mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: σ : Type u_1
R : Type u_3
S : Type u_4
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
φ : MvPolynomial σ R
hφ : φ.IsSymmetric
f : R →+* S
e : Perm σ
⊢ (MvPolynomial.map f) φ = (MvPolynomial.map f) φ

Tactic:
]

Goal state after:
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 ≤ δ

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 ≤ δ
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: σ : Type u_1
τ : Type u_2
R : Type u_3
inst✝ : CommSemiring R
φ : MvPolynomial σ R
hφ : φ.IsSymmetric
e : σ ≃ τ
x✝ : Perm τ
⊢ φ = φ

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: case neg.inr
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 ∈ q.aroots (p * q).SplittingField
this : Fact (map (algebraMap F (p * q).SplittingField) q).Splits
key :
  x = (algebraMap q.SplittingField (p * q).SplittingField) ↑((rootsEquivRoots q (p * q).SplittingField).invFun ⟨x, ⋯⟩)
⊢ (algebraMap q.SplittingField (p * q).SplittingField)
      ((AlgEquiv.restrictNormal f q.SplittingField) ↑((rootsEquivRoots q (p * q).SplittingField).invFun ⟨x, ⋯⟩)) =
    (algebraMap q.SplittingField (p * q).SplittingField)
      ((AlgEquiv.restrictNormal g q.SplittingField) ↑((rootsEquivRoots q (p * q).SplittingField).invFun ⟨x, ⋯⟩))

Tactic:
]

Goal state after:
case neg.inr
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 ∈ q.aroots (p * q).SplittingField
this : Fact (map (algebraMap F (p * q).SplittingField) q).Splits
key :
  x = (algebraMap q.SplittingField (p * q).SplittingField) ↑((rootsEquivRoots q (p * q).SplittingField).invFun ⟨x, ⋯⟩)
⊢ (algebraMap q.SplittingField (p * q).SplittingField)
      ((AlgEquiv.restrictNormal f q.SplittingField) ↑((rootsEquivRoots q (p * q).SplittingField).invFun ⟨x, ⋯⟩)) =
    (algebraMap q.SplittingField (p * q).SplittingField)
      ((AlgEquiv.restrictNormal g q.SplittingField) ↑((rootsEquivRoots q (p * q).SplittingField).invFun ⟨x, ⋯⟩))
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝ : Field F
p₁ q₁ p₂ q₂ : F[X]
hq₁ : q₁ ≠ 0
hq₂ : q₂ ≠ 0
h₁ : (map (algebraMap F q₁.SplittingField) p₁).Splits
h₂ : (map (algebraMap F q₂.SplittingField) p₂).Splits
⊢ (map (algebraMap F (q₁ * q₂).SplittingField) p₁ * map (algebraMap F (q₁ * q₂).SplittingField) p₂).Splits

Tactic:
]

Goal state after:
F : Type u_1
inst✝ : Field F
p₁ q₁ p₂ q₂ : F[X]
hq₁ : q₁ ≠ 0
hq₂ : q₂ ≠ 0
h₁ : (map (algebraMap F q₁.SplittingField) p₁).Splits
h₂ : (map (algebraMap F q₂.SplittingField) p₂).Splits
⊢ (map (algebraMap F (q₁ * q₂).SplittingField) p₁ * map (algebraMap F (q₁ * q₂).SplittingField) p₂).Splits
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: case hf
F : Type u_1
inst✝ : Field F
p₁ q₁ p₂ q₂ : F[X]
hq₁ : q₁ ≠ 0
hq₂ : q₂ ≠ 0
h₁ : (map (algebraMap F q₁.SplittingField) p₁).Splits
h₂ : (map (algebraMap F q₂.SplittingField) p₂).Splits
⊢ (map (↑(SplittingField.lift q₁ ⋯)) (map (algebraMap F q₁.SplittingField) p₁)).Splits

Tactic:
]

Goal state after:
case hf
F : Type u_1
inst✝ : Field F
p₁ q₁ p₂ q₂ : F[X]
hq₁ : q₁ ≠ 0
hq₂ : q₂ ≠ 0
h₁ : (map (algebraMap F q₁.SplittingField) p₁).Splits
h₂ : (map (algebraMap F q₂.SplittingField) p₂).Splits
⊢ (map (↑(SplittingField.lift q₁ ⋯)) (map (algebraMap F q₁.SplittingField) p₁)).Splits
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: case hg
F : Type u_1
inst✝ : Field F
p₁ q₁ p₂ q₂ : F[X]
hq₁ : q₁ ≠ 0
hq₂ : q₂ ≠ 0
h₁ : (map (algebraMap F q₁.SplittingField) p₁).Splits
h₂ : (map (algebraMap F q₂.SplittingField) p₂).Splits
⊢ (map (↑(SplittingField.lift q₂ ⋯)) (map (algebraMap F q₂.SplittingField) p₂)).Splits

Tactic:
]

Goal state after:
case hg
F : Type u_1
inst✝ : Field F
p₁ q₁ p₂ q₂ : F[X]
hq₁ : q₁ ≠ 0
hq₂ : q₂ ≠ 0
h₁ : (map (algebraMap F q₁.SplittingField) p₁).Splits
h₂ : (map (algebraMap F q₂.SplittingField) p₂).Splits
⊢ (map (↑(SplittingField.lift q₂ ⋯)) (map (algebraMap F q₂.SplittingField) p₂)).Splits
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r ↦ (map (algebraMap F (r.comp q).SplittingField) r).Splits
r : F[X]
hr : Irreducible r
hr' : ¬r.natDegree = 0
h : (r.comp q).degree = 0
⊢ (r.comp q).degree = ↑0

Tactic:
]

Goal state after:
F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r ↦ (map (algebraMap F (r.comp q).SplittingField) r).Splits
r : F[X]
hr : Irreducible r
hr' : ¬r.natDegree = 0
h : (r.comp q).degree = 0
⊢ (r.comp q).degree = ↑0
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: case neg
F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r ↦ (map (algebraMap F (r.comp q).SplittingField) r).Splits
r : F[X]
hr : Irreducible r
hr' : ¬r.natDegree = 0
x : (r.comp q).SplittingField
hx : (aeval ((aeval x) q)) r = 0
⊢ P r

Tactic:
]

Goal state after:
case neg
F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r ↦ (map (algebraMap F (r.comp q).SplittingField) r).Splits
r : F[X]
hr : Irreducible r
hr' : ¬r.natDegree = 0
x : (r.comp q).SplittingField
hx : (aeval ((aeval x) q)) r = 0
⊢ P r
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: τ : Type u_2
σ : Type u_5
R : Type u_6
inst✝² : CommSemiring R
inst✝¹ : Fintype σ
inst✝ : Fintype τ
n : ℕ
e : σ ≃ τ
⊢ ∑ t ∈ powersetCard n univ, ∏ i ∈ t, X i = ∑ t ∈ powersetCard n univ, ∏ i ∈ t, X i

Tactic:
]

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

Goal state before:
pp: σ : Type u_5
R : Type u_6
inst✝¹ : CommSemiring R
inst✝ : Fintype σ
n : ℕ
e✝ : Perm σ
⊢ esymm σ R n = esymm σ R n

Tactic:
]

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

Goal state before:
pp: σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
⊢ (∑ t ∈ powersetCard n univ, (monomial (∑ i ∈ t, Finsupp.single i 1)) 1).support =
    (powersetCard n univ).biUnion fun t ↦ (Finsupp.single (∑ i ∈ t, Finsupp.single i 1) 1).support

Tactic:
]

Goal state after:
σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
⊢ (∑ t ∈ powersetCard n univ, (monomial (∑ i ∈ t, Finsupp.single i 1)) 1).support =
    (powersetCard n univ).biUnion fun t ↦ (Finsupp.single (∑ i ∈ t, Finsupp.single i 1) 1).support
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
h0✝ : 0 ≤ R
f : ℂ → E
c : ℂ
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R)
hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z
hE : CompleteSpace E
h0 : 0 < R
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
s t : Finset σ
hst : s ≠ t
⊢ ∀ ⦃a : σ →₀ ℕ⦄, a ∈ {∑ i ∈ s, Finsupp.single i 1} → a ∉ (Finsupp.single (∑ i ∈ t, Finsupp.single i 1) 1).support

Tactic:
]

Goal state after:
σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
s t : Finset σ
hst : s ≠ t
⊢ ∀ ⦃a : σ →₀ ℕ⦄, a ∈ {∑ i ∈ s, Finsupp.single i 1} → a ∉ (Finsupp.single (∑ i ∈ t, Finsupp.single i 1) 1).support
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
s t : Finset σ
hst : s ≠ t
⊢ ∀ ⦃a : σ →₀ ℕ⦄, a ∈ {∑ i ∈ s, Finsupp.single i 1} → a ∉ {∑ i ∈ t, Finsupp.single i 1}

Tactic:
]

Goal state after:
σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
s t : Finset σ
hst : s ≠ t
⊢ ∀ ⦃a : σ →₀ ℕ⦄, a ∈ {∑ i ∈ s, Finsupp.single i 1} → a ∉ {∑ i ∈ t, Finsupp.single i 1}
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: case pos.inl
F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r ↦ (map (algebraMap F (r.comp q).SplittingField) r).Splits
key1 : ∀ {r : F[X]}, Irreducible r → P r
p₁ p₂ : F[X]
hp₁ : P p₁
hp₂ : P p₂
h₁ : p₁.comp q = 0
h : p₁ = 0
⊢ P 0

Tactic:
]

Goal state after:
case pos.inl
F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r ↦ (map (algebraMap F (r.comp q).SplittingField) r).Splits
key1 : ∀ {r : F[X]}, Irreducible r → P r
p₁ p₂ : F[X]
hp₁ : P p₁
hp₂ : P p₂
h₁ : p₁.comp q = 0
h : p₁ = 0
⊢ P 0
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: case refine_2
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 : Fin (n + 1) → ℝ
hx : x ∈ (Set.univ.pi fun i ↦ Set.Ioo (eL a i) (eL b i)) \ ⇑eL.symm ⁻¹' s
i : Fin (n + 1)
⊢ eL.symm x ∈ interior (⇑eL ⁻¹' Set.Icc (eL a) (eL b))

Tactic:
]

Goal state after:
case refine_2
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 : Fin (n + 1) → ℝ
hx : x ∈ (Set.univ.pi fun i ↦ Set.Ioo (eL a i) (eL b i)) \ ⇑eL.symm ⁻¹' s
i : Fin (n + 1)
⊢ eL.symm x ∈ interior (⇑eL ⁻¹' Set.Icc (eL a) (eL b))
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r ↦ (map (algebraMap F (r.comp q).SplittingField) r).Splits
key1 : ∀ {r : F[X]}, Irreducible r → P r
p₁ p₂ : F[X]
hp₁ : P p₁
hp₂ : P p₂
h₁ : p₁.comp q = 0
h : eval (q.coeff 0) p₁ = 0 ∧ q = C (q.coeff 0)
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
s t : Finset σ
hst : s ≠ t
h : ∑ i ∈ t, Finsupp.single i 1 = ∑ i ∈ s, Finsupp.single i 1
this : (t.biUnion fun i ↦ (Finsupp.single i 1).support) = s.biUnion fun i ↦ (Finsupp.single i 1).support
⊢ False

Tactic:
]

Goal state after:
σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
s t : Finset σ
hst : s ≠ t
h : ∑ i ∈ t, Finsupp.single i 1 = ∑ i ∈ s, Finsupp.single i 1
this : (t.biUnion fun i ↦ (Finsupp.single i 1).support) = s.biUnion fun i ↦ (Finsupp.single i 1).support
⊢ False
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
⊢ ((powersetCard n univ).biUnion fun t ↦ (Finsupp.single (∑ i ∈ t, Finsupp.single i 1) 1).support) =
    (powersetCard n univ).biUnion fun t ↦ {∑ i ∈ t, Finsupp.single i 1}

Tactic:
]

Goal state after:
σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
⊢ ((powersetCard n univ).biUnion fun t ↦ (Finsupp.single (∑ i ∈ t, Finsupp.single i 1) 1).support) =
    (powersetCard n univ).biUnion fun t ↦ {∑ i ∈ t, Finsupp.single i 1}
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
⊢ ((powersetCard n univ).biUnion fun t ↦ {∑ i ∈ t, Finsupp.single i 1}) =
    image (fun t ↦ ∑ i ∈ t, Finsupp.single i 1) (powersetCard n univ)

Tactic:
]

Goal state after:
σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
⊢ ((powersetCard n univ).biUnion fun t ↦ {∑ i ∈ t, Finsupp.single i 1}) =
    image (fun t ↦ ∑ i ∈ t, Finsupp.single i 1) (powersetCard n univ)
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r ↦ (map (algebraMap F (r.comp q).SplittingField) r).Splits
key1 : ∀ {r : F[X]}, Irreducible r → P r
p₁ p₂ : F[X]
hp₁ : P p₁
hp₂ : P p₂
h₁ : ¬p₁.comp q = 0
h₂ : p₂.comp q = 0
h : eval (q.coeff 0) p₂ = 0 ∧ q = C (q.coeff 0)
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: σ : Type u_5
R : Type u_6
inst✝² : CommSemiring R
inst✝¹ : Fintype σ
inst✝ : Nontrivial R
n : ℕ
hpos : 0 < n
hn : n ≤ Fintype.card σ
this : (⇑Finsupp.toMultiset ∘ fun t ↦ ∑ i ∈ t, Finsupp.single i 1) = val
⊢ (powersetCard n univ).sup val = univ.val

Tactic:
]

Goal state after:
σ : Type u_5
R : Type u_6
inst✝² : CommSemiring R
inst✝¹ : Fintype σ
inst✝ : Nontrivial R
n : ℕ
hpos : 0 < n
hn : n ≤ Fintype.card σ
this : (⇑Finsupp.toMultiset ∘ fun t ↦ ∑ i ∈ t, Finsupp.single i 1) = val
⊢ (powersetCard n univ).sup val = univ.val
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
F : ℂ → E := dslope f w
hws : (insert w s).Countable
hcF : ContinuousOn F (closedBall c R)
hdF : ∀ z ∈ ball c R \ insert w s, DifferentiableAt ℂ F z
HI : ∮ (z : ℂ) in C(c, R), F z = 0
hne : ∀ z ∈ sphere c R, z ≠ w
hFeq : EqOn F (fun z ↦ (z - w)⁻¹ • f z - (z - w)⁻¹ • f w) (sphere c R)
hc' : ContinuousOn (fun z ↦ (z - w)⁻¹) (sphere c R)
⊢ ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z = (2 * ↑π * I) • f w

Tactic:
← circleIntegral.integral_sub_inv_of_mem_ball hw.1,

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
F : ℂ → E := dslope f w
hws : (insert w s).Countable
hcF : ContinuousOn F (closedBall c R)
hdF : ∀ z ∈ ball c R \ insert w s, DifferentiableAt ℂ F z
HI : ∮ (z : ℂ) in C(c, R), F z = 0
hne : ∀ z ∈ sphere c R, z ≠ w
hFeq : EqOn F (fun z ↦ (z - w)⁻¹ • f z - (z - w)⁻¹ • f w) (sphere c R)
hc' : ContinuousOn (fun z ↦ (z - w)⁻¹) (sphere c R)
⊢ ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z = (∮ (z : ℂ) in C(c, R), (z - w)⁻¹) • f w
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: case neg
F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r ↦ (map (algebraMap F (r.comp q).SplittingField) r).Splits
key1 : ∀ {r : F[X]}, Irreducible r → P r
p₁ p₂ : F[X]
hp₁ : P p₁
hp₂ : P p₂
h₁ : ¬p₁.comp q = 0
h₂ : ¬p₂.comp q = 0
key : (map (algebraMap F ((p₁ * p₂).comp q).SplittingField) (p₁ * p₂)).Splits
⊢ P (p₁ * p₂)

Tactic:
]

Goal state after:
case neg
F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r ↦ (map (algebraMap F (r.comp q).SplittingField) r).Splits
key1 : ∀ {r : F[X]}, Irreducible r → P r
p₁ p₂ : F[X]
hp₁ : P p₁
hp₂ : P p₂
h₁ : ¬p₁.comp q = 0
h₂ : ¬p₂.comp q = 0
key : (map (algebraMap F ((p₁ * p₂).comp q).SplittingField) (p₁ * p₂)).Splits
⊢ P (p₁ * p₂)
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: σ : Type u_5
R : Type u_6
inst✝² : CommSemiring R
inst✝¹ : Fintype σ
inst✝ : Nontrivial R
n : ℕ
hpos : 0 < n
hn : n ≤ Fintype.card σ
this✝ : (⇑Finsupp.toMultiset ∘ fun t ↦ ∑ i ∈ t, Finsupp.single i 1) = val
this : ((powersetCard n univ).sup fun x ↦ x).val = (powersetCard n univ).sup val
⊢ ((powersetCard n univ).sup fun x ↦ x).val = univ.val

Tactic:
]

Goal state after:
σ : Type u_5
R : Type u_6
inst✝² : CommSemiring R
inst✝¹ : Fintype σ
inst✝ : Nontrivial R
n : ℕ
hpos : 0 < n
hn : n ≤ Fintype.card σ
this✝ : (⇑Finsupp.toMultiset ∘ fun t ↦ ∑ i ∈ t, Finsupp.single i 1) = val
this : ((powersetCard n univ).sup fun x ↦ x).val = (powersetCard n univ).sup val
⊢ ((powersetCard n univ).sup fun x ↦ x).val = univ.val
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
⊢ (map (algebraMap F (p.comp q).SplittingField) p).Splits

Tactic:
let P : F[X] → Prop := fun r => (r.map (algebraMap F (r.comp q).SplittingField)).Splits
  have key1 : ∀ {r : F[X]}, Irreducible r → P r := by
    intro r hr
    by_cases hr' : natDegree r = 0
    · exact Splits.of_natDegree_le_one <| natDegree_map_le.trans (hr'.trans_le zero_le_one)
    obtain ⟨x, hx⟩ :=
      Splits.exists_eval_eq_zero (SplittingField.splits (r.comp q)) fun h =>
        hr' ((mul_eq_zero.mp (natDegree_comp.symm.trans (natDegree_eq_of_degree_eq_some
          (by rwa [degree_map] at h)))).resolve_right hq)
    rw [eval_map_algebraMap, aeval_comp] at hx
    have h_normal : Normal F (r.comp q).SplittingField := SplittingField.instNormal (r.comp q)
    have qx_int := Normal.isIntegral h_normal (aeval x q)
    exact (h_normal.splits _).of_dvd (map_ne_zero (minpoly.ne_zero (h_normal.isIntegral _)))
      ((map_dvd_map' _).mpr ((minpoly.irreducible qx_int).dvd_symm hr (minpoly.dvd F _ hx)))
  have key2 : ∀ {p₁ p₂ : F[X]}, P p₁ → P p₂ → P (p₁ * p₂) := by
    intro p₁ p₂ hp₁ hp₂
    by_cases h₁ : p₁.comp q = 0
    · rcases comp_eq_zero_iff.mp h₁ with h | h
      · rw [h, zero_mul]
        simp [P]
      · exact False.elim (hq (by rw [h.2, natDegree_C]))
    by_cases h₂ : p₂.comp q = 0
    · rcases comp_eq_zero_iff.mp h₂ with h | h
      · simp [h, P]
      · exact False.elim (hq (by rw [h.2, natDegree_C]))
    have key := mul_splits_in_splittingField_of_mul h₁ h₂ hp₁ hp₂
    rwa [← mul_comp] at key
  exact
    WfDvdMonoid.induction_on_irreducible p (by simp) (fun _ hu => hu.splits.map _)
      fun _ _ _ h => key2 (key1 h)

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
p : F[X]
inst✝ : CharZero F
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
⊢ p.natDegree ∣ finrank F p.SplittingField

Tactic:
]

Goal state after:
F : Type u_1
inst✝¹ : Field F
p : F[X]
inst✝ : CharZero F
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
⊢ p.natDegree ∣ finrank F p.SplittingField
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
p : F[X]
inst✝ : CharZero F
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
hp : p.degree ≠ 0
⊢ p.degree ≠ 0

Tactic:
]

Goal state after:
F : Type u_1
inst✝¹ : Field F
p : F[X]
inst✝ : CharZero F
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
hp : p.degree ≠ 0
⊢ p.degree ≠ 0
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: case refine_3
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
⊢ IntegrableOn ((fun x ↦ ∑ i, (f' i (eL.symm x) ∘SL ↑eL.symm) (e i)) ∘ ⇑eL) (Set.Icc a b) volume

Tactic:
]

Goal state after:
case refine_3
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
⊢ IntegrableOn ((fun x ↦ ∑ i, (f' i (eL.symm x) ∘SL ↑eL.symm) (e i)) ∘ ⇑eL) (Set.Icc a b) volume
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
p : F[X]
inst✝ : CharZero F
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
hp : p.degree ≠ 0
α : p.SplittingField := rootOfSplits ⋯ ⋯
hα : IsIntegral F α
this : (minpoly F α).natDegree = p.natDegree
x✝ : AddCommGroup ↥F⟮α⟯ := F⟮α⟯.toField.toDivisionRing.toAddCommGroup
⊢ p.natDegree * finrank (↥F⟮α⟯) p.SplittingField = p.natDegree * finrank (↥F⟮α⟯) p.SplittingField

Tactic:
]

Goal state after: