mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: case inr
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : 0 ≤ B
⊢ ‖(map f p).coeff i‖ ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)

Tactic:
]

Goal state after:
case inr
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : 0 ≤ B
⊢ ‖(map f p).coeff i‖ ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: case e'_2
R : Type u_1
inst✝ : CommRing R
s : Multiset R
k : ℕ
h : k ≤ s.card
⊢ (map (fun x ↦ X + C (-x)) s).prod.coeff k = (map ((fun r ↦ X + C r) ∘ fun t ↦ -t) s).prod.coeff ?convert_1

Tactic:
]

Goal state after:
case e'_2
R : Type u_1
inst✝ : CommRing R
s : Multiset R
k : ℕ
h : k ≤ s.card
⊢ (map (fun x ↦ X + C (-x)) s).prod.coeff k = (map ((fun r ↦ X + C r) ∘ fun t ↦ -t) s).prod.coeff ?convert_1
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: case e'_3
R : Type u_1
inst✝ : CommRing R
s : Multiset R
k : ℕ
h : k ≤ s.card
⊢ (-1) ^ (s.card - k) * s.esymm (s.card - k) = (-1) ^ (s.card - k) * s.esymm (s.card - k)

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
R : ℝ≥0
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c ↑R)
hd : ∀ z ∈ ball c ↑R \ s, DifferentiableAt ℂ f z
hR : 0 < R
w : ℂ
hw : w ∈ eball 0 ↑R
hw' : c + w ∈ ball c ↑R
⊢ HasSum (fun n ↦ (cauchyPowerSeries f c (↑R) n) fun x ↦ w)
    ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - (c + w))⁻¹ • f z)

Tactic:
]

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ≥0
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c ↑R)
hd : ∀ z ∈ ball c ↑R \ s, DifferentiableAt ℂ f z
hR : 0 < R
w : ℂ
hw : w ∈ eball 0 ↑R
hw' : c + w ∈ ball c ↑R
⊢ HasSum (fun n ↦ (cauchyPowerSeries f c (↑R) n) fun x ↦ w)
    ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - (c + w))⁻¹ • f z)
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: case convert_2
R : Type u_1
inst✝ : CommRing R
s : Multiset R
k : ℕ
h : k ≤ s.card
⊢ k ≤ s.card

Tactic:
]

Goal state after:
case convert_2
R : Type u_1
inst✝ : CommRing R
s : Multiset R
k : ℕ
h : k ≤ s.card
⊢ k ≤ s.card
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: case inr.inl
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : 0 ≤ B
hi : (map f p).natDegree < i
⊢ 0 ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)

Tactic:
]

Goal state after:
case inr.inl
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : 0 ≤ B
hi : (map f p).natDegree < i
⊢ 0 ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: case inr
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f f' : ℝ → E
a b : ℝ
s : Set ℝ
hs : s.Countable
Hi : IntervalIntegrable f' volume a b
hab : b ≤ a
Hc : ContinuousOn f (Set.Icc b a)
Hd : ∀ x ∈ Set.Ioo b a \ s, HasDerivAt f (f' x) x
⊢ ∫ (x : ℝ) in b..a, f' x = f a - f b

Tactic:
]

Goal state after:
case inr
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f f' : ℝ → E
a b : ℝ
s : Set ℝ
hs : s.Countable
Hi : IntervalIntegrable f' volume a b
hab : b ≤ a
Hc : ContinuousOn f (Set.Icc b a)
Hd : ∀ x ∈ Set.Ioo b a \ s, HasDerivAt f (f' x) x
⊢ ∫ (x : ℝ) in b..a, f' x = f a - f b
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hroots : p.roots.card = p.natDegree
k : ℕ
h : k ≤ p.natDegree
⊢ p.leadingCoeff * (map (fun a ↦ X - C a) p.roots).prod.coeff k =
    p.leadingCoeff * ((-1) ^ (p.natDegree - k) * p.roots.esymm (p.natDegree - k))

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hroots : p.roots.card = p.natDegree
k : ℕ
h : k ≤ p.natDegree
⊢ p.leadingCoeff * (map (fun a ↦ X - C a) p.roots).prod.coeff k =
    p.leadingCoeff * ((-1) ^ (p.natDegree - k) * p.roots.esymm (p.natDegree - k))
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hroots : p.roots.card = p.natDegree
k : ℕ
h : k ≤ p.natDegree
⊢ k ≤ p.natDegree

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hroots : p.roots.card = p.natDegree
k : ℕ
h : k ≤ p.natDegree
⊢ k ≤ p.natDegree
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
s : Set ℂ
o : IsOpen s
⊢ AnalyticOnNhd ℂ f s ↔ DifferentiableOn ℂ f s

Tactic:
]

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
s : Set ℂ
o : IsOpen s
⊢ AnalyticOnNhd ℂ f s ↔ DifferentiableOn ℂ f s
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: case e'_5
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hroots : p.roots.card = p.natDegree
k : ℕ
h : k ≤ p.natDegree
this : k ≤ p.roots.card
⊢ p.natDegree - k = p.natDegree - k

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: case e'_6
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hroots : p.roots.card = p.natDegree
k : ℕ
h : k ≤ p.natDegree
this : k ≤ p.roots.card
⊢ p.natDegree - k = p.natDegree - k

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
σ : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Fintype σ
s : Multiset (MvPolynomial σ R) := Multiset.map (fun i ↦ X i) univ.val
⊢ univ.val.card = univ.val.card

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
f : ℂ → E
⊢ AnalyticOnNhd ℂ f univ ↔ Differentiable ℂ f

Tactic:
]

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
⊢ AnalyticOnNhd ℂ f univ ↔ Differentiable ℂ f
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: case inr.inr
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : 0 ≤ B
hi : i ≤ (map f p).natDegree
⊢ ‖(map f p).roots.esymm ((map f p).natDegree - i)‖ ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)

Tactic:
]

Goal state after:
case inr.inr
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : 0 ≤ B
hi : i ≤ (map f p).natDegree
⊢ ‖(map f p).roots.esymm ((map f p).natDegree - i)‖ ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
σ : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Fintype σ
k : ℕ
h : k ≤ Fintype.card σ
s : Multiset (MvPolynomial σ R) := Multiset.map (fun i ↦ X i) univ.val
⊢ univ.val.card = univ.val.card

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
σ : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Fintype σ
k : ℕ
s : Multiset (MvPolynomial σ R) := Multiset.map (fun i ↦ X i) univ.val
h : k ≤ s.card
this : Fintype.card σ = s.card
⊢ (∏ i, (Polynomial.X + Polynomial.C (X i))).coeff k = esymm σ R (s.card - k)

Tactic:
]

Goal state after:
R : Type u_1
σ : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Fintype σ
k : ℕ
s : Multiset (MvPolynomial σ R) := Multiset.map (fun i ↦ X i) univ.val
h : k ≤ s.card
this : Fintype.card σ = s.card
⊢ (∏ i, (Polynomial.X + Polynomial.C (X i))).coeff k = esymm σ R (s.card - k)
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: case inr.inr
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : 0 ≤ B
hi : i ≤ (map f p).natDegree
⊢ ↑((map f p).natDegree.choose i) * ?inr.inr.a ≤ ↑((map f p).natDegree.choose i) * B ^ ((map f p).natDegree - i)

pp: case inr.inr.a
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : 0 ≤ B
hi : i ≤ (map f p).natDegree
⊢ ℝ

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
σ : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Fintype σ
k : ℕ
s : Multiset (MvPolynomial σ R) := Multiset.map (fun i ↦ X i) univ.val
h : k ≤ s.card
this : Fintype.card σ = s.card
⊢ (Multiset.map (fun i ↦ Polynomial.X + Polynomial.C (X i)) univ.val).prod.coeff k =
    (Multiset.map X univ.val).esymm (s.card - k)

Tactic:
]

Goal state after:
R : Type u_1
σ : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Fintype σ
k : ℕ
s : Multiset (MvPolynomial σ R) := Multiset.map (fun i ↦ X i) univ.val
h : k ≤ s.card
this : Fintype.card σ = s.card
⊢ (Multiset.map (fun i ↦ Polynomial.X + Polynomial.C (X i)) univ.val).prod.coeff k =
    (Multiset.map X univ.val).esymm (s.card - k)
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
f : ℂ → E
c : ℂ
s : Set ℂ
h0 : 0 < R
n : ℕ
hs : s.Countable
hc : ContinuousOn f (closedBall c R)
hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z
this : (n.factorial • (cauchyPowerSeries f c (↑(NNReal.mk R ⋯)) n) fun x ↦ 1) = iteratedDeriv n f c
⊢ ∮ (z : ℂ) in C(c, R), (1 / (z - c) ^ (n + 1)) • f z = (2 * ↑π * I / ↑n.factorial) • iteratedDeriv n f c

Tactic:
]

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
f : ℂ → E
c : ℂ
s : Set ℂ
h0 : 0 < R
n : ℕ
hs : s.Countable
hc : ContinuousOn f (closedBall c R)
hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z
this : (n.factorial • (cauchyPowerSeries f c (↑(NNReal.mk R ⋯)) n) fun x ↦ 1) = iteratedDeriv n f c
⊢ ∮ (z : ℂ) in C(c, R), (1 / (z - c) ^ (n + 1)) • f z = (2 * ↑π * I / ↑n.factorial) • iteratedDeriv n f c
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : 0 ≤ B
hi : i ≤ (map f p).natDegree
s : Multiset K
hs : s ≤ (map f p).roots ∧ s.card = (map f p).natDegree - i
⊢ ‖s.prod‖ ≤ B ^ ((map f p).natDegree - i)

Tactic:
]

Goal state after:
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : 0 ≤ B
hi : i ≤ (map f p).natDegree
s : Multiset K
hs : s ≤ (map f p).roots ∧ s.card = (map f p).natDegree - i
⊢ ‖s.prod‖ ≤ B ^ ((map f p).natDegree - i)
Module: Mathlib.Analysis.Complex.Polynomial.Basic

Goal state before:
pp: p : ℚ[X]
hp : p = 0
⊢ IsEmpty ↑∅

Tactic:
]

Goal state after:
p : ℚ[X]
hp : p = 0
⊢ IsEmpty ↑∅
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
hi : i ≤ (map f p).natDegree
s : Multiset K
hs : s ≤ (map f p).roots ∧ s.card = (map f p).natDegree - i
B : ℝ≥0
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ ↑B
⊢ (Multiset.map (⇑↑nnnormHom) s).prod ≤ B ^ ((map f p).natDegree - i)

Tactic:
]

Goal state after:
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
hi : i ≤ (map f p).natDegree
s : Multiset K
hs : s ≤ (map f p).roots ∧ s.card = (map f p).natDegree - i
B : ℝ≥0
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ ↑B
⊢ (Multiset.map (⇑↑nnnormHom) s).prod ≤ B ^ ((map f p).natDegree - i)
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
hi : i ≤ (map f p).natDegree
s : Multiset K
hs : s ≤ (map f p).roots ∧ s.card = (map f p).natDegree - i
B : ℝ≥0
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ ↑B
⊢ B ^ ((map f p).natDegree - i) = B ^ ((map f p).natDegree - i)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.Basic

Goal state before:
pp: case neg
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
⊢ (p.rootSet ℂ).toFinset.card =
    (Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset).card +
      (Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support).card

Tactic:
]

Goal state after:
case neg
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
⊢ (p.rootSet ℂ).toFinset.card =
    (Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset).card +
      (Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support).card
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
f : ℂ → E
c : ℂ
s : Set ℂ
h0 : 0 < R
n : ℕ
hs : s.Countable
hc : ContinuousOn f (closedBall c R)
hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z
this : (n.factorial • (cauchyPowerSeries f c (↑(NNReal.mk R ⋯)) n) fun x ↦ 1) = iteratedDeriv n f c
⊢ ∮ (z : ℂ) in C(c, R), (1 / (z - c) ^ (n + 1)) • f z =
    1 • ∮ (z : ℂ) in C(c, ↑(NNReal.mk R ⋯)), (1 / (z - c)) ^ n • (z - c)⁻¹ • f z

Tactic:
]

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
f : ℂ → E
c : ℂ
s : Set ℂ
h0 : 0 < R
n : ℕ
hs : s.Countable
hc : ContinuousOn f (closedBall c R)
hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z
this : (n.factorial • (cauchyPowerSeries f c (↑(NNReal.mk R ⋯)) n) fun x ↦ 1) = iteratedDeriv n f c
⊢ ∮ (z : ℂ) in C(c, R), (1 / (z - c) ^ (n + 1)) • f z =
    1 • ∮ (z : ℂ) in C(c, ↑(NNReal.mk R ⋯)), (1 / (z - c)) ^ n • (z - c)⁻¹ • f z
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: case refine_4
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ × ℝ → E
f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E
a b : ℝ × ℝ
hle : a ≤ b
s : Set (ℝ × ℝ)
hs : s.Countable
Hcf : ContinuousOn f (Set.Icc a b)
Hcg : ContinuousOn g (Set.Icc a b)
Hdf : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt f (f' x) x
Hdg : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt g (g' x) x
Hi : IntegrableOn (fun x ↦ (f' x) (1, 0) + (g' x) (0, 1)) (Set.Icc a b) volume
e : (ℝ × ℝ) ≃L[ℝ] Fin 2 → ℝ := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm
x : ℝ × ℝ
hx : x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s
⊢ ∀ (i : Fin (1 + 1)), HasFDerivAt (![f, g] i) (![f', g'] i x) x

Tactic:
]

Goal state after:
case refine_4
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ × ℝ → E
f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E
a b : ℝ × ℝ
hle : a ≤ b
s : Set (ℝ × ℝ)
hs : s.Countable
Hcf : ContinuousOn f (Set.Icc a b)
Hcg : ContinuousOn g (Set.Icc a b)
Hdf : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt f (f' x) x
Hdg : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt g (g' x) x
Hi : IntegrableOn (fun x ↦ (f' x) (1, 0) + (g' x) (0, 1)) (Set.Icc a b) volume
e : (ℝ × ℝ) ≃L[ℝ] Fin 2 → ℝ := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm
x : ℝ × ℝ
hx : x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s
⊢ ∀ (i : Fin (1 + 1)), HasFDerivAt (![f, g] i) (![f', g'] i x) x
Module: Mathlib.Analysis.Complex.Polynomial.Basic

Goal state before:
pp: p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
z : ℂ
⊢ (aeval z) p = 0 ↔ (aeval z) p = 0

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: case refine_5
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ × ℝ → E
f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E
a b : ℝ × ℝ
hle : a ≤ b
s : Set (ℝ × ℝ)
hs : s.Countable
Hcf : ContinuousOn f (Set.Icc a b)
Hcg : ContinuousOn g (Set.Icc a b)
Hdf : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt f (f' x) x
Hdg : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt g (g' x) x
Hi : IntegrableOn (fun x ↦ (f' x) (1, 0) + (g' x) (0, 1)) (Set.Icc a b) volume
e : (ℝ × ℝ) ≃L[ℝ] Fin 2 → ℝ := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm
x : ℝ × ℝ
⊢ (f' x) (1, 0) + (g' x) (0, 1) = (![f', g'] 0 x) (e.symm (Pi.single 0 1)) + (![f', g'] 1 x) (e.symm (Pi.single 1 1))

Tactic:
]

Goal state after:
case refine_5
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ × ℝ → E
f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E
a b : ℝ × ℝ
hle : a ≤ b
s : Set (ℝ × ℝ)
hs : s.Countable
Hcf : ContinuousOn f (Set.Icc a b)
Hcg : ContinuousOn g (Set.Icc a b)
Hdf : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt f (f' x) x
Hdg : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt g (g' x) x
Hi : IntegrableOn (fun x ↦ (f' x) (1, 0) + (g' x) (0, 1)) (Set.Icc a b) volume
e : (ℝ × ℝ) ≃L[ℝ] Fin 2 → ℝ := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm
x : ℝ × ℝ
⊢ (f' x) (1, 0) + (g' x) (0, 1) = (![f', g'] 0 x) (e.symm (Pi.single 0 1)) + (![f', g'] 1 x) (e.symm (Pi.single 1 1))
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: case inr
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
B : ℝ
d : ℕ
f : F →+* K
p : F[X]
h1 : p.Monic
h2 : (map f p).Splits
h3 : p.natDegree ≤ d
h4 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
i : ℕ
hB : B < 0
⊢ ‖if i = 0 then 1 else 0‖ ≤ max B 1 ^ d * ↑(d.choose (d / 2))

Tactic:
]

Goal state after:
case inr
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
B : ℝ
d : ℕ
f : F →+* K
p : F[X]
h1 : p.Monic
h2 : (map f p).Splits
h3 : p.natDegree ≤ d
h4 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
i : ℕ
hB : B < 0
⊢ ‖if i = 0 then 1 else 0‖ ≤ max B 1 ^ d * ↑(d.choose (d / 2))
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f✝ g : ℝ × ℝ → E
f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E
a✝ b✝ : ℝ × ℝ
hle : a✝ ≤ b✝
s : Set (ℝ × ℝ)
hs : s.Countable
Hcf : ContinuousOn f✝ (Set.Icc a✝ b✝)
Hcg : ContinuousOn g (Set.Icc a✝ b✝)
Hdf : ∀ x ∈ Set.Ioo a✝.1 b✝.1 ×ˢ Set.Ioo a✝.2 b✝.2 \ s, HasFDerivAt f✝ (f' x) x
Hdg : ∀ x ∈ Set.Ioo a✝.1 b✝.1 ×ˢ Set.Ioo a✝.2 b✝.2 \ s, HasFDerivAt g (g' x) x
Hi : IntegrableOn (fun x ↦ (f' x) (1, 0) + (g' x) (0, 1)) (Set.Icc a✝ b✝) volume
e : (ℝ × ℝ) ≃L[ℝ] Fin 2 → ℝ := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm
a b : Fin 1 → ℝ
f : (Fin 1 → ℝ) → E
⊢ ∫ (x : Fin 1 → ℝ) in Set.Icc a b, f x = ∫ (x : ℝ) in Set.Icc (a 0) (b 0), f fun x_1 ↦ x

Tactic:
by
        convert!
          (((volume_preserving_funUnique (Fin 1) ℝ).symm _).setIntegral_preimage_emb
              (MeasurableEquiv.measurableEmbedding _) f _).symm
        exact ((OrderIso.funUnique (Fin 1) ℝ).symm.preimage_Icc a b).symm

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.Basic

Goal state before:
pp: p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
w : ℝ
hw : (aeval w) p = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.Basic

Goal state before:
pp: case a
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
z : ℂ
hz1 : (aeval z) p = 0
hz2 : z.im = 0
⊢ ((IsScalarTower.toAlgHom ℚ ℝ ℂ) z.re).im = 0

Tactic:
]

Goal state after:
case a
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
z : ℂ
hz1 : (aeval z) p = 0
hz2 : z.im = 0
⊢ ((IsScalarTower.toAlgHom ℚ ℝ ℂ) z.re).im = 0
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
A : Type u_2
inst✝³ : Field A
inst✝² : CharZero A
inst✝¹ : NumberField K
inst✝ : IsAlgClosed A
⊢ finrank ℚ K = finrank ℚ K

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
A : Type u_2
inst✝³ : Field A
inst✝² : CharZero A
inst✝¹ : NumberField K
inst✝ : IsAlgClosed A
⊢ 0 < finrank ℚ K

Tactic:
]

Goal state after:
K : Type u_1
inst✝⁴ : Field K
A : Type u_2
inst✝³ : Field A
inst✝² : CharZero A
inst✝¹ : NumberField K
inst✝ : IsAlgClosed A
⊢ 0 < finrank ℚ K
Module: Mathlib.Analysis.Complex.Polynomial.Basic

Goal state before:
pp: p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
z : ℂ
hz1 : (aeval z) p = 0
hz2 : z.im = 0
key : (IsScalarTower.toAlgHom ℚ ℝ ℂ) z.re = z
⊢ (aeval z) p = 0

Tactic:
]

Goal state after:
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
z : ℂ
hz1 : (aeval z) p = 0
hz2 : z.im = 0
key : (IsScalarTower.toAlgHom ℚ ℝ ℂ) z.re = z
⊢ (aeval z) p = 0
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
A : Type u_2
inst✝² : NormedField A
inst✝¹ : IsAlgClosed A
inst✝ : NormedAlgebra ℚ A
B : ℝ
x : K
h : ∀ (φ : K →+* A), ‖φ x‖ ≤ B
i : ℕ
hx : IsIntegral ℚ x
⊢ ‖(map (algebraMap ℚ A) (minpoly ℚ x)).coeff i‖ ≤ max B 1 ^ finrank ℚ K * ↑((finrank ℚ K).choose (finrank ℚ K / 2))

Tactic:
]

Goal state after:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
A : Type u_2
inst✝² : NormedField A
inst✝¹ : IsAlgClosed A
inst✝ : NormedAlgebra ℚ A
B : ℝ
x : K
h : ∀ (φ : K →+* A), ‖φ x‖ ≤ B
i : ℕ
hx : IsIntegral ℚ x
⊢ ‖(map (algebraMap ℚ A) (minpoly ℚ x)).coeff i‖ ≤ max B 1 ^ finrank ℚ K * ↑((finrank ℚ K).choose (finrank ℚ K / 2))