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

Goal state before:
pp: f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
⊢ IntegrableOn (fun x ↦ f |x|) (Iic 0) (Measure.map Neg.neg volume)

Tactic:
]

Goal state after:
f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
⊢ IntegrableOn (fun x ↦ f |x|) (Iic 0) (Measure.map Neg.neg volume)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Integral

Goal state before:
pp: f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
m : MeasurableEmbedding fun x ↦ -x := Homeomorph.measurableEmbedding (Homeomorph.neg ℝ)
⊢ IntegrableOn ((fun x ↦ f |x|) ∘ fun x ↦ -x) ((fun x ↦ -x) ⁻¹' Iic 0) volume

Tactic:
]

Goal state after:
f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
m : MeasurableEmbedding fun x ↦ -x := Homeomorph.measurableEmbedding (Homeomorph.neg ℝ)
⊢ IntegrableOn ((fun x ↦ f |x|) ∘ fun x ↦ -x) ((fun x ↦ -x) ⁻¹' Iic 0) volume
Module: Mathlib.Analysis.Fourier.FourierTransform

Goal state before:
pp: case e_f.e_f
𝕜 : Type u_1
inst✝²³ : CommRing 𝕜
V : Type u_2
inst✝²² : AddCommGroup V
inst✝²¹ : Module 𝕜 V
inst✝²⁰ : MeasurableSpace V
W : Type u_3
inst✝¹⁹ : AddCommGroup W
inst✝¹⁸ : Module 𝕜 W
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace ℂ E
inst✝¹⁵ : NormedAddCommGroup F
inst✝¹⁴ : NormedSpace ℂ F
inst✝¹³ : NormedAddCommGroup G
inst✝¹² : NormedSpace ℂ G
inst✝¹¹ : TopologicalSpace 𝕜
inst✝¹⁰ : IsTopologicalRing 𝕜
inst✝⁹ : TopologicalSpace V
inst✝⁸ : BorelSpace V
inst✝⁷ : TopologicalSpace W
inst✝⁶ : MeasurableSpace W
inst✝⁵ : BorelSpace W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜
ν : Measure W
inst✝⁴ : SigmaFinite μ
inst✝³ : SigmaFinite ν
inst✝² : SecondCountableTopologyEither W V
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
f : V → E
g : W → F
M : E →L⋆[ℂ] F →L[ℂ] G
he : Continuous[inst✝¹¹, _] ⇑e
hL : Continuous[instTopologicalSpaceProd, inst✝¹¹] fun p ↦ (L p.1) p.2
hf : Integrable f μ
hg : Integrable g ν
hG : CompleteSpace G
x : V
ξ : W
⊢ (M.flip (g ξ)) ((e (-(L x) ξ) • 1) • f x) = (M (f x)) ((e ((L.flip ξ) x) • 1) • g ξ)

Tactic:
]

Goal state after:
case e_f.e_f
𝕜 : Type u_1
inst✝²³ : CommRing 𝕜
V : Type u_2
inst✝²² : AddCommGroup V
inst✝²¹ : Module 𝕜 V
inst✝²⁰ : MeasurableSpace V
W : Type u_3
inst✝¹⁹ : AddCommGroup W
inst✝¹⁸ : Module 𝕜 W
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace ℂ E
inst✝¹⁵ : NormedAddCommGroup F
inst✝¹⁴ : NormedSpace ℂ F
inst✝¹³ : NormedAddCommGroup G
inst✝¹² : NormedSpace ℂ G
inst✝¹¹ : TopologicalSpace 𝕜
inst✝¹⁰ : IsTopologicalRing 𝕜
inst✝⁹ : TopologicalSpace V
inst✝⁸ : BorelSpace V
inst✝⁷ : TopologicalSpace W
inst✝⁶ : MeasurableSpace W
inst✝⁵ : BorelSpace W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜
ν : Measure W
inst✝⁴ : SigmaFinite μ
inst✝³ : SigmaFinite ν
inst✝² : SecondCountableTopologyEither W V
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
f : V → E
g : W → F
M : E →L⋆[ℂ] F →L[ℂ] G
he : Continuous[inst✝¹¹, _] ⇑e
hL : Continuous[instTopologicalSpaceProd, inst✝¹¹] fun p ↦ (L p.1) p.2
hf : Integrable f μ
hg : Integrable g ν
hG : CompleteSpace G
x : V
ξ : W
⊢ (M.flip (g ξ)) ((e (-(L x) ξ) • 1) • f x) = (M (f x)) ((e ((L.flip ξ) x) • 1) • g ξ)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Integral

Goal state before:
pp: f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
int_Iic : IntegrableOn (fun x ↦ f |x|) (Iic 0) volume
⊢ ∫ (x : ℝ), f |x| = ∫ (x : ℝ), f |x|

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: H : Type u_8
V : Type u_9
inst✝⁷ : NormedAddCommGroup H
inst✝⁶ : NormedSpace ℝ H
inst✝⁵ : FiniteDimensional ℝ H
inst✝⁴ : MeasurableSpace H
inst✝³ : BorelSpace H
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℂ V
f g : 𝓢(H, V)
μ : Measure H
inst✝ : μ.HasTemperateGrowth
hf_ae : ↑↑(f.toLp 2 μ) =ᵐ[μ] ⇑f
hg_ae : ↑↑(g.toLp 2 μ) =ᵐ[μ] ⇑g
a✝ : H
hf : ↑↑(f.toLp 2 μ) a✝ = f a✝
hg : ↑↑(g.toLp 2 μ) a✝ = g a✝
⊢ inner ℂ (f a✝) (g a✝) = inner ℂ (f a✝) (g a✝)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Integral

Goal state before:
pp: f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
int_Iic : IntegrableOn (fun x ↦ f |x|) (Iic 0) volume
⊢ (∫ (x : ℝ) in Iic 0, f |x|) + ∫ (x : ℝ) in Ioi 0, f x = (∫ (x : ℝ) in Ioi 0, f x) + ∫ (x : ℝ) in Ioi 0, f x

Tactic:
]

Goal state after:
f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
int_Iic : IntegrableOn (fun x ↦ f |x|) (Iic 0) volume
⊢ (∫ (x : ℝ) in Iic 0, f |x|) + ∫ (x : ℝ) in Ioi 0, f x = (∫ (x : ℝ) in Ioi 0, f x) + ∫ (x : ℝ) in Ioi 0, f x
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: c : ℝ
⊢ 𝓝 (rexp c) = 𝓝 (rexp c)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Integral

Goal state before:
pp: f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
int_Iic : IntegrableOn (fun x ↦ f |x|) (Iic 0) volume
⊢ ∫ (x : ℝ) in Iic 0, f |x| = ∫ (x : ℝ) in Iic 0, f (-x)

Tactic:
]

Goal state after:
f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
int_Iic : IntegrableOn (fun x ↦ f |x|) (Iic 0) volume
⊢ ∫ (x : ℝ) in Iic 0, f |x| = ∫ (x : ℝ) in Iic 0, f (-x)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Integral

Goal state before:
pp: f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
int_Iic : IntegrableOn (fun x ↦ f |x|) (Iic 0) volume
x✝ : ℝ
hx : x✝ ∈ Iic 0
⊢ f (-x✝) = f (-x✝)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Integral

Goal state before:
pp: case neg
f : ℝ → ℝ
eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x
hf : ¬IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume
this : ¬Integrable (fun x ↦ f |x|) volume
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: a : ℝ
ha : a < 0
c : ℝ
⊢ -(Complex.exp (↑a * ↑c)).re / a = -(Complex.exp (↑a * ↑c)).re / a

pp: a : ℝ
ha : a < 0
c : ℝ
⊢ Integrable (fun x ↦ Complex.exp (↑a * ↑x)) (volume.restrict (Ioi c))

Tactic:
]

Goal state after:
a : ℝ
ha : a < 0
c : ℝ
⊢ Integrable (fun x ↦ Complex.exp (↑a * ↑x)) (volume.restrict (Ioi c))
Module: Mathlib.Analysis.Fourier.FourierTransform

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F : Type u_4
V : Type u_5
W : Type u_6
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup V
inst✝⁸ : NormedSpace 𝕜 V
inst✝⁷ : MeasurableSpace V
inst✝⁶ : BorelSpace V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace 𝕜 W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →L[𝕜] W →L[𝕜] 𝕜
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : V → F →L[ℝ] E
a : F
w : W
he : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑e
hf : Integrable f μ
⊢ ∫ (x : V), (e (-((ContinuousLinearMap.toLinearMap₁₂ L) x) w) • f x) a ∂μ =
    fourierIntegral e μ (ContinuousLinearMap.toLinearMap₁₂ L) (fun x ↦ (f x) a) w

pp: case φ_int
𝕜 : Type u_1
E : Type u_3
F : Type u_4
V : Type u_5
W : Type u_6
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup V
inst✝⁸ : NormedSpace 𝕜 V
inst✝⁷ : MeasurableSpace V
inst✝⁶ : BorelSpace V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace 𝕜 W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →L[𝕜] W →L[𝕜] 𝕜
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : V → F →L[ℝ] E
a : F
w : W
he : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑e
hf : Integrable f μ
⊢ Integrable (fun v ↦ e (-((ContinuousLinearMap.toLinearMap₁₂ L) v) w) • f v) μ

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_3
F : Type u_4
V : Type u_5
W : Type u_6
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup V
inst✝⁸ : NormedSpace 𝕜 V
inst✝⁷ : MeasurableSpace V
inst✝⁶ : BorelSpace V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace 𝕜 W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →L[𝕜] W →L[𝕜] 𝕜
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : V → F →L[ℝ] E
a : F
w : W
he : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑e
hf : Integrable f μ
⊢ ∫ (x : V), (e (-((ContinuousLinearMap.toLinearMap₁₂ L) x) w) • f x) a ∂μ =
    fourierIntegral e μ (ContinuousLinearMap.toLinearMap₁₂ L) (fun x ↦ (f x) a) w

case φ_int
𝕜 : Type u_1
E : Type u_3
F : Type u_4
V : Type u_5
W : Type u_6
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup V
inst✝⁸ : NormedSpace 𝕜 V
inst✝⁷ : MeasurableSpace V
inst✝⁶ : BorelSpace V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace 𝕜 W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →L[𝕜] W →L[𝕜] 𝕜
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : V → F →L[ℝ] E
a : F
w : W
he : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑e
hf : Integrable f μ
⊢ Integrable (fun v ↦ e (-((ContinuousLinearMap.toLinearMap₁₂ L) v) w) • f v) μ
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a c m : ℝ
ha : a < -1
hc : -m < c
hd : ∀ x ∈ Ici c, HasDerivAt (fun t ↦ (t + m) ^ (a + 1) / (a + 1)) ((x + m) ^ a) x
⊢ Tendsto (fun t ↦ (t + m) ^ (- -(a + 1)) / - -(a + 1)) atTop (𝓝 (0 / - -(a + 1)))

Tactic:
]

Goal state after:
a c m : ℝ
ha : a < -1
hc : -m < c
hd : ∀ x ∈ Ici c, HasDerivAt (fun t ↦ (t + m) ^ (a + 1) / (a + 1)) ((x + m) ^ a) x
⊢ Tendsto (fun t ↦ (t + m) ^ (- -(a + 1)) / - -(a + 1)) atTop (𝓝 (0 / - -(a + 1)))
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: s t : ℝ
ht : 0 < t
h : -1 ≤ s
H : IntegrableOn (fun x ↦ x ^ s) (Ioi t) volume
H' : IntegrableOn (fun x ↦ x ^ s) (Ioi (max 1 t)) volume
x : ℝ
hx : x ∈ Ioi (max 1 t)
x_one : 1 ≤ x
⊢ x ^ (-1) ≤ x ^ s

Tactic:
]

Goal state after:
s t : ℝ
ht : 0 < t
h : -1 ≤ s
H : IntegrableOn (fun x ↦ x ^ s) (Ioi t) volume
H' : IntegrableOn (fun x ↦ x ^ s) (Ioi (max 1 t)) volume
x : ℝ
hx : x ∈ Ioi (max 1 t)
x_one : 1 ≤ x
⊢ x ^ (-1) ≤ x ^ s
Module: Mathlib.Analysis.Fourier.FourierTransform

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : Type u_3
V : Type u_5
W : Type u_6
inst✝¹¹ : Fintype ι
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup V
inst✝⁸ : NormedSpace 𝕜 V
inst✝⁷ : MeasurableSpace V
inst✝⁶ : BorelSpace V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace 𝕜 W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →L[𝕜] W →L[𝕜] 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
M : ι → Type u_7
inst✝¹ : (i : ι) → NormedAddCommGroup (M i)
inst✝ : (i : ι) → NormedSpace ℝ (M i)
f : V → ContinuousMultilinearMap ℝ M E
m : (i : ι) → M i
w : W
he : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑e
hf : Integrable f μ
⊢ ∫ (x : V), (e (-((ContinuousLinearMap.toLinearMap₁₂ L) x) w) • f x) m ∂μ =
    fourierIntegral e μ (ContinuousLinearMap.toLinearMap₁₂ L) (fun x ↦ (f x) m) w

pp: case φ_int
𝕜 : Type u_1
ι : Type u_2
E : Type u_3
V : Type u_5
W : Type u_6
inst✝¹¹ : Fintype ι
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup V
inst✝⁸ : NormedSpace 𝕜 V
inst✝⁷ : MeasurableSpace V
inst✝⁶ : BorelSpace V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace 𝕜 W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →L[𝕜] W →L[𝕜] 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
M : ι → Type u_7
inst✝¹ : (i : ι) → NormedAddCommGroup (M i)
inst✝ : (i : ι) → NormedSpace ℝ (M i)
f : V → ContinuousMultilinearMap ℝ M E
m : (i : ι) → M i
w : W
he : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑e
hf : Integrable f μ
⊢ Integrable (fun v ↦ e (-((ContinuousLinearMap.toLinearMap₁₂ L) v) w) • f v) μ

Tactic:
]

Goal state after:
𝕜 : Type u_1
ι : Type u_2
E : Type u_3
V : Type u_5
W : Type u_6
inst✝¹¹ : Fintype ι
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup V
inst✝⁸ : NormedSpace 𝕜 V
inst✝⁷ : MeasurableSpace V
inst✝⁶ : BorelSpace V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace 𝕜 W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →L[𝕜] W →L[𝕜] 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
M : ι → Type u_7
inst✝¹ : (i : ι) → NormedAddCommGroup (M i)
inst✝ : (i : ι) → NormedSpace ℝ (M i)
f : V → ContinuousMultilinearMap ℝ M E
m : (i : ι) → M i
w : W
he : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑e
hf : Integrable f μ
⊢ ∫ (x : V), (e (-((ContinuousLinearMap.toLinearMap₁₂ L) x) w) • f x) m ∂μ =
    fourierIntegral e μ (ContinuousLinearMap.toLinearMap₁₂ L) (fun x ↦ (f x) m) w

case φ_int
𝕜 : Type u_1
ι : Type u_2
E : Type u_3
V : Type u_5
W : Type u_6
inst✝¹¹ : Fintype ι
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup V
inst✝⁸ : NormedSpace 𝕜 V
inst✝⁷ : MeasurableSpace V
inst✝⁶ : BorelSpace V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace 𝕜 W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →L[𝕜] W →L[𝕜] 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
M : ι → Type u_7
inst✝¹ : (i : ι) → NormedAddCommGroup (M i)
inst✝ : (i : ι) → NormedSpace ℝ (M i)
f : V → ContinuousMultilinearMap ℝ M E
m : (i : ι) → M i
w : W
he : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑e
hf : Integrable f μ
⊢ Integrable (fun v ↦ e (-((ContinuousLinearMap.toLinearMap₁₂ L) v) w) • f v) μ
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: case inl
s : ℝ
h : IntegrableOn (fun x ↦ x ^ s) (Ioi 0) volume
hs : s ≤ -1
this : -1 < s
⊢ False

Tactic:
]

Goal state after:
case inl
s : ℝ
h : IntegrableOn (fun x ↦ x ^ s) (Ioi 0) volume
hs : s ≤ -1
this : -1 < s
⊢ False
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: case inr
s : ℝ
h : IntegrableOn (fun x ↦ x ^ s) (Ioi 0) volume
hs : -1 < s
this : s < -1
⊢ False

Tactic:
]

Goal state after:
case inr
s : ℝ
h : IntegrableOn (fun x ↦ x ^ s) (Ioi 0) volume
hs : -1 < s
this : s < -1
⊢ False
Module: Mathlib.MeasureTheory.Integral.Asymptotics

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
g : α → F
l : Filter α
ι : Type u_4
inst✝² : MeasurableSpace ι
f : ι × α → E
s : Set ι
μ : Measure ι
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedAddCommGroup F
hf : f =O[𝓟 s ×ˢ l] (g ∘ Prod.snd)
hμ : μ s < ⊤
C : ℝ
hC : ∀ᶠ (x : ι × α) in 𝓟 s ×ˢ l, ‖f x‖ ≤ C * ‖(g ∘ Prod.snd) x‖
t : Set (ι × α)
htl : t ∈ 𝓟 s ×ˢ l
ht : ∀ y ∈ t, ‖f y‖ ≤ C * ‖(g ∘ Prod.snd) y‖
u : Set ι
hu : u ∈ 𝓟 s
v : Set α
hv : v ∈ l
huv : u ×ˢ v ⊆ t
x : α
hx : x ∈ v
⊢ C * ‖g x‖ * μ.real s = C * μ.real s * ‖g x‖

Tactic:
by ring

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

Goal state before:
pp: a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
x : ℝ
hx : x ∈ Ioi c
⊢ (fun t ↦ t ^ a.re) x = x ^ a.re

Tactic:
]

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

Goal state before:
pp: s : ℂ
t : ℝ
ht : 0 < t
h : IntegrableOn (fun x ↦ ‖↑x ^ s‖) (Ioi t) volume
a : ℝ
ha : a ∈ Ioi t
⊢ a ^ s.re = a ^ s.re

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : V → E
hf : Integrable f volume
h'f : Differentiable ℝ f
hf' : Integrable (fderiv ℝ f) volume
⊢ 𝓕 (fderiv ℝ f) = fourierSMulRight (-(innerSL ℝ).flip) (𝓕 f)

Tactic:
]

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : V → E
hf : Integrable f volume
h'f : Differentiable ℝ f
hf' : Integrable (fderiv ℝ f) volume
⊢ 𝓕 (fderiv ℝ f) = fourierSMulRight (-(innerSL ℝ).flip) (𝓕 f)
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: s : ℂ
t : ℝ
ht : 0 < t
hs : s.re < 0
⊢ s.re < 0

Tactic:
]

Goal state after:
s : ℂ
t : ℝ
ht : 0 < t
hs : s.re < 0
⊢ s.re < 0
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: s : ℂ
t : ℝ
ht : 0 < t
hs : s.re < 0
h : IntegrableOn (fun x ↦ s * ↑x ^ (s - 1)) (Ioi t) volume
x : ℝ
hx : x ∈ Ioi t
⊢ (fun x ↦ s * ↑x ^ (s - 1)) x = s * ↑x ^ (s - 1)

Tactic:
]

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

Goal state before:
pp: s : ℂ
t : ℝ
ht : 0 < t
hs : s.re ≤ 0
x : ℝ
hx : x ∈ Ioi t
⊢ s.re * x ^ (s.re - 1) = ?m.36 x

Tactic:
]

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

Goal state before:
pp: s : ℂ
t : ℝ
ht : 0 < t
hs : s.re ≤ 0
⊢ IntegrableOn (fun x ↦ s.re * x ^ (s.re - 1)) (Ioi t) volume

Tactic:
]

Goal state after:
s : ℂ
t : ℝ
ht : 0 < t
hs : s.re ≤ 0
⊢ IntegrableOn (fun x ↦ s.re * x ^ (s.re - 1)) (Ioi t) volume
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : V → E
N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) volume
n : ℕ
hn : ↑n ≤ N
⊢ 𝓕 (iteratedFDeriv ℝ n f) = fun w ↦ fourierPowSMulRight (-(innerSL ℝ).flip) (𝓕 f) w n

Tactic:
]

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : V → E
N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) volume
n : ℕ
hn : ↑n ≤ N
⊢ 𝓕 (iteratedFDeriv ℝ n f) = fun w ↦ fourierPowSMulRight (-(innerSL ℝ).flip) (𝓕 f) w n
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: s : ℂ
t : ℝ
ht : 0 < t
hs✝ : s.re ≤ 0
hs : s.re < 0
⊢ s.re < 0

Tactic:
]

Goal state after:
s : ℂ
t : ℝ
ht : 0 < t
hs✝ : s.re ≤ 0
hs : s.re < 0
⊢ s.re < 0
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: case inl
s : ℂ
h : IntegrableOn (fun x ↦ ↑x ^ s) (Ioi 0) volume
hs : s.re ≤ -1
this : -1 < s.re
⊢ False

Tactic:
]

Goal state after:
case inl
s : ℂ
h : IntegrableOn (fun x ↦ ↑x ^ s) (Ioi 0) volume
hs : s.re ≤ -1
this : -1 < s.re
⊢ False
Module: Mathlib.MeasureTheory.Integral.Asymptotics

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
f : α → E
g : α → F
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : SecondCountableTopology α
inst✝⁸ : MeasurableSpace α
μ : Measure α
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : AddCommGroup α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsOrderedAddMonoid α
inst✝³ : CompactIccSpace α
inst✝² : atTop.IsMeasurablyGenerated
inst✝¹ : MeasurableNeg α
inst✝ : μ.IsNegInvariant
hf : LocallyIntegrable f μ
hsymm : norm ∘ f =ᵐ[μ] norm ∘ f ∘ Neg.neg
ho : f =O[atTop] g
hg : IntegrableAtFilter g atTop μ
h_int : IntegrableOn f (Ici 0) μ
⊢ IntegrableOn f (Iic ?m.106) μ ∧ IntegrableOn f (Ici ?m.106) μ

pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
f : α → E
g : α → F
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : SecondCountableTopology α
inst✝⁸ : MeasurableSpace α
μ : Measure α
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : AddCommGroup α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsOrderedAddMonoid α
inst✝³ : CompactIccSpace α
inst✝² : atTop.IsMeasurablyGenerated
inst✝¹ : MeasurableNeg α
inst✝ : μ.IsNegInvariant
hf : LocallyIntegrable f μ
hsymm : norm ∘ f =ᵐ[μ] norm ∘ f ∘ Neg.neg
ho : f =O[atTop] g
hg : IntegrableAtFilter g atTop μ
h_int : IntegrableOn f (Ici 0) μ
⊢ α

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
f : α → E
g : α → F
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : SecondCountableTopology α
inst✝⁸ : MeasurableSpace α
μ : Measure α
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : AddCommGroup α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsOrderedAddMonoid α
inst✝³ : CompactIccSpace α
inst✝² : atTop.IsMeasurablyGenerated
inst✝¹ : MeasurableNeg α
inst✝ : μ.IsNegInvariant
hf : LocallyIntegrable f μ
hsymm : norm ∘ f =ᵐ[μ] norm ∘ f ∘ Neg.neg
ho : f =O[atTop] g
hg : IntegrableAtFilter g atTop μ
h_int : IntegrableOn f (Ici 0) μ
⊢ IntegrableOn f (Iic ?m.106) μ ∧ IntegrableOn f (Ici ?m.106) μ

α : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
f : α → E
g : α → F
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : SecondCountableTopology α
inst✝⁸ : MeasurableSpace α
μ : Measure α
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : AddCommGroup α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsOrderedAddMonoid α
inst✝³ : CompactIccSpace α
inst✝² : atTop.IsMeasurablyGenerated
inst✝¹ : MeasurableNeg α
inst✝ : μ.IsNegInvariant
hf : LocallyIntegrable f μ
hsymm : norm ∘ f =ᵐ[μ] norm ∘ f ∘ Neg.neg
ho : f =O[atTop] g
hg : IntegrableAtFilter g atTop μ
h_int : IntegrableOn f (Ici 0) μ
⊢ α
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: case inr
s : ℂ
h : IntegrableOn (fun x ↦ ↑x ^ s) (Ioi 0) volume
hs : -1 < s.re
this : s.re < -1
⊢ False

Tactic:
]

Goal state after:
case inr
s : ℂ
h : IntegrableOn (fun x ↦ ↑x ^ s) (Ioi 0) volume
hs : -1 < s.re
this : s.re < -1
⊢ False
Module: Mathlib.Analysis.Fourier.FourierTransform

Goal state before:
pp: V : Type u_1
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : FiniteDimensional ℝ V
f₁ f₂ : V → E
hf : f₁ =ᵐ[volume] f₂
x : V
⊢ (fun a ↦ 𝐞 (-((innerₗ V) a) x) • f₁ a) =ᵐ[volume] fun a ↦ 𝐞 (-((innerₗ V) a) x) • f₂ a

Tactic:
filter_upwards [hf] with _ hf'

Goal state after:
V : Type u_1
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : FiniteDimensional ℝ V
f₁ f₂ : V → E
hf : f₁ =ᵐ[volume] f₂
x a✝ : V
hf' : f₁ a✝ = f₂ a✝
⊢ 𝐞 (-((innerₗ V) a✝) x) • f₁ a✝ = 𝐞 (-((innerₗ V) a✝) x) • f₂ a✝
Module: Mathlib.Analysis.Fourier.FourierTransform

Goal state before:
pp: V : Type u_1
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : FiniteDimensional ℝ V
f₁ f₂ : V → E
hf : f₁ =ᵐ[volume] f₂
x a✝ : V
hf' : f₁ a✝ = f₂ a✝
⊢ 𝐞 (-((innerₗ V) a✝) x) • f₂ a✝ = 𝐞 (-((innerₗ V) a✝) x) • f₂ a✝

Tactic:
]

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

Goal state before:
pp: a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
this : Tendsto (fun x ↦ (↑x ^ (a + 1) - ↑c ^ (a + 1)) / (a + 1)) atTop (𝓝 (-↑c ^ (a + 1) / (a + 1)))
x : ℝ
hx : 0 < x
⊢ (↑x ^ (a + 1) - ↑c ^ (a + 1)) / (a + 1) = (↑x ^ (a + 1) - ↑c ^ (a + 1)) / (a + 1)

pp: a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
this : Tendsto (fun x ↦ (↑x ^ (a + 1) - ↑c ^ (a + 1)) / (a + 1)) atTop (𝓝 (-↑c ^ (a + 1) / (a + 1)))
x : ℝ
hx : 0 < x
⊢ -1 < a.re ∨ a ≠ -1 ∧ 0 ∉ uIcc c (id x)

Tactic:
]

Goal state after:
a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
this : Tendsto (fun x ↦ (↑x ^ (a + 1) - ↑c ^ (a + 1)) / (a + 1)) atTop (𝓝 (-↑c ^ (a + 1) / (a + 1)))
x : ℝ
hx : 0 < x
⊢ -1 < a.re ∨ a ≠ -1 ∧ 0 ∉ uIcc c (id x)
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
this : Tendsto (fun x ↦ (↑x ^ (a + 1) - ↑c ^ (a + 1)) / (a + 1)) atTop (𝓝 (-↑c ^ (a + 1) / (a + 1)))
x : ℝ
hx : 0 < x
⊢ a.re ≠ -1

Tactic:
]

Goal state after:
a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
this : Tendsto (fun x ↦ (↑x ^ (a + 1) - ↑c ^ (a + 1)) / (a + 1)) atTop (𝓝 (-↑c ^ (a + 1) / (a + 1)))
x : ℝ
hx : 0 < x
⊢ a.re ≠ -1