mathlib-initiative/mathlib-tactics huggingface dataset:
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}
hzne : ∀ z ∈ sphere c r, z ≠ c
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • y‖ =
    ‖(∮ (z : ℂ) in C(c, r), (z - c)⁻¹ • f z) - ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ • y‖

Tactic:
by

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

Goal state before:
pp: E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
X : Type u_5
Y : Type u_6
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
⊢ ∫ (x : X), ∫ (y : Y), f x y ∂ν ∂μ = ∫ (y : Y), ∫ (x : X), f x y ∂μ ∂ν

Tactic:
by

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
z : ℂ
hz : z ∈ sphere c R
⊢ ‖(z - c)⁻¹ ^ (n + 1) • f z‖ ≤ C / (R ^ n * R)

Tactic:
by

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)
⊢ 2 * ↑π * I / ↑n.factorial ≠ 0

Tactic:
by

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)
⊢ iteratedDeriv n f c = n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z

Tactic:
by

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
⊢ σ = τ

Tactic:
by

Goal state after:
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
x : p.SplittingField
⊢ f x = 1 x

Tactic:
by

Goal state after:
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
⊢ Splits 0

Tactic:
by

Goal state after:
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
x : F
⊢ (C x).Splits

Tactic:
by

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
⊢ ‖iteratedDeriv n f c‖ = ‖n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z‖

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: 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
⊢ Function.Bijective (mapRoots p E)

Tactic:
by

Goal state after:
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
x✝² x✝¹ : p.Gal
x✝ : ↑(p.rootSet p.SplittingField)
⊢ (x✝² * x✝¹) • x✝ = x✝² • x✝¹ • x✝

Tactic:
by

Goal state after:
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
x✝ : ↑(p.rootSet p.SplittingField)
⊢ 1 • x✝ = x✝

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝³ : Field F
p q : F[X]
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Fact (map (algebraMap F E) p).Splits
x✝² x✝¹ : p.Gal
x✝ : ↑(p.rootSet E)
⊢ (x✝² * x✝¹) • x✝ = x✝² • x✝¹ • x✝

Tactic:
by

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
⊢ 0 < ↑n.factorial

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝³ : Field F
p q : F[X]
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Fact (map (algebraMap F E) p).Splits
x✝ : ↑(p.rootSet E)
⊢ 1 • x✝ = x✝

Tactic:
by

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
hp : Irreducible p
⊢ MulAction.IsPretransitive p.Gal ↑(p.rootSet E)

Tactic:
by

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
⊢ ‖n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z‖ ≤
    ↑n.factorial * (R * (C / R ^ (n + 1)))

Tactic:
by

Goal state after:
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
⊢ ∫ (x : Fin (n + 1) → ℝ) in Set.Icc a b, ∑ i, (f' x) (e i) i =
    ∑ i,
      ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f (i.insertNth (b i) x) i) -
        ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f (i.insertNth (a i) x) i)

Tactic:
by

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
⊢ ↑n.factorial * (R * (C / R ^ (n + 1))) = ↑n.factorial * C / R ^ n

Tactic:
by

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
ϕ : Gal(E/F)
x : ↑(p.rootSet E)
⊢ ↑((restrict p E) ϕ • x) = ϕ ↑x

Tactic:
by

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
⊢ ‖iteratedDeriv n f c‖ ≤ ↑n.factorial * C / R ^ n

Tactic:
by

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}
hzne : ∀ z ∈ sphere c r, z ≠ c
⊢ ‖(∮ (z : ℂ) in C(c, r), (z - c)⁻¹ • f z) - ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ • y‖ =
    ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ • (f z - y)‖

Tactic:
by

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
hR : 0 < R
hf : DiffContOnCl ℂ f (ball c R)
hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
⊢ ‖deriv f c‖ ≤ C / R

Tactic:
by

Goal state after:
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
⊢ ⇑eL ⁻¹' Set.Icc (eL a) (eL b) = Set.Icc a b

Tactic:
by

Goal state after:
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‖ = ‖deriv (⇑e ∘ f) c‖

Tactic:
by

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
⊢ Function.Injective ⇑(galActionHom p E)

Tactic:
by

Goal state after:
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
⊢ ‖deriv f c‖ ≤ C / R

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
p q : F[X]
inst✝ : Decidable (q = 0)
hpq : p ∣ q
⊢ restrictDvd hpq = if hq : q = 0 then 1 else restrict p q.SplittingField

Tactic:
by

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

Goal state before:
pp: F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
c : ℂ
⊢ ∃ C > 0, ∀ (z : ℂ), ‖f z‖ ≤ C

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝ : Field F
p q : F[X]
hpq : p ∣ q
hq : q ≠ 0
⊢ Function.Surjective ⇑(restrictDvd hpq)

Tactic:
by

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

Goal state before:
pp: F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
z w : ℂ
⊢ f z = f w

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

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

Tactic:
by

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

Goal state before:
pp: E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
z w : E
g : ℂ → F := f ∘ fun t ↦ t • (w - z) + z
this : g 0 = g 1
⊢ f z = f w

Tactic:
by

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

Goal state before:
pp: E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
z w : E
⊢ f z = f w

Tactic:
by

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

Goal state before:
pp: E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
hf : Differentiable ℂ f
c : F
hb : Tendsto f (cocompact E) (𝓝 c)
s : Set E
hs : s ∈ cocompact E
hs_bdd : Bornology.IsBounded (f '' s)
t : Set E
ht : IsCompact t
hts : tᶜ ⊆ s
⊢ t ∪ tᶜ ⊆ t ∪ s

Tactic:
by

Goal state after:
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) = ⇑eL.symm ⁻¹' Set.Icc a b

Tactic:
by

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

Goal state before:
pp: E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
hf : Differentiable ℂ f
c : F
hb : Tendsto f (cocompact E) (𝓝 c)
⊢ Bornology.IsBounded (range f)

Tactic:
by

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

Goal state before:
pp: E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
hf : Differentiable ℂ f
c : F
hb : Tendsto f (cocompact E) (𝓝 c)
h_bdd : Bornology.IsBounded (range f)
c' : F
hc' : f = const E c'
⊢ Tendsto f (cocompact E) (𝓝 c')

Tactic:
by

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

Goal state before:
pp: E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
hf : Differentiable ℂ f
c : F
hb : Tendsto f (cocompact E) (𝓝 c)
⊢ f = const E c

Tactic:
by

Goal state after:
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, DF x = ∫ (x : F) in Set.Icc a b, ∑ i, (f' i x) (eL.symm (e i))

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
σ : Type u_2
f : σ → R
s : Finset σ
n : ℕ
⊢ (map f s.val).esymm n = ∑ t ∈ Finset.powersetCard n s, t.prod f

Tactic:
by

Goal state after:
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
⊢ s ^ n • m.esymm n = (map (fun x ↦ s • x) m).esymm n

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
x y : R
⊢ (x ::ₘ {y}).esymm 1 = x + y

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
x y : R
⊢ (x ::ₘ {y}).esymm 2 = x * y

Tactic:
by

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
⊢ (rename ⇑e) (a✝ * b✝) = a✝ * b✝

Tactic:
by

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
⊢ (rename ⇑e) (a✝ + b✝) = a✝ + b✝

Tactic:
by

Goal state after:
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
⊢ (rename ⇑e) ((MvPolynomial.map f) φ) = (MvPolynomial.map f) φ

Tactic:
by

Goal state after:
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 : Fin (n + 1) → ℝ) in Set.Icc (eL a) (eL b), ∑ i, (f' i (eL.symm x)) (eL.symm (e i))

Tactic:
by

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
inst✝ : CommSemiring R
φ : MvPolynomial σ R
hφ : φ.IsSymmetric
e : σ ≃ τ
x✝ : Perm τ
⊢ (rename ⇑x✝) ((rename ⇑e) φ) = (rename ⇑e) φ

Tactic:
by

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
inst✝ : CommSemiring R
φ : MvPolynomial σ R
e : σ ≃ τ
h : ((rename ⇑e) φ).IsSymmetric
⊢ φ.IsSymmetric

Tactic:
by

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}
hzne : ∀ z ∈ sphere c r, z ≠ c
⊢ ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ • (f z - y)‖ ≤ 2 * π * r * (r⁻¹ * (ε / (2 * π)))

Tactic:
by

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
inst✝ : CommSemiring R
e : σ ≃ τ
p : ↥(symmetricSubalgebra τ R)
⊢ ↑(((((rename ⇑e).comp (symmetricSubalgebra σ R).val).codRestrict (symmetricSubalgebra τ R) ⋯).comp
          (((rename ⇑e.symm).comp (symmetricSubalgebra τ R).val).codRestrict (symmetricSubalgebra σ R) ⋯))
        p) =
    ↑((AlgHom.id R ↥(symmetricSubalgebra τ R)) p)

Tactic:
by

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
inst✝ : CommSemiring R
e : σ ≃ τ
p : ↥(symmetricSubalgebra σ R)
⊢ ↑(((((rename ⇑e.symm).comp (symmetricSubalgebra τ R).val).codRestrict (symmetricSubalgebra σ R) ⋯).comp
          (((rename ⇑e).comp (symmetricSubalgebra σ R).val).codRestrict (symmetricSubalgebra τ R) ⋯))
        p) =
    ↑((AlgHom.id R ↥(symmetricSubalgebra σ R)) p)

Tactic:
by

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
⊢ esymm σ R = (Multiset.map X univ.val).esymm

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝ : Field F
p q : F[X]
⊢ Function.Injective ⇑(restrictProd p q)

Tactic:
by

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}
hzne : ∀ z ∈ sphere c r, z ≠ c
⊢ 2 * π * r * (r⁻¹ * (ε / (2 * π))) = ε

Tactic:
by

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

Goal state before:
pp: S : Type u_4
σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : CommSemiring S
inst✝¹ : Fintype σ
inst✝ : Algebra R S
n : ℕ
f : σ → S
⊢ (aeval f) (esymm σ R n) = (Multiset.map f univ.val).esymm n

Tactic:
by

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
⊢ esymm σ R n = ∑ t ∈ powersetCard n univ, (monomial (∑ i ∈ t, Finsupp.single i 1)) 1

Tactic:
by

Goal state after: