₁'' f₂'' f₃'' (α ≫ β) n₀ n₁ n₂ hn₁ hn₂ =
    X.map f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ hn₁ hn₂ ≫ X.map f₁' f₂' f₃' f₁'' f₂'' f₃'' β n₀ n₁ n₂ hn₁ hn₂

Tactic:
dsimp only [map]

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
i' j' k' l' : ι
f₁' : i' ⟶ j'
f₂' : j' ⟶ k'
f₃' : k' ⟶ l'
i'' j'' k'' l'' : ι
f₁'' : i'' ⟶ j''
f₂'' : j'' ⟶ k''
f₃'' : k'' ⟶ l''
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
β : mk₃ f₁' f₂' f₃' ⟶ mk₃ f₁'' f₂'' f₃''
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ ShortComplex.homologyMap (X.shortComplexMap f₁ f₂ f₃ f₁'' f₂'' f₃'' (α ≫ β) n₀ n₁ n₂ ⋯ ⋯) =
    ShortComplex.homologyMap (X.shortComplexMap f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ ⋯ ⋯) ≫
      ShortComplex.homologyMap (X.shortComplexMap f₁' f₂' f₃' f₁'' f₂'' f₃'' β n₀ n₁ n₂ ⋯ ⋯)
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
E : SpectralObject C EInt
r r' : ℤ
pq : ℤ × ℤ
hpq : ∀ (pq' : ℤ × ℤ), ¬(ComplexShape.up' (r, 1 - r)).Rel pq pq'
n : ℤ
hn : n = coreE₂Cohomological.deg pq + 1
hrr' : r + 1 = r'
hr : 2 ≤ r
⊢ IsZero ((E.H n).obj (mk₁ (homOfLE ⋯)))

Tactic:
exfalso

Goal state after:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
E : SpectralObject C EInt
r r' : ℤ
pq : ℤ × ℤ
hpq : ∀ (pq' : ℤ × ℤ), ¬(ComplexShape.up' (r, 1 - r)).Rel pq pq'
n : ℤ
hn : n = coreE₂Cohomological.deg pq + 1
hrr' : r + 1 = r'
hr : 2 ≤ r
⊢ False
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
E : SpectralObject C EInt
r r' : ℤ
pq : ℤ × ℤ
hpq : ∀ (pq' : ℤ × ℤ), ¬(ComplexShape.up' (r, 1 - r)).Rel pq' pq
n : ℤ
hn : n = coreE₂Cohomological.deg pq - 1
hrr' : r + 1 = r'
hr : 2 ≤ r
⊢ IsZero ((E.H n).obj (mk₁ (homOfLE ⋯)))

Tactic:
exfalso

Goal state after:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
E : SpectralObject C EInt
r r' : ℤ
pq : ℤ × ℤ
hpq : ∀ (pq' : ℤ × ℤ), ¬(ComplexShape.up' (r, 1 - r)).Rel pq' pq
n : ℤ
hn : n = coreE₂Cohomological.deg pq - 1
hrr' : r + 1 = r'
hr : 2 ≤ r
⊢ False
Module: Mathlib.Order.Filter.Subsingleton

Goal state before:
pp: case inl
α : Type u_1
hl : ⊥.Subsingleton
⊢ ⊥.IsCountablyGenerated

pp: case inr
α : Type u_1
x : α
hl : (pure x).Subsingleton
⊢ (pure x).IsCountablyGenerated

Tactic:
· exact isCountablyGenerated_bot

Goal state after:
case inr
α : Type u_1
x : α
hl : (pure x).Subsingleton
⊢ (pure x).IsCountablyGenerated
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 r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
⊢ pageD X data r pq pq' ⋯ ≫
      (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯).hom ≫
        X.mapFourδ₄Toδ₃' i₀ i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ =
    0

Tactic:
pageD_eq X data r hr pq pq' h (homOfLE (data.le₀₁' r hr pq' hi₀ hi₁))
      (homOfLE (data.le₁₂' pq' hi₁ hi₂)) (homOfLE (data.le₂₃' r hr pq' hi₂ hi₃))
      (homOfLE (data.le₃₃' hrr' hr pq' hi₃ hi₃'))
      (homOfLE (by simpa only [hi₃', data.i₃_next r r' _ _ h] using data.le₂₃ r pq))
      hi₀ hi₁ (by rw [hi₂, data.hc₀₂ r _ _ h])
      (by rw [hi₃, data.hc₁₃ r _ _ h]) (by rw [hi₃', data.i₃_next r r' _ _ h]) rfl
      (n₀ - 1) n₀ n₁ n₂ (by have := data.hc r pq pq' h; lia) (by simp) hn₁ hn₂,

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 r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
⊢ ((pageXIso X data r ⋯ pq i₂ i₃ i₃' (data.i₃ r pq ⋯) ⋯ ⋯ ⋯ ⋯ (n₀ - 1) n₀ n₁ ⋯ ⋯ hn₁).hom ≫
        X.d (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (n₀ - 1) n₀ n₁ n₂ ⋯ hn₁ hn₂ ≫
          (pageXIso X data r ⋯ pq' i₀ i₁ i₂ i₃ hi₀ hi₁ ⋯ ⋯ n₀ n₁ n₂ ⋯ hn₁ hn₂).inv) ≫
      (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯).hom ≫
        X.mapFourδ₄Toδ₃' i₀ i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ =
    0
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
m n : M →ₗ[R] R
⊢ { toFun := fun m_1 ↦ -(m + n) ⁅x, m_1⁆, map_add' := ⋯, map_smul' := ⋯ } =
    { toFun := fun m_1 ↦ -m ⁅x, m_1⁆, map_add' := ⋯, map_smul' := ⋯ } +
      { toFun := fun m ↦ -n ⁅x, m⁆, map_add' := ⋯, map_smul' := ⋯ }

Tactic:
ext p

Goal state after:
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
m n : M →ₗ[R] R
p : M
⊢ { toFun := fun m_1 ↦ -(m + n) ⁅x, m_1⁆, map_add' := ⋯, map_smul' := ⋯ } p =
    ({ toFun := fun m_1 ↦ -m ⁅x, m_1⁆, map_add' := ⋯, map_smul' := ⋯ } +
        { toFun := fun m ↦ -n ⁅x, m⁆, map_add' := ⋯, map_smul' := ⋯ })
      p
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
m n : M →ₗ[R] R
⊢ { toFun := fun m_1 ↦ -(m + n) ⁅x, m_1⁆, map_add' := ⋯, map_smul' := ⋯ } =
    { toFun := fun m_1 ↦ -m ⁅x, m_1⁆, map_add' := ⋯, map_smul' := ⋯ } +
      { toFun := fun m ↦ -n ⁅x, m⁆, map_add' := ⋯, map_smul' := ⋯ }

Tactic:
ext p; simp [-neg_add_rev, neg_add]

Goal state after:
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 m : L
n : M →ₗ[R] R
⊢ { toFun := fun m_1 ↦ -{ toFun := fun m_2 ↦ -n ⁅m, m_2⁆, map_add' := ⋯, map_smul' := ⋯ } ⁅x, m_1⁆, map_add' := ⋯,
      map_smul' := ⋯ } =
    { toFun := fun m_1 ↦ -n ⁅⁅x, m⁆, m_1⁆, map_add' := ⋯, map_smul' := ⋯ } +
      { toFun := fun m_1 ↦ -{ toFun := fun m ↦ -n ⁅x, m⁆, map_add' := ⋯, map_smul' := ⋯ } ⁅m, m_1⁆, map_add' := ⋯,
        map_smul' := ⋯ }

Tactic:
ext p

Goal state after:
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 m : L
n : M →ₗ[R] R
p : M
⊢ { toFun := fun m_1 ↦ -{ toFun := fun m_2 ↦ -n ⁅m, m_2⁆, map_add' := ⋯, map_smul' := ⋯ } ⁅x, m_1⁆, map_add' := ⋯,
        map_smul' := ⋯ }
      p =
    ({ toFun := fun m_1 ↦ -n ⁅⁅x, m⁆, m_1⁆, map_add' := ⋯, map_smul' := ⋯ } +
        { toFun := fun m_1 ↦ -{ toFun := fun m ↦ -n ⁅x, m⁆, map_add' := ⋯, map_smul' := ⋯ } ⁅m, m_1⁆, map_add' := ⋯,
          map_smul' := ⋯ })
      p
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
ι : Type u_1
κ : Type u_3
s : Finset ι
t : Finset κ
f : ι → L
g : κ → M
⊢ ⁅∑ i ∈ s, f i, ∑ j ∈ t, g j⁆ = ∑ i ∈ s, ∑ j ∈ t, ⁅f i, g j⁆

Tactic:
by
  simp_rw [sum_lie, lie_sum]

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
⊢ Injective DFunLike.coe

Tactic:
by
  rintro ⟨⟨⟨f, _⟩, _⟩, _⟩ ⟨⟨⟨g, _⟩, _⟩, _⟩ h
  congr

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

Goal state before:
pp: ι : Type u_2
inst✝ : Finite ι
⊢ ∀ {α β : Type u_2} (a : α ≃ β),
    (∀ {R : α → Type u_1} [inst : (i : α) → Semiring (R i)] [∀ (i : α), IsArtinianRing (R i)],
        IsArtinianRing ((i : α) → R i)) →
      ∀ {R : β → Type u_1} [inst : (i : β) → Semiring (R i)] [∀ (i : β), IsArtinianRing (R i)],
        IsArtinianRing ((i : β) → R i)

pp: ι : Type u_2
inst✝ : Finite ι
⊢ ∀ {R : PEmpty.{u_2 + 1} → Type u_1} [inst : (i : PEmpty.{u_2 + 1}) → Semiring (R i)]
    [∀ (i : PEmpty.{u_2 + 1}), IsArtinianRing (R i)], IsArtinianRing ((i : PEmpty.{u_2 + 1}) → R i)

pp: ι : Type u_2
inst✝ : Finite ι
⊢ ∀ {α : Type u_2} [Fintype α],
    (∀ {R : α → Type u_1} [inst : (i : α) → Semiring (R i)] [∀ (i : α), IsArtinianRing (R i)],
        IsArtinianRing ((i : α) → R i)) →
      ∀ {R : Option α → Type u_1} [inst : (i : Option α) → Semiring (R i)] [∀ (i : Option α), IsArtinianRing (R i)],
        IsArtinianRing ((i : Option α) → R i)

Tactic:
· exact fun e h ↦ RingEquiv.isArtinianRing (.piCongrLeft _ e)

Goal state after:
ι : Type u_2
inst✝ : Finite ι
⊢ ∀ {R : PEmpty.{u_2 + 1} → Type u_1} [inst : (i : PEmpty.{u_2 + 1}) → Semiring (R i)]
    [∀ (i : PEmpty.{u_2 + 1}), IsArtinianRing (R i)], IsArtinianRing ((i : PEmpty.{u_2 + 1}) → R i)

ι : Type u_2
inst✝ : Finite ι
⊢ ∀ {α : Type u_2} [Fintype α],
    (∀ {R : α → Type u_1} [inst : (i : α) → Semiring (R i)] [∀ (i : α), IsArtinianRing (R i)],
        IsArtinianRing ((i : α) → R i)) →
      ∀ {R : Option α → Type u_1} [inst : (i : Option α) → Semiring (R i)] [∀ (i : Option α), IsArtinianRing (R i)],
        IsArtinianRing ((i : Option α) → R i)