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

Goal state before:
pp: Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}
Gs_open :
  ∀ (J : Set ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P :
  ∀ (G : Set Ω),
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G → {Q | ↑P G < ↑Q G + ENNReal.ofReal (ε / 3)} ∈ 𝓝 P
Q : ProbabilityMeasure Ω
hQ :
  ∀ i ⊆ Iio N,
    ↑P (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) < ↑Q (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) + ENNReal.ofReal (ε / 3)
δ : ℝ
B : Set Ω
δ_gt : 2 * (ε / 3) < δ
x✝ : MeasurableSet B
JB : Set ℕ := {i | (B ∩ Es i).Nonempty ∧ i ∈ Iio N}
B_subset : B ⊆ (⋃ i ∈ JB, thickening (ε / 3) (Es i)) ∪ (⋃ j ∈ Iio N, Es j)ᶜ
ω : Ω
k : ℕ
ω_in_thEk : ∃ z ∈ Es k, dist ω z < ε / 3
B_intersects : (B ∩ Es k).Nonempty
right✝ : k ∈ Iio N
⊢ ∃ z ∈ B, dist ω z < δ

Tactic:
]

Goal state after:
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}
Gs_open :
  ∀ (J : Set ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P :
  ∀ (G : Set Ω),
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G → {Q | ↑P G < ↑Q G + ENNReal.ofReal (ε / 3)} ∈ 𝓝 P
Q : ProbabilityMeasure Ω
hQ :
  ∀ i ⊆ Iio N,
    ↑P (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) < ↑Q (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) + ENNReal.ofReal (ε / 3)
δ : ℝ
B : Set Ω
δ_gt : 2 * (ε / 3) < δ
x✝ : MeasurableSet B
JB : Set ℕ := {i | (B ∩ Es i).Nonempty ∧ i ∈ Iio N}
B_subset : B ⊆ (⋃ i ∈ JB, thickening (ε / 3) (Es i)) ∪ (⋃ j ∈ Iio N, Es j)ᶜ
ω : Ω
k : ℕ
ω_in_thEk : ∃ z ∈ Es k, dist ω z < ε / 3
B_intersects : (B ∩ Es k).Nonempty
right✝ : k ∈ Iio N
⊢ ∃ z ∈ B, dist ω z < δ
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
⊢ Integrable (fun a ↦ ‖Function.uncurry (fun x y ↦ cexp (↑x * ↑y * I)) a‖) ((volume.restrict (Set.Ioc (-r) r)).prod μ)

Tactic:
]

Goal state after:
μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
⊢ Integrable (fun a ↦ ‖Function.uncurry (fun x y ↦ cexp (↑x * ↑y * I)) a‖) ((volume.restrict (Set.Ioc (-r) r)).prod μ)
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
this : (fun a ↦ ‖Function.uncurry (fun x y ↦ cexp (↑x * ↑y * I)) a‖) = fun x ↦ 1
⊢ Integrable (fun x ↦ 1) ((volume.restrict (Set.Ioc (-r) r)).prod μ)

Tactic:
]

Goal state after:
μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
this : (fun a ↦ ‖Function.uncurry (fun x y ↦ cexp (↑x * ↑y * I)) a‖) = fun x ↦ 1
⊢ Integrable (fun x ↦ 1) ((volume.restrict (Set.Ioc (-r) r)).prod μ)
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
p : ℝ × ℝ
⊢ ‖Function.uncurry (fun x y ↦ cexp (↑x * ↑y * I)) (p.1, p.2)‖ = 1

Tactic:
]

Goal state after:
μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
p : ℝ × ℝ
⊢ ‖Function.uncurry (fun x y ↦ cexp (↑x * ↑y * I)) (p.1, p.2)‖ = 1
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
h_int : Integrable (Function.uncurry fun x y ↦ cexp (↑x * ↑y * I)) ((volume.restrict (Set.uIoc (-r) r)).prod μ)
⊢ ∫ (y : ℝ), ∫ (x : ℝ) in -r..r, cexp (↑x * ↑y * I) ∂μ = ∫ (y : ℝ), ∫ (x : ℝ) in -r..r, cexp (↑x * ↑y * I) ∂μ

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case pos
μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
h_int : Integrable (Function.uncurry fun x y ↦ cexp (↑x * ↑y * I)) ((volume.restrict (Set.uIoc (-r) r)).prod μ)
y : ℝ
hy : y = 0
⊢ ∫ (x : ℝ) in -r..r, cexp (↑x * ↑y * I) =
    if r * y = 0 then 2 * ↑r else ↑y⁻¹ * ∫ (x : ℝ) in -(y * r)..y * r, cexp (↑x * I)

Tactic:
simp [hy, two_mul]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}
Gs_open :
  ∀ (J : Set ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P :
  ∀ (G : Set Ω),
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G → {Q | ↑P G < ↑Q G + ENNReal.ofReal (ε / 3)} ∈ 𝓝 P
Q : ProbabilityMeasure Ω
δ : ℝ
B : Set Ω
δ_gt : 2 * (ε / 3) < δ
x✝ : MeasurableSet B
JB : Set ℕ := {i | (B ∩ Es i).Nonempty ∧ i ∈ Iio N}
hQ : ↑P (⋃ i ∈ JB, thickening (ε / 3) (Es i)) < ↑Q (⋃ i ∈ JB, thickening (ε / 3) (Es i)) + ENNReal.ofReal (ε / 3)
B_subset : B ⊆ (⋃ i ∈ JB, thickening (ε / 3) (Es i)) ∪ (⋃ j ∈ Iio N, Es j)ᶜ
subset_thickB : ⋃ i ∈ JB, thickening (ε / 3) (Es i) ⊆ thickening δ B
⊢ ↑Q (⋃ i ∈ JB, thickening (ε / 3) (Es i)) + ENNReal.ofReal (2 * (ε / 3)) ≤
    ↑(ofMeasure Q).toMeasure (thickening δ B) + ENNReal.ofReal δ

Tactic:
]

Goal state after:
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}
Gs_open :
  ∀ (J : Set ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P :
  ∀ (G : Set Ω),
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G → {Q | ↑P G < ↑Q G + ENNReal.ofReal (ε / 3)} ∈ 𝓝 P
Q : ProbabilityMeasure Ω
δ : ℝ
B : Set Ω
δ_gt : 2 * (ε / 3) < δ
x✝ : MeasurableSet B
JB : Set ℕ := {i | (B ∩ Es i).Nonempty ∧ i ∈ Iio N}
hQ : ↑P (⋃ i ∈ JB, thickening (ε / 3) (Es i)) < ↑Q (⋃ i ∈ JB, thickening (ε / 3) (Es i)) + ENNReal.ofReal (ε / 3)
B_subset : B ⊆ (⋃ i ∈ JB, thickening (ε / 3) (Es i)) ∪ (⋃ j ∈ Iio N, Es j)ᶜ
subset_thickB : ⋃ i ∈ JB, thickening (ε / 3) (Es i) ⊆ thickening δ B
⊢ ↑Q (⋃ i ∈ JB, thickening (ε / 3) (Es i)) + ENNReal.ofReal (2 * (ε / 3)) ≤
    ↑(ofMeasure Q).toMeasure (thickening δ B) + ENNReal.ofReal δ
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case neg.refine_2
μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
h_int : Integrable (Function.uncurry fun x y ↦ cexp (↑x * ↑y * I)) ((volume.restrict (Set.uIoc (-r) r)).prod μ)
y : ℝ
hy : ¬y = 0
h : ↑y * ∫ (x : ℝ) in -r..r, cexp (↑y * ↑x * I) = ∫ (x : ℝ) in -(y * r)..y * r, cexp (↑x * I)
⊢ ∫ (x : ℝ) in -r..r, cexp (↑x * ↑y * I) = (↑y)⁻¹ * ↑y * ∫ (x : ℝ) in -r..r, cexp (↑y * ↑x * I)

Tactic:
]

Goal state after:
case neg.refine_2
μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
h_int : Integrable (Function.uncurry fun x y ↦ cexp (↑x * ↑y * I)) ((volume.restrict (Set.uIoc (-r) r)).prod μ)
y : ℝ
hy : ¬y = 0
h : ↑y * ∫ (x : ℝ) in -r..r, cexp (↑y * ↑x * I) = ∫ (x : ℝ) in -(y * r)..y * r, cexp (↑x * I)
⊢ ∫ (x : ℝ) in -r..r, cexp (↑x * ↑y * I) = (↑y)⁻¹ * ↑y * ∫ (x : ℝ) in -r..r, cexp (↑y * ↑x * I)
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case e_f
μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
h_int : Integrable (Function.uncurry fun x y ↦ cexp (↑x * ↑y * I)) ((volume.restrict (Set.uIoc (-r) r)).prod μ)
y : ℝ
⊢ (if r * y = 0 then 2 * ↑r else ↑y⁻¹ * (2 * ↑(y * r) * ↑(sinc (y * r)))) = 2 * ↑r * ↑(sinc (r * y))

Tactic:
]

Goal state after:
case e_f
μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
h_int : Integrable (Function.uncurry fun x y ↦ cexp (↑x * ↑y * I)) ((volume.restrict (Set.uIoc (-r) r)).prod μ)
y : ℝ
⊢ (if r * y = 0 then 2 * ↑r else ↑y⁻¹ * (2 * ↑(y * r) * ↑(sinc (y * r)))) = 2 * ↑r * ↑(sinc (r * y))
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case inl
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Subsingleton E
⊢ μ {0} = μ {0}

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
h_int : Integrable (Function.uncurry fun x y ↦ cexp (↑x * ↑y * I)) ((volume.restrict (Set.uIoc (-r) r)).prod μ)
⊢ ∫ (x : ℝ), 2 * r * sinc (r * x) ∂μ = ∫ (a : ℝ), 2 * r * sinc (r * a) ∂μ

Tactic:
]

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

Goal state before:
pp: E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Nontrivial E
this : 0 < ↑(finrank ℝ E)
⊢ 1 / p * Real.Gamma (↑(finrank ℝ E) / p) / (↑(finrank ℝ E) / p * Real.Gamma (↑(finrank ℝ E) / p)) * ↑(finrank ℝ E) = 1

Tactic:
]

Goal state after:
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Nontrivial E
this : 0 < ↑(finrank ℝ E)
⊢ 1 / p * Real.Gamma (↑(finrank ℝ E) / p) / (↑(finrank ℝ E) / p * Real.Gamma (↑(finrank ℝ E) / p)) * ↑(finrank ℝ E) = 1
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case e_s
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
x : ℝ
⊢ r < |x| ↔ r < |x|

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ ∫ (x : ℝ) in {x | 2 < |2 * r⁻¹ * x|}, 1 ∂μ = ∫ (a : ℝ) in {x | 2 < |2 * r⁻¹ * x|}, 2 * 2⁻¹ ∂μ

Tactic:
]

Goal state after:
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ ∫ (x : ℝ) in {x | 2 < |2 * r⁻¹ * x|}, 1 ∂μ = ∫ (a : ℝ) in {x | 2 < |2 * r⁻¹ * x|}, 2 * 2⁻¹ ∂μ
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case e_f
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
x✝ : ℝ
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: case inr
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Nontrivial E
this✝ : 0 < ↑(finrank ℝ E)
this :
  (∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) / Real.Gamma (↑(finrank ℝ E) / p + 1) *
      ↑(finrank ℝ E) =
    1
⊢ μ (Metric.ball 0 1) = μ (Metric.ball 0 1)

pp: E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Nontrivial E
this✝ : 0 < ↑(finrank ℝ E)
this :
  (∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) / Real.Gamma (↑(finrank ℝ E) / p + 1) *
      ↑(finrank ℝ E) =
    1
⊢ μ (Metric.ball 0 1) ≠ ∞

Tactic:
]

Goal state after:
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Nontrivial E
this✝ : 0 < ↑(finrank ℝ E)
this :
  (∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) / Real.Gamma (↑(finrank ℝ E) / p + 1) *
      ↑(finrank ℝ E) =
    1
⊢ μ (Metric.ball 0 1) ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p✝ : ℝ
hp : 0 < p✝
F : Type u_1 := E
p : AddGroupNorm F := { toFun := g, map_zero' := h1, add_le' := h3, neg' := h2, eq_zero_of_map_eq_zero' := ⋯ }
this : NormedAddCommGroup F := p.toNormedAddCommGroup
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p✝) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p✝ + 1))

Tactic:
let : NormedSpace ℝ F := { norm_smul_le := fun _ _ ↦ h5 _ _ }

Goal state after:
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p✝ : ℝ
hp : 0 < p✝
F : Type u_1 := E
p : AddGroupNorm F := { toFun := g, map_zero' := h1, add_le' := h3, neg' := h2, eq_zero_of_map_eq_zero' := ⋯ }
this✝ : NormedAddCommGroup F := p.toNormedAddCommGroup
this : NormedSpace ℝ F := { toModule := inst✝⁶, norm_smul_le := ⋯ }
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p✝) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p✝ + 1))
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case hbc.refine_3
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
x : ℝ
hx : x ∈ {x | 2 < |2 * r⁻¹ * x|}
hx_ne : 2 * r⁻¹ * x ≠ 0
⊢ sinc (2 * r⁻¹ * x) ≤ 1 - 2⁻¹

Tactic:
]

Goal state after:
case hbc.refine_3
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
x : ℝ
hx : x ∈ {x | 2 < |2 * r⁻¹ * x|}
hx_ne : 2 * r⁻¹ * x ≠ 0
⊢ sinc (2 * r⁻¹ * x) ≤ 1 - 2⁻¹
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case hbc.refine_3
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
x : ℝ
hx : x ∈ {x | 2 < |2 * r⁻¹ * x|}
hx_ne : 2 * r⁻¹ * x ≠ 0
⊢ sinc (2 * r⁻¹ * x) ≤ 2⁻¹

Tactic:
]

Goal state after:
case hbc.refine_3
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
x : ℝ
hx : x ∈ {x | 2 < |2 * r⁻¹ * x|}
hx_ne : 2 * r⁻¹ * x ≠ 0
⊢ sinc (2 * r⁻¹ * x) ≤ 2⁻¹
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ 2 * ∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ ≤ 2 * ∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ

pp: μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
x : ℝ
⊢ 0 x ≤ 1 - sinc (2 * r⁻¹ * x)

Tactic:
]

Goal state after:
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
x : ℝ
⊢ 0 x ≤ 1 - sinc (2 * r⁻¹ * x)
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ 2 * ‖∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ‖ = 2⁻¹ * r * 2 * (2 * r⁻¹) * ‖1 - ∫ (x : ℝ), sinc (2 * r⁻¹ * x) ∂μ‖

Tactic:
]

Goal state after:
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ 2 * ‖∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ‖ = 2⁻¹ * r * 2 * (2 * r⁻¹) * ‖1 - ∫ (x : ℝ), sinc (2 * r⁻¹ * x) ∂μ‖
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case e_a
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ 2 = 2

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case e_a
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ ∫ (a : ℝ), 1 ∂μ - ∫ (a : ℝ), sinc (2 * r⁻¹ * a) ∂μ = 1 - ∫ (x : ℝ), sinc (2 * r⁻¹ * x) ∂μ

Tactic:
]

Goal state after:
case e_a
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ ∫ (a : ℝ), 1 ∂μ - ∫ (a : ℝ), sinc (2 * r⁻¹ * a) ∂μ = 1 - ∫ (x : ℝ), sinc (2 * r⁻¹ * x) ∂μ
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ 2⁻¹ * r * ‖2 * (↑r)⁻¹ + 2 * ↑r⁻¹ - 2 * (2 * ↑r⁻¹) * ↑(∫ (x : ℝ), sinc (2 * r⁻¹ * x) ∂μ)‖ =
    2⁻¹ * r * ‖(∫ (x : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1) - 2 * ↑(2 * r⁻¹) * ↑(∫ (x : ℝ), sinc (2 * r⁻¹ * x) ∂μ)‖

Tactic:
]

Goal state after:
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ 2⁻¹ * r * ‖2 * (↑r)⁻¹ + 2 * ↑r⁻¹ - 2 * (2 * ↑r⁻¹) * ↑(∫ (x : ℝ), sinc (2 * r⁻¹ * x) ∂μ)‖ =
    2⁻¹ * r * ‖(∫ (x : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1) - 2 * ↑(2 * r⁻¹) * ↑(∫ (x : ℝ), sinc (2 * r⁻¹ * x) ∂μ)‖
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case e'_3
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure E
inst✝ : IsProbabilityMeasure μ
L : StrongDual ℝ E
r : ℝ
hr : 0 < r
this : IsProbabilityMeasure (Measure.map (⇑L) μ)
⊢ μ.real {x | r < |L x|} = μ.real (⇑L ⁻¹' {x | r < |x|})

pp: case e'_3
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure E
inst✝ : IsProbabilityMeasure μ
L : StrongDual ℝ E
r : ℝ
hr : 0 < r
this : IsProbabilityMeasure (Measure.map (⇑L) μ)
⊢ MeasurableSet {x | r < |x|}

Tactic:
]

Goal state after:
case e'_3
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure E
inst✝ : IsProbabilityMeasure μ
L : StrongDual ℝ E
r : ℝ
hr : 0 < r
this : IsProbabilityMeasure (Measure.map (⇑L) μ)
⊢ μ.real {x | r < |L x|} = μ.real (⇑L ⁻¹' {x | r < |x|})

case e'_3
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure E
inst✝ : IsProbabilityMeasure μ
L : StrongDual ℝ E
r : ℝ
hr : 0 < r
this : IsProbabilityMeasure (Measure.map (⇑L) μ)
⊢ MeasurableSet {x | r < |x|}
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case e'_4
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure E
inst✝ : IsProbabilityMeasure μ
L : StrongDual ℝ E
r : ℝ
hr : 0 < r
this : IsProbabilityMeasure (Measure.map (⇑L) μ)
x : ℝ
⊢ charFunDual μ (x • L) = charFunDual μ (x • L)

Tactic:
]

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

Goal state before:
pp: case e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p✝ : ℝ
hp : 0 < p✝
F : Type u_1 := E
p : AddGroupNorm F := { toFun := g, map_zero' := h1, add_le' := h3, neg' := h2, eq_zero_of_map_eq_zero' := ⋯ }
this✝⁴ : NormedAddCommGroup F := p.toNormedAddCommGroup
this✝³ : NormedSpace ℝ F := { toModule := inst✝⁶, norm_smul_le := ⋯ }
this✝² : TopologicalSpace F := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
ψ : E ≃ᵐ F := φ.toHomeomorph.toMeasurableEquiv
this : MeasurePreserving (⇑ψ) μ ν
⊢ ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p✝) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p✝ + 1)) =
    ENNReal.ofReal ((∫ (x : E), Real.exp (-‖ψ x‖ ^ p✝) ∂μ) / Real.Gamma (↑(finrank ℝ F) / p✝ + 1))

Tactic:
]

Goal state after:
case e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p✝ : ℝ
hp : 0 < p✝
F : Type u_1 := E
p : AddGroupNorm F := { toFun := g, map_zero' := h1, add_le' := h3, neg' := h2, eq_zero_of_map_eq_zero' := ⋯ }
this✝⁴ : NormedAddCommGroup F := p.toNormedAddCommGroup
this✝³ : NormedSpace ℝ F := { toModule := inst✝⁶, norm_smul_le := ⋯ }
this✝² : TopologicalSpace F := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
ψ : E ≃ᵐ F := φ.toHomeomorph.toMeasurableEquiv
this : MeasurePreserving (⇑ψ) μ ν
⊢ ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p✝) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p✝ + 1)) =
    ENNReal.ofReal ((∫ (x : E), Real.exp (-‖ψ x‖ ^ p✝) ∂μ) / Real.Gamma (↑(finrank ℝ F) / p✝ + 1))