mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsOpenPosMeasure
a✝ : Nontrivial E
U : Set ↑(sphere 0 1)
hUo : IsOpen U
hU : U.Nonempty
⊢ 0 ∉ Ioo 0 1

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsOpenPosMeasure
a✝ : Nontrivial E
U : Set ↑(sphere 0 1)
hUo : IsOpen U
hU : U.Nonempty
⊢ (Ioo 0 1 • Subtype.val '' U).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsOpenPosMeasure
⊢ ∀ (U : Set ↑(sphere 0 1)), IsOpen U → U.Nonempty → μ.toSphere U ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
⊢ μ.toSphere univ = ↑(dim E) * μ (ball 0 1)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
⊢ μ.toSphere.real univ = ↑(dim E) * μ.real (ball 0 1)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
⊢ μ.toSphere = 0 ↔ dim E = 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : Nontrivial E
⊢ μ.toSphere ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
⊢ μ.toSphere univ < ∞

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: n : ℕ
x : ↑(Ioi 0)
⊢ (volumeIoiPow n) (Iio x) = ENNReal.ofReal (↑x ^ (n + 1) / (↑n + 1))

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
n k : ℕ
⊢ (volumeIoiPow n) (Iio ⟨↑k + 1, ⋯⟩) < ∞

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral

Goal state before:
pp: ε : ℝ
E : Type u_2
inst✝⁶ : MeasurableSpace E
inst✝⁵ : PseudoEMetricSpace E
inst✝⁴ : BorelSpace E
inst✝³ : CompleteSpace E
inst✝² : SecondCountableTopology E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
f : E →ᵇ ℝ
A : Subalgebra ℝ (E →ᵇ ℝ)
hA : (Subalgebra.map (toContinuousMapₐ ℝ) A).SeparatesPoints
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
hε : 0 < ε
hPP' : ¬(P = 0 ∧ P' = 0)
const : ℝ := max (P.real Set.univ) (P'.real Set.univ)
pos_of_measure : 0 < const
KP : Set E
left✝¹ : KP ⊆ Set.univ
hKPco : IsCompact KP
hKPcl : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] KP
KP' : Set E
left✝ : KP' ⊆ Set.univ
hKP'co : IsCompact KP'
hKP'cl : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] KP'
K : Set E := KP ∪ KP'
hKco : IsCompact (KP ∪ KP')
hKcl : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (KP ∪ KP')
hKP : P KPᶜ < ENNReal.ofReal ε
hKP' : P' KP'ᶜ < ENNReal.ofReal ε
hKPbound : P (KP ∪ KP')ᶜ < ↑ε.toNNReal
hKP'bound : P' (KP ∪ KP')ᶜ < ↑ε.toNNReal
g' : C(E, ℝ)
hg'approx : ∀ x ∈ KP ∪ KP', ‖g' x - ↑f x‖ < √ε * const⁻¹
hg'A : ∃ x ∈ A, (toContinuousMapₐ ℝ) x = g'
g : E →ᵇ ℝ := hg'A.choose
hgA : g ∈ A
⊢ ∀ x ∈ K, ‖g x - f x‖ < √ε * const⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
s : Set ↑(sphere 0 1)
hs : s ∈ {s | MeasurableSet s}
r : ↑(Ioi 0)
⊢ Ioo 0 ↑r = ↑r • Ioo 0 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Integral

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω ^ p) ∂μ =
    ENNReal.ofReal p * ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: M : Type u_1
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
L : PeriodPair
⊢ Fintype.card (Fin 2) = finrank ℝ ℂ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ L.basis 0 = L.ω₁

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ L.basis 1 = L.ω₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
x : ℂ
⊢ x ∈ L.lattice ↔ ∃ m n, ↑m * L.ω₁ + ↑n * L.ω₂ = x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ L.ω₁ ∈ {L.ω₁, L.ω₂}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ L.ω₂ ∈ {L.ω₁, L.ω₂}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Integral

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω ^ p) ∂μ =
    ENNReal.ofReal p * ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
α β : ℚ
H : ↑α * L.ω₁ + ↑β * L.ω₂ ∈ L.lattice
m n : ℤ
e : ↑m * L.ω₁ + ↑n * L.ω₂ = ↑α * L.ω₁ + ↑β * L.ω₂
⊢ (↑m - ↑α) • L.ω₁ + (↑n - ↑β) • L.ω₂ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Integral

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedAddCommGroup F
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hd : 1 ≤ Module.finrank ℝ E
f : E → F
C α r : ℝ
hα : α < ↑(Module.finrank ℝ E)
h_decay : ∀ᵐ (x : E) ∂μ.restrict (ball 0 r), ‖f x‖ ≤ C * ‖x‖ ^ (-α)
h_meas : AEStronglyMeasurable f μ
⊢ Nontrivial E

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
α β : ℚ
⊢ ↑α * L.ω₁ + ↑β * L.ω₂ ∈ L.lattice ↔ α.den = 1 ∧ β.den = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ ¬((1 / 2).den = 1 ∧ Rat.den 0 = 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ L.ω₁ / 2 ∉ L.lattice

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ ¬(Rat.den 0 = 1 ∧ (1 / 2).den = 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ L.ω₂ / 2 ∉ L.lattice

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ L.lattice = Submodule.span ℤ (Set.range ⇑L.basis)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: M : Type u_1
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
L : PeriodPair
⊢ IsZLattice ℝ L.lattice

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
s : Set ℂ
hs : s ⊆ ↑L.lattice
⊢ IsClosed s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
x : ℂ
⊢ x ∈ (↑L.lattice \ {x})ᶜ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: M : Type u_1
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
L : PeriodPair
⊢ Submodule.span ℤ (Set.range ![L.ω₁, L.ω₂]) = L.lattice

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ ↑(L.latticeBasis 0) = L.ω₁

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ ↑(L.latticeBasis 1) = L.ω₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
x : ℤ × ℤ
⊢ ↑(L.latticeEquivProd.symm x) = ↑x.1 * L.ω₁ + ↑x.2 * L.ω₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
f : ↥L.lattice → ℂ → ℂ
u : ℝ → ↥L.lattice → ℝ
hu : ∀ r > 0, Summable (u r)
hf : ∀ r > 0, ∀ᶠ (R : ℝ) in atTop, ∀ (x : ℂ), ‖x‖ < r → ∀ (l : ↥L.lattice), ‖↑l‖ = R → ‖f l x‖ ≤ u r l
x : ℂ
⊢ 0 < ‖x‖ + 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
f : ↥L.lattice → ℂ → ℂ
u : ℝ → ↥L.lattice → ℝ
hu : ∀ r > 0, Summable (u r)
hf : ∀ r > 0, ∀ᶠ (R : ℝ) in atTop, ∀ (x : ℂ), ‖x‖ < r → ∀ (l : ↥L.lattice), ‖↑l‖ = R → ‖f l x‖ ≤ u r l
x : ℂ
⊢ x ∈ Metric.ball 0 (‖x‖ + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Integral

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedAddCommGroup F
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hd : 1 ≤ Module.finrank ℝ E
f : E → F
C α r : ℝ
hα : α < ↑(Module.finrank ℝ E)
h_decay : ∀ᵐ (x : E) ∂μ.restrict (ball 0 r), ‖f x‖ ≤ C * ‖x‖ ^ (-α)
h_meas : AEStronglyMeasurable f μ
this : Nontrivial E
⊢ IntegrableOn (fun y ↦ y ^ (↑(Module.finrank ℝ E) - 1 - α)) (Ioo 0 r) volume

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Integral

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedAddCommGroup F
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hd : 1 ≤ Module.finrank ℝ E
f : E → F
C α r : ℝ
hα : α < ↑(Module.finrank ℝ E)
h_decay : ∀ᵐ (x : E) ∂μ.restrict (ball 0 r), ‖f x‖ ≤ C * ‖x‖ ^ (-α)
h_meas : AEStronglyMeasurable f μ
this : Nontrivial E
h_rpow : IntegrableOn (fun y ↦ y ^ (↑(Module.finrank ℝ E) - 1 - α)) (Ioo 0 r) volume
y : ℝ
hy : 0 < y
right✝ : y < r
⊢ (fun x ↦ C * x ^ (↑(Module.finrank ℝ E) - 1 - α)) y = (fun y ↦ y ^ (Module.finrank ℝ E - 1) * (C * y ^ (-α))) y

Tactic:
simp only

Goal state after:
E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedAddCommGroup F
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hd : 1 ≤ Module.finrank ℝ E
f : E → F
C α r : ℝ
hα : α < ↑(Module.finrank ℝ E)
h_decay : ∀ᵐ (x : E) ∂μ.restrict (ball 0 r), ‖f x‖ ≤ C * ‖x‖ ^ (-α)
h_meas : AEStronglyMeasurable f μ
this : Nontrivial E
h_rpow : IntegrableOn (fun y ↦ y ^ (↑(Module.finrank ℝ E) - 1 - α)) (Ioo 0 r) volume
y : ℝ
hy : 0 < y
right✝ : y < r
⊢ C * y ^ (↑(Module.finrank ℝ E) - 1 - α) = y ^ (Module.finrank ℝ E - 1) * (C * y ^ (-α))
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
⊢ MeasurePreserving (⇑(homeomorphUnitSphereProd E)) (comap Subtype.val μ) (μ.toSphere.prod (volumeIoiPow (dim E - 1)))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral

Goal state before:
pp: ε : ℝ
E : Type u_2
inst✝⁶ : MeasurableSpace E
inst✝⁵ : PseudoEMetricSpace E
inst✝⁴ : BorelSpace E
inst✝³ : CompleteSpace E
inst✝² : SecondCountableTopology E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
f : E →ᵇ ℝ
A : Subalgebra ℝ (E →ᵇ ℝ)
hA : (Subalgebra.map (toContinuousMapₐ ℝ) A).SeparatesPoints
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
hε : 0 < ε
hPP' : ¬(P = 0 ∧ P' = 0)
const : ℝ := max (P.real Set.univ) (P'.real Set.univ)
pos_of_measure : 0 < const
KP : Set E
left✝¹ : KP ⊆ Set.univ
hKPco : IsCompact KP
hKPcl : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] KP
KP' : Set E
left✝ : KP' ⊆ Set.univ
hKP'co : IsCompact KP'
hKP'cl : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] KP'
K : Set E := KP ∪ KP'
hKco : IsCompact (KP ∪ KP')
hKcl : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (KP ∪ KP')
hKP : P KPᶜ < ENNReal.ofReal ε
hKP' : P' KP'ᶜ < ENNReal.ofReal ε
hKPbound : P (KP ∪ KP')ᶜ < ↑ε.toNNReal
hKP'bound : P' (KP ∪ KP')ᶜ < ↑ε.toNNReal
g' : C(E, ℝ)
hg'approx : ∀ x ∈ KP ∪ KP', ‖g' x - ↑f x‖ < √ε * const⁻¹
hg'A : ∃ x ∈ A, (toContinuousMapₐ ℝ) x = g'
g : E →ᵇ ℝ := hg'A.choose
hgA : g ∈ A
hgapprox : ∀ x ∈ K, ‖g x - f x‖ < √ε * const⁻¹
line1 : |∫ (x : E), ε.mulExpNegMulSq (f x) ∂P - ∫ (x : E) in K, ε.mulExpNegMulSq (f x) ∂P| < √ε
⊢ |∫ (x : E) in K, ε.mulExpNegMulSq (g x) ∂P - ∫ (x : E), ε.mulExpNegMulSq (g x) ∂P| < √ε

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Integral

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedAddCommGroup F
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hd : 1 ≤ Module.finrank ℝ E
f : E → F
C α r : ℝ
hα : α < ↑(Module.finrank ℝ E)
h_decay : ∀ᵐ (x : E) ∂μ.restrict (ball 0 r), ‖f x‖ ≤ C * ‖x‖ ^ (-α)
h_meas : AEStronglyMeasurable f μ
this : Nontrivial E
⊢ IntegrableOn (fun y ↦ y ^ (Module.finrank ℝ E - 1) • (C * y ^ (-α))) (Ioo 0 r) volume

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
⊢ 0 ≤ 1 - ε / 4

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Integral

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedAddCommGroup F
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hd : 1 ≤ Module.finrank ℝ E
f : E → F
C α r : ℝ
hα : α < ↑(Module.finrank ℝ E)
h_decay : ∀ᵐ (x : E) ∂μ.restrict (ball 0 r), ‖f x‖ ≤ C * ‖x‖ ^ (-α)
h_meas : AEStronglyMeasurable f μ
⊢ IntegrableOn f (ball 0 r) μ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
f : ↥L.lattice → ℂ → ℂ
u : ℝ → ↥L.lattice → ℝ
hu : ∀ r > 0, Summable (u r)
hf : ∀ r > 0, ∀ᶠ (R : ℝ) in atTop, ∀ (x : ℂ), ‖x‖ < r → ∀ (l : ↥L.lattice), ‖↑l‖ = R → ‖f l x‖ ≤ u r l
x : ℂ
r : ℝ
hr : 0 < r
hr' : 𝓝 x ≤ 𝓟 (Metric.ball 0 r)
R : ℝ
hR : ∀ (b : ℝ), R ≤ b → ∀ (x : ℂ), ‖x‖ < r → ∀ (l : ↥L.lattice), ‖↑l‖ = b → ‖f l x‖ ≤ u r l
l : ↥L.lattice
hl : l ∈ {x | (fun n ↦ ∀ x ∈ Metric.ball 0 r, ‖f n x‖ ≤ u r n) x}ᶜ
⊢ ∃ x, ‖x‖ < r ∧ u r l < ‖f l x‖

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
habs : |1 - ε / 4| = 1 - ε / 4
hy : dist 0 ((1 - ε / 4) • x) < ε / 4
⊢ 1 - ε / 4 < ε / 4

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
habs : |1 - ε / 4| = 1 - ε / 4
⊢ y ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
habs : |1 - ε / 4| = 1 - ε / 4
hy₀ : y ≠ 0
⊢ ‖y‖ ≤ dist y ((1 - ε / 4) • x) + ‖(1 - ε / 4) • x‖

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
habs : |1 - ε / 4| = 1 - ε / 4
hy₀ : y ≠ 0
⊢ dist y ((1 - ε / 4) • x) + ‖(1 - ε / 4) • x‖ < ε / 4 + ‖(1 - ε / 4) • x‖

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
habs : |1 - ε / 4| = 1 - ε / 4
hy₀ : y ≠ 0
⊢ ε / 4 + ‖(1 - ε / 4) • x‖ = 1

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
habs : |1 - ε / 4| = 1 - ε / 4
hy₀ : y ≠ 0
hy₁ : ‖y‖ < 1
u : E := ‖y‖⁻¹ • y
⊢ ‖u‖ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Integral

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedAddCommGroup F
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hdim : 1 ≤ Module.finrank ℝ E
f : E → F
C α : ℝ
hα : α < ↑(Module.finrank ℝ E)
h_decay : ∀ᵐ (x : E) ∂μ, ‖f x‖ ≤ C * ‖x‖ ^ (-α)
h_meas : AEStronglyMeasurable f μ
⊢ LocallyIntegrable f μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
habs : |1 - ε / 4| = 1 - ε / 4
hy₀ : y ≠ 0
hy₁ : ‖y‖ < 1
u : E := ‖y‖⁻¹ • y
hu₁ : ‖u‖ = 1
⊢ 0 < ‖y‖

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
habs : |1 - ε / 4| = 1 - ε / 4
hy₀ : y ≠ 0
hy₁ : ‖y‖ < 1
u : E := ‖y‖⁻¹ • y
hu₁ : ‖u‖ = 1
⊢ u ∈ sphere 0 1

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
habs : |1 - ε / 4| = 1 - ε / 4
hy₀ : y ≠ 0
hy₁ : ‖y‖ < 1
u : E := ‖y‖⁻¹ • y
hu₁ : ‖u‖ = 1
⊢ (fun x1 x2 ↦ x1 • x2) ‖y‖ u = y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: n : ℕ
h : n ≠ 0
⊢ n.nthRoot 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: n : ℕ
⊢ n.nthRoot 1 = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: n a guess : ℕ
hle : guess ≤ 0
⊢ guess = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: n a fuel : ℕ
ih : ∀ {guess : ℕ}, guess ≤ fuel → go n a fuel guess ^ (n + 2) ≤ a
guess : ℕ
hle : guess ≤ fuel + 1
h : ¬(a / guess ^ (n + 1) + (n + 1) * guess) / (n + 2) < guess
⊢ guess ≤ a / guess ^ (n + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
f : ↥L.lattice → ℂ → ℂ
u : ℝ → ↥L.lattice → ℝ
hu : ∀ r > 0, Summable (u r)
hf : ∀ r > 0, ∀ᶠ (R : ℝ) in atTop, ∀ (x : ℂ), ‖x‖ < r → ∀ (l : ↥L.lattice), ‖↑l‖ = R → ‖f l x‖ ≤ u r l
⊢ HasSumLocallyUniformly f fun x ↦ ∑' (j : ↥L.lattice), f j x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: r : ℝ
hr : 0 < r
s : ℂ
hs : ‖s‖ < r
l : ℂ
h : 2 * r ≤ ‖l‖
⊢ s ≠ l

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
habs : |1 - ε / 4| = 1 - ε / 4
hy₀ : y ≠ 0
hy₁ : ‖y‖ < 1
u : E := ‖y‖⁻¹ • y
hu₁ : ‖u‖ = 1
⊢ dist y ((1 - ε / 4) • x) + dist ((1 - ε / 4) • x) x < ε / 4 + dist ((1 - ε / 4) • x) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: r : ℝ
hr : 0 < r
s : ℂ
hs : ‖s‖ < r
l : ℂ
h : 2 * r ≤ ‖l‖
this : s ≠ l
⊢ 0 < ‖l‖

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
habs : |1 - ε / 4| = 1 - ε / 4
hy₀ : y ≠ 0
hy₁ : ‖y‖ < 1
u : E := ‖y‖⁻¹ • y
hu₁ : ‖u‖ = 1
⊢ ε / 4 + dist ((1 - ε / 4) • x) x = ε / 4 + ε / 4

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: guess fuel : ℕ
hle : guess ≤ fuel
n a : ℕ
⊢ go n a fuel guess ^ (n + 2) ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
habs : |1 - ε / 4| = 1 - ε / 4
hy₀ : y ≠ 0
hy₁ : ‖y‖ < 1
u : E := ‖y‖⁻¹ • y
hu₁ : ‖u‖ = 1
⊢ ε / 4 + ε / 4 = ε / 2

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
habs : |1 - ε / 4| = 1 - ε / 4
hy₀ : y ≠ 0
hy₁ : ‖y‖ < 1
u : E := ‖y‖⁻¹ • y
hu₁ : ‖u‖ = 1
hyx : dist y x < ε / 2
⊢ u - y = (1 - ‖y‖) • u

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: n a : ℕ
⊢ n.nthRoot a ^ n ≤ a ↔ n ≠ 0 ∨ a ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: n a b : ℕ
hb : b ≠ 0
⊢ 0 < n + 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: n a b : ℕ
hb : b ≠ 0
⊢ 0 < b ^ n

Tactic:
by

Goal state after: