mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: case inl
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
i : Fin (n + 1)
hi : a i = b i
hi' : Set.Ioc (a i) (b i) = ∅
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
⊢ 0 = 0

pp: case inl
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
i : Fin (n + 1)
hi : a i = b i
hi' : Set.Ioc (a i) (b i) = ∅
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
⊢ ∀ x ∈ Finset.univ,
    (∫ (x_1 : Fin n → ℝ) in Set.Icc (a ∘ x.succAbove) (b ∘ x.succAbove), f (x.insertNth (b x) x_1) x) -
        ∫ (x_1 : Fin n → ℝ) in Set.Icc (a ∘ x.succAbove) (b ∘ x.succAbove), f (x.insertNth (a x) x_1) x =
      0

Tactic:
]

Goal state after:
case inl
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
i : Fin (n + 1)
hi : a i = b i
hi' : Set.Ioc (a i) (b i) = ∅
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
⊢ ∀ x ∈ Finset.univ,
    (∫ (x_1 : Fin n → ℝ) in Set.Icc (a ∘ x.succAbove) (b ∘ x.succAbove), f (x.insertNth (b x) x_1) x) -
        ∫ (x_1 : Fin n → ℝ) in Set.Icc (a ∘ x.succAbove) (b ∘ x.succAbove), f (x.insertNth (a x) x_1) x =
      0
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • y‖ ≤ 0

Tactic:
]

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • y‖ ≤ 0
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
j : Fin (n + 1)
i : Fin n
hi : a (j.succAbove i) = b (j.succAbove i)
hi' : Set.Ioc (a (j.succAbove i)) (b (j.succAbove i)) = ∅
hne : j.succAbove i ≠ j
⊢ 0 = 0

pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
j : Fin (n + 1)
i : Fin n
hi : a (j.succAbove i) = b (j.succAbove i)
hi' : Set.Ioc (a (j.succAbove i)) (b (j.succAbove i)) = ∅
hne : j.succAbove i ≠ j
⊢ ENNReal.ofReal ((b ∘ j.succAbove) i - (a ∘ j.succAbove) i) = 0

Tactic:
]

Goal state after:
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
j : Fin (n + 1)
i : Fin n
hi : a (j.succAbove i) = b (j.succAbove i)
hi' : Set.Ioc (a (j.succAbove i)) (b (j.succAbove i)) = ∅
hne : j.succAbove i ≠ j
⊢ ENNReal.ofReal ((b ∘ j.succAbove) i - (a ∘ j.succAbove) i) = 0
Module: Mathlib.Analysis.Complex.Liouville

Goal state before:
pp: F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
R C : ℝ
f : ℂ → F
n : ℕ
hR : 0 < R
hf : DiffContOnCl ℂ f (ball c R)
hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
hp : ∀ z ∈ sphere c R, ‖(z - c)⁻¹ ^ (n + 1) • f z‖ ≤ C / (R ^ n * R)
this : 2 * ↑π * I / ↑n.factorial ≠ 0
⊢ n.factorial • (2 * ↑π * I)⁻¹ • (2 * ↑π * I / ↑n.factorial) • iteratedDeriv n f c =
    n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z

Tactic:
]

Goal state after:
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
R C : ℝ
f : ℂ → F
n : ℕ
hR : 0 < R
hf : DiffContOnCl ℂ f (ball c R)
hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
hp : ∀ z ∈ sphere c R, ‖(z - c)⁻¹ ^ (n + 1) • f z‖ ≤ C / (R ^ n * R)
this : 2 * ↑π * I / ↑n.factorial ≠ 0
⊢ n.factorial • (2 * ↑π * I)⁻¹ • (2 * ↑π * I / ↑n.factorial) • iteratedDeriv n f c =
    n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: case inl.inr
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
this✝ : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
j : Fin (n + 1)
i : Fin n
hi : a (j.succAbove i) = b (j.succAbove i)
hi' : Set.Ioc (a (j.succAbove i)) (b (j.succAbove i)) = ∅
hne : j.succAbove i ≠ j
this : Set.Icc (a ∘ j.succAbove) (b ∘ j.succAbove) =ᵐ[volume] ∅
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝ : Field F
p : F[X]
σ τ : p.Gal
h : ∀ x ∈ p.rootSet p.SplittingField, σ x = τ x
x : p.SplittingField
⊢ p.rootSet p.SplittingField ⊆ ↑((↑σ).equalizer ↑τ)

Tactic:
]

Goal state after:
F : Type u_1
inst✝ : Field F
p : F[X]
σ τ : p.Gal
h : ∀ x ∈ p.rootSet p.SplittingField, σ x = τ x
x : p.SplittingField
⊢ p.rootSet p.SplittingField ⊆ ↑((↑σ).equalizer ↑τ)
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝² : Field F
p q : F[X]
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
h : p.Splits
f : p.Gal
y : F
⊢ (algebraMap F p.SplittingField) y = (algebraMap F p.SplittingField) y

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: case right
F : Type u_1
inst✝² : Field F
p : F[X]
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
h : Fact (map (algebraMap F E) p).Splits
y : ↑(p.rootSet E)
key :
  (map (algebraMap F E) p).roots =
    Multiset.map (⇑↑(IsScalarTower.toAlgHom F p.SplittingField E)) (map (algebraMap F p.SplittingField) p).roots
⊢ ∃ a, mapRoots p E a = y

Tactic:
]

Goal state after:
case right
F : Type u_1
inst✝² : Field F
p : F[X]
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
h : Fact (map (algebraMap F E) p).Splits
y : ↑(p.rootSet E)
key :
  (map (algebraMap F E) p).roots =
    Multiset.map (⇑↑(IsScalarTower.toAlgHom F p.SplittingField E)) (map (algebraMap F p.SplittingField) p).roots
⊢ ∃ a, mapRoots p E a = y
Module: Mathlib.Analysis.Complex.Liouville

Goal state before:
pp: F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
R C : ℝ
f : ℂ → F
n : ℕ
hR : 0 < R
hf : DiffContOnCl ℂ f (ball c R)
hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
hp : ∀ z ∈ sphere c R, ‖(z - c)⁻¹ ^ (n + 1) • f z‖ ≤ C / (R ^ n * R)
hq : iteratedDeriv n f c = n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z
⊢ ‖n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z‖ =
    ‖n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z‖

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Complex.Liouville

Goal state before:
pp: F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
R C : ℝ
f : ℂ → F
n : ℕ
hR : 0 < R
hf : DiffContOnCl ℂ f (ball c R)
hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
hp : ∀ z ∈ sphere c R, ‖(z - c)⁻¹ ^ (n + 1) • f z‖ ≤ C / (R ^ n * R)
hq : iteratedDeriv n f c = n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z
⊢ ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z‖ ≤ R * (C / R ^ (n + 1))

Tactic:
]

Goal state after:
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
R C : ℝ
f : ℂ → F
n : ℕ
hR : 0 < R
hf : DiffContOnCl ℂ f (ball c R)
hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
hp : ∀ z ∈ sphere c R, ‖(z - c)⁻¹ ^ (n + 1) • f z‖ ≤ C / (R ^ n * R)
hq : iteratedDeriv n f c = n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z
⊢ ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z‖ ≤ R * (C / R ^ (n + 1))
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝³ : Field F
p : F[X]
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Fact (map (algebraMap F E) p).Splits
ϕ : Gal(E/F)
x : ↑(p.rootSet E)
ψ : p.SplittingField ≃ₐ[F] ↥(IsScalarTower.toAlgHom F p.SplittingField E).range :=
  AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F p.SplittingField E)
⊢ ↑(↑((↑ϕ).restrictNormalAux p.SplittingField)
        (↑↑(AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F p.SplittingField E)) ↑((rootsEquivRoots p E).symm x))) =
    ϕ ↑x

Tactic:
]

Goal state after:
F : Type u_1
inst✝³ : Field F
p : F[X]
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Fact (map (algebraMap F E) p).Splits
ϕ : Gal(E/F)
x : ↑(p.rootSet E)
ψ : p.SplittingField ≃ₐ[F] ↥(IsScalarTower.toAlgHom F p.SplittingField E).range :=
  AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F p.SplittingField E)
⊢ ↑(↑((↑ϕ).restrictNormalAux p.SplittingField)
        (↑↑(AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F p.SplittingField E)) ↑((rootsEquivRoots p E).symm x))) =
    ϕ ↑x
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝³ : Field F
p : F[X]
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Fact (map (algebraMap F E) p).Splits
ϕ : Gal(E/F)
x : ↑(p.rootSet E)
ψ : p.SplittingField ≃ₐ[F] ↥(IsScalarTower.toAlgHom F p.SplittingField E).range :=
  AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F p.SplittingField E)
⊢ ϕ ↑x = ϕ ↑x

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝³ : Field F
p : F[X]
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Fact (map (algebraMap F E) p).Splits
⊢ ∀ (a : p.Gal), (galActionHom p E) a = 1 → a = 1

Tactic:
]

Goal state after:
F : Type u_1
inst✝³ : Field F
p : F[X]
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Fact (map (algebraMap F E) p).Splits
⊢ ∀ (a : p.Gal), (galActionHom p E) a = 1 → a = 1
Module: Mathlib.Analysis.Complex.Liouville

Goal state before:
pp: F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
c : ℂ
R C : ℝ
f : ℂ → F
hR : 0 < R
hd : DiffContOnCl ℂ f (ball c R)
hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL
this : HasDerivAt (⇑e ∘ f) (e (deriv f c)) c
⊢ ‖deriv f c‖ = ‖e (deriv f c)‖

Tactic:
]

Goal state after:
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
c : ℂ
R C : ℝ
f : ℂ → F
hR : 0 < R
hd : DiffContOnCl ℂ f (ball c R)
hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL
this : HasDerivAt (⇑e ∘ f) (e (deriv f c)) c
⊢ ‖deriv f c‖ = ‖e (deriv f c)‖
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝³ : Field F
p : F[X]
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Fact (map (algebraMap F E) p).Splits
ϕ : p.Gal
hϕ : (galActionHom p E) ϕ = 1
x : p.SplittingField
hx : x ∈ p.rootSet p.SplittingField
key : (rootsEquivRoots p E) (ϕ • ⟨x, hx⟩) = (rootsEquivRoots p E) ⟨x, hx⟩
⊢ ϕ x = 1 x

Tactic:
]

Goal state after:
F : Type u_1
inst✝³ : Field F
p : F[X]
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Fact (map (algebraMap F E) p).Splits
ϕ : p.Gal
hϕ : (galActionHom p E) ϕ = 1
x : p.SplittingField
hx : x ∈ p.rootSet p.SplittingField
key : (rootsEquivRoots p E) (ϕ • ⟨x, hx⟩) = (rootsEquivRoots p E) ⟨x, hx⟩
⊢ ϕ x = 1 x
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
δ : ℝ
δ0 : δ > 0
hδ : ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π)
r : ℝ
hr0 : 0 < r
hrδ : r ≤ δ
hrR : r ≤ R
hsub : closedBall c R \ ball c r ⊆ closedBall c R \ {c}
hsub' : ball c R \ closedBall c r ⊆ ball c R \ {c}
z : ℂ
hzne : z ≠ c
hz : ‖z - c‖ = r
⊢ ‖(z - c)⁻¹ • (f z - y)‖ ≤ r⁻¹ * (ε / (2 * π))

Tactic:
]

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
δ : ℝ
δ0 : δ > 0
hδ : ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π)
r : ℝ
hr0 : 0 < r
hrδ : r ≤ δ
hrR : r ≤ R
hsub : closedBall c R \ ball c r ⊆ closedBall c R \ {c}
hsub' : ball c R \ closedBall c r ⊆ ball c R \ {c}
z : ℂ
hzne : z ≠ c
hz : ‖z - c‖ = r
⊢ ‖(z - c)⁻¹ • (f z - y)‖ ≤ r⁻¹ * (ε / (2 * π))
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝ : Field F
p q : F[X]
hpq : p * q = 0
⊢ Unique (Gal 0)

Tactic:
]

Goal state after:
F : Type u_1
inst✝ : Field F
p q : F[X]
hpq : p * q = 0
⊢ Unique (Gal 0)
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: E : Type u
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
n : ℕ
F : Type u_1
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : Preorder F
inst✝¹ : MeasureSpace F
inst✝ : BorelSpace F
eL : F ≃L[ℝ] Fin (n + 1) → ℝ
he_ord : ∀ (x y : F), eL x ≤ eL y ↔ x ≤ y
he_vol : MeasurePreserving (⇑eL) volume volume
f : Fin (n + 1) → F → E
f' : Fin (n + 1) → F → F →L[ℝ] E
s : Set F
hs : s.Countable
a b : F
hle : a ≤ b
Hc : ∀ (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)
Hd : ∀ x ∈ interior (Set.Icc a b) \ s, ∀ (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x
DF : F → E
hDF : ∀ (x : F), DF x = ∑ i, (f' i x) (eL.symm (e i))
Hi : IntegrableOn DF (Set.Icc a b) volume
he_emb : MeasurableEmbedding ⇑eL
hIcc : ⇑eL ⁻¹' Set.Icc (eL a) (eL b) = Set.Icc a b
⊢ Set.Icc (eL a) (eL b) = Set.Icc (eL a) (eL b)

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: case neg
F : Type u_1
inst✝ : Field F
p q : F[X]
hpq : ¬p * q = 0
f g : (p * q).Gal
hfg :
  (restrict p (p * q).SplittingField) f = (restrict p (p * q).SplittingField) g ∧
    (restrict q (p * q).SplittingField) f = (restrict q (p * q).SplittingField) g
x : (p * q).SplittingField
hx : x ∈ ↑(p.aroots (p * q).SplittingField + q.aroots (p * q).SplittingField).toFinset
⊢ f x = g x

Tactic:
]

Goal state after:
case neg
F : Type u_1
inst✝ : Field F
p q : F[X]
hpq : ¬p * q = 0
f g : (p * q).Gal
hfg :
  (restrict p (p * q).SplittingField) f = (restrict p (p * q).SplittingField) g ∧
    (restrict q (p * q).SplittingField) f = (restrict q (p * q).SplittingField) g
x : (p * q).SplittingField
hx : x ∈ ↑(p.aroots (p * q).SplittingField + q.aroots (p * q).SplittingField).toFinset
⊢ f x = g x
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
δ : ℝ
δ0 : δ > 0
hδ : ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π)
r : ℝ
hr0 : 0 < r
hrδ : r ≤ δ
hrR : r ≤ R
hsub : closedBall c R \ ball c r ⊆ closedBall c R \ {c}
hsub' : ball c R \ closedBall c r ⊆ ball c R \ {c}
z : ℂ
hzne : z ≠ c
hz : ‖z - c‖ = r
⊢ r⁻¹ * dist (f z) y ≤ r⁻¹ * (ε / (2 * π))

Tactic:
]

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
δ : ℝ
δ0 : δ > 0
hδ : ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π)
r : ℝ
hr0 : 0 < r
hrδ : r ≤ δ
hrR : r ≤ R
hsub : closedBall c R \ ball c r ⊆ closedBall c R \ {c}
hsub' : ball c R \ closedBall c r ⊆ ball c R \ {c}
z : ℂ
hzne : z ≠ c
hz : ‖z - c‖ = r
⊢ r⁻¹ * dist (f z) y ≤ r⁻¹ * (ε / (2 * π))
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: E : Type u
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
n : ℕ
F : Type u_1
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : Preorder F
inst✝¹ : MeasureSpace F
inst✝ : BorelSpace F
eL : F ≃L[ℝ] Fin (n + 1) → ℝ
he_ord : ∀ (x y : F), eL x ≤ eL y ↔ x ≤ y
he_vol : MeasurePreserving (⇑eL) volume volume
f : Fin (n + 1) → F → E
f' : Fin (n + 1) → F → F →L[ℝ] E
s : Set F
hs : s.Countable
a b : F
hle : a ≤ b
Hc : ∀ (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)
Hd : ∀ x ∈ interior (Set.Icc a b) \ s, ∀ (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x
DF : F → E
hDF : ∀ (x : F), DF x = ∑ i, (f' i x) (eL.symm (e i))
Hi : IntegrableOn DF (Set.Icc a b) volume
he_emb : MeasurableEmbedding ⇑eL
hIcc : ⇑eL ⁻¹' Set.Icc (eL a) (eL b) = Set.Icc a b
hIcc' : Set.Icc (eL a) (eL b) = ⇑eL.symm ⁻¹' Set.Icc a b
⊢ ∫ (x : F) in Set.Icc a b, ∑ i, (f' i x) (eL.symm (e i)) =
    ∫ (x : F) in ⇑eL ⁻¹' Set.Icc (eL a) (eL b), ∑ i, (f' i (eL.symm (eL x))) (eL.symm (e i))

Tactic:
]

Goal state after:
E : Type u
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
n : ℕ
F : Type u_1
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : Preorder F
inst✝¹ : MeasureSpace F
inst✝ : BorelSpace F
eL : F ≃L[ℝ] Fin (n + 1) → ℝ
he_ord : ∀ (x y : F), eL x ≤ eL y ↔ x ≤ y
he_vol : MeasurePreserving (⇑eL) volume volume
f : Fin (n + 1) → F → E
f' : Fin (n + 1) → F → F →L[ℝ] E
s : Set F
hs : s.Countable
a b : F
hle : a ≤ b
Hc : ∀ (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)
Hd : ∀ x ∈ interior (Set.Icc a b) \ s, ∀ (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x
DF : F → E
hDF : ∀ (x : F), DF x = ∑ i, (f' i x) (eL.symm (e i))
Hi : IntegrableOn DF (Set.Icc a b) volume
he_emb : MeasurableEmbedding ⇑eL
hIcc : ⇑eL ⁻¹' Set.Icc (eL a) (eL b) = Set.Icc a b
hIcc' : Set.Icc (eL a) (eL b) = ⇑eL.symm ⁻¹' Set.Icc a b
⊢ ∫ (x : F) in Set.Icc a b, ∑ i, (f' i x) (eL.symm (e i)) =
    ∫ (x : F) in ⇑eL ⁻¹' Set.Icc (eL a) (eL b), ∑ i, (f' i (eL.symm (eL x))) (eL.symm (e i))
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : Monoid S
inst✝² : DistribMulAction S R
inst✝¹ : IsScalarTower S R R
inst✝ : SMulCommClass S R R
s : S
n : ℕ
m : Multiset R
⊢ (map ((fun x ↦ s ^ n • x) ∘ prod) (powersetCard n m)).sum = (map (fun x ↦ s • x) m).esymm n

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : Monoid S
inst✝² : DistribMulAction S R
inst✝¹ : IsScalarTower S R R
inst✝ : SMulCommClass S R R
s : S
n : ℕ
m : Multiset R
⊢ (map ((fun x ↦ s ^ n • x) ∘ prod) (powersetCard n m)).sum = (map (fun x ↦ s • x) m).esymm n
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : Monoid S
inst✝² : DistribMulAction S R
inst✝¹ : IsScalarTower S R R
inst✝ : SMulCommClass S R R
s : S
n : ℕ
m x : Multiset R
hx : x ∈ powersetCard n m
⊢ s ^ n • x.prod = s ^ n • x.prod

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: σ✝ : Type u_1
τ : Type u_2
R✝ : Type u_3
S : Type u_4
σ : Type u_5
R : Type u_6
inst✝ : CommSemiring R
a✝ b✝ : MvPolynomial σ R
ha : a✝ ∈ Set.ofPred IsSymmetric
hb : b✝ ∈ Set.ofPred IsSymmetric
e : Perm σ
⊢ a✝ * b✝ = a✝ * b✝

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: σ✝ : Type u_1
τ : Type u_2
R✝ : Type u_3
S : Type u_4
σ : Type u_5
R : Type u_6
inst✝ : CommSemiring R
a✝ b✝ : MvPolynomial σ R
ha : a✝ ∈ Set.ofPred IsSymmetric
hb : b✝ ∈ Set.ofPred IsSymmetric
e : Perm σ
⊢ a✝ + b✝ = a✝ + b✝

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: case neg.inl
F : Type u_1
inst✝ : Field F
p q : F[X]
hpq : ¬p * q = 0
f g : (p * q).Gal
hfg :
  (restrict p (p * q).SplittingField) f = (restrict p (p * q).SplittingField) g ∧
    (restrict q (p * q).SplittingField) f = (restrict q (p * q).SplittingField) g
x : (p * q).SplittingField
hx : x ∈ ↑(p.aroots (p * q).SplittingField + q.aroots (p * q).SplittingField).toFinset
h : x ∈ p.aroots (p * q).SplittingField
this : Fact (map (algebraMap F (p * q).SplittingField) p).Splits
key :
  x = (algebraMap p.SplittingField (p * q).SplittingField) ↑((rootsEquivRoots p (p * q).SplittingField).invFun ⟨x, ⋯⟩)
⊢ (algebraMap p.SplittingField (p * q).SplittingField)
      ((AlgEquiv.restrictNormal f p.SplittingField) ↑((rootsEquivRoots p (p * q).SplittingField).invFun ⟨x, ⋯⟩)) =
    (algebraMap p.SplittingField (p * q).SplittingField)
      ((AlgEquiv.restrictNormal g p.SplittingField) ↑((rootsEquivRoots p (p * q).SplittingField).invFun ⟨x, ⋯⟩))

Tactic:
]

Goal state after:
case neg.inl
F : Type u_1
inst✝ : Field F
p q : F[X]
hpq : ¬p * q = 0
f g : (p * q).Gal
hfg :
  (restrict p (p * q).SplittingField) f = (restrict p (p * q).SplittingField) g ∧
    (restrict q (p * q).SplittingField) f = (restrict q (p * q).SplittingField) g
x : (p * q).SplittingField
hx : x ∈ ↑(p.aroots (p * q).SplittingField + q.aroots (p * q).SplittingField).toFinset
h : x ∈ p.aroots (p * q).SplittingField
this : Fact (map (algebraMap F (p * q).SplittingField) p).Splits
key :
  x = (algebraMap p.SplittingField (p * q).SplittingField) ↑((rootsEquivRoots p (p * q).SplittingField).invFun ⟨x, ⋯⟩)
⊢ (algebraMap p.SplittingField (p * q).SplittingField)
      ((AlgEquiv.restrictNormal f p.SplittingField) ↑((rootsEquivRoots p (p * q).SplittingField).invFun ⟨x, ⋯⟩)) =
    (algebraMap p.SplittingField (p * q).SplittingField)
      ((AlgEquiv.restrictNormal g p.SplittingField) ↑((rootsEquivRoots p (p * q).SplittingField).invFun ⟨x, ⋯⟩))