awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
n : ℕ
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ (ae (F := Measure ℝ) (Measure.restrict volume (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹)))).EventuallyLE (β := ℝ)
    (fun (x : ℝ) ↦ ‖1 - charFun (μ n) (x • z)‖) fun (x : ℝ) ↦ 2

Tactic:
ae_of_all _ fun _ ↦ norm_one_sub_charFun_le_two

Premises used: MeasureTheory.norm_one_sub_charFun_le_two, MeasureTheory.ae_of_all, Filter.EventuallyLE
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
n : ℕ
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ (ae (F := Measure ℝ) (Measure.restrict volume (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹)))).EventuallyLE (β := ℝ)
    (fun (x : ℝ) ↦ ‖1 - charFun (μ n) (x • z)‖) fun (x : ℝ) ↦ 2

Tactic:
exact ae_of_all _ fun _ ↦ norm_one_sub_charFun_le_two

Premises used: MeasureTheory.norm_one_sub_charFun_le_two, MeasureTheory.ae_of_all
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
n : ℕ
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
simp : 2⁻¹ * r * ((2 * r⁻¹ + 2 * r⁻¹) * 2) ≤ 4
⊢ 2⁻¹ * r * ∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 2 ≤ 4

Tactic:
simp only [intervalIntegral.integral_const, sub_neg_eq_add, smul_eq_mul]

Premises used: smul_eq_mul, sub_neg_eq_add, intervalIntegral.integral_const
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
n : ℕ
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
field_simp : 2 * HAdd.hAdd (α := ℝ) 1 1 ≤ 4
⊢ 2⁻¹ * r * ((2 * r⁻¹ + 2 * r⁻¹) * 2) ≤ 4

Tactic:
field_simp
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
n : ℕ
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ 2 * HAdd.hAdd (α := ℝ) 1 1 ≤ 4

Tactic:
norm_num
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
n : ℕ
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
calc :
  2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤
    2⁻¹ * r * ∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, ‖1 - charFun (μ n) (t • z)‖
calc₁ :
  2⁻¹ * r * ∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, ‖1 - charFun (μ n) (t • z)‖ ≤
    2⁻¹ * r * ∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 2
calc₂ : 2⁻¹ * r * ∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 2 ≤ 4
⊢ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4

Tactic:
calc 2⁻¹ * r * ‖∫ t in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
    _ ≤ 2⁻¹ * r * ∫ t in -(2 * r⁻¹)..2 * r⁻¹, ‖1 - charFun (μ n) (t • z)‖ := by
      grw [neg_mul, intervalIntegral.norm_integral_le_integral_norm hr']
    _ ≤ 2⁻¹ * r * ∫ t in -(2 * r⁻¹)..2 * r⁻¹, 2 := by
      gcongr
      rw [intervalIntegral.integral_of_le hr', intervalIntegral.integral_of_le hr']
      refine integral_mono_of_nonneg ?_ (by fun_prop) ?_
      · exact ae_of_all _ fun _ ↦ by positivity
      · exact ae_of_all _ fun _ ↦ norm_one_sub_charFun_le_two
    _ ≤ 4 := by
      simp only [intervalIntegral.integral_const, sub_neg_eq_add, smul_eq_mul]
      field_simp
      norm_num

Premises used: HMul.hMul, Inv.inv, OfNat.ofNat
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
⊢ ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4

Tactic:
have h_le_4 n r (hr : 0 < r) :
      2⁻¹ * r * ‖∫ t in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4 := by
    have hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹ := by rw [neg_le_self_iff]; positivity
    calc 2⁻¹ * r * ‖∫ t in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
    _ ≤ 2⁻¹ * r * ∫ t in -(2 * r⁻¹)..2 * r⁻¹, ‖1 - charFun (μ n) (t • z)‖ := by
      grw [neg_mul, intervalIntegral.norm_integral_le_integral_norm hr']
    _ ≤ 2⁻¹ * r * ∫ t in -(2 * r⁻¹)..2 * r⁻¹, 2 := by
      gcongr
      rw [intervalIntegral.integral_of_le hr', intervalIntegral.integral_of_le hr']
      refine integral_mono_of_nonneg ?_ (by fun_prop) ?_
      · exact ae_of_all _ fun _ ↦ by positivity
      · exact ae_of_all _ fun _ ↦ norm_one_sub_charFun_le_two
    _ ≤ 4 := by
      simp only [intervalIntegral.integral_const, sub_neg_eq_add, smul_eq_mul]
      field_simp
      norm_num
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
⊢ ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))

Tactic:
have h_le n r (hr : 0 < r) : (μ n).real {x | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * ‖∫ t in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ :=
    measureReal_abs_inner_gt_le_integral_charFun hr
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
⊢ LE.le (α := ℝ) (limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop)
    (limsup (fun (n : ℕ) ↦ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)))
      atTop)

Tactic:
limsup_le_limsup (.of_forall fun n ↦ h_le n r hr) ?_ ?_

Premises used: Filter.Eventually.of_forall, Filter.limsup_le_limsup
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
refine_1 : IsCoboundedUnder (fun (x1 x2 : ℝ) ↦ x1 ≤ x2) atTop fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}
refine_2 :
  IsBoundedUnder (fun (x1 x2 : ℝ) ↦ x1 ≤ x2) atTop fun (n : ℕ) ↦
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
⊢ LE.le (α := ℝ) (limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop)
    (limsup (fun (n : ℕ) ↦ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)))
      atTop)

Tactic:
refine limsup_le_limsup (.of_forall fun n ↦ h_le n r hr) ?_ ?_

Premises used: Filter.Eventually.of_forall, Filter.limsup_le_limsup
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
⊢ IsCoboundedUnder (fun (x1 x2 : ℝ) ↦ x1 ≤ x2) atTop fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}

Tactic:
IsCoboundedUnder.of_frequently_ge <| .of_forall fun _ ↦ ENNReal.toReal_nonneg

Premises used: Filter.Frequently.of_forall, ENNReal.toReal_nonneg, Filter.IsCoboundedUnder.of_frequently_ge
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
⊢ IsCoboundedUnder (fun (x1 x2 : ℝ) ↦ x1 ≤ x2) atTop fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}

Tactic:
exact IsCoboundedUnder.of_frequently_ge <| .of_forall fun _ ↦ ENNReal.toReal_nonneg

Premises used: Filter.IsCoboundedUnder.of_frequently_ge
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
⊢ IsBoundedUnder (fun (x1 x2 : ℝ) ↦ x1 ≤ x2) atTop fun (n : ℕ) ↦
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))

Tactic:
⟨4, ?_⟩

Premises used: Filter.IsBounded, Exists.intro, OfNat.ofNat, Filter.IsBoundedUnder
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
refine_2 :
  ∀ᶠ (x : ℝ) in
    Filter.map (fun (n : ℕ) ↦ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)))
      atTop,
    (fun (x1 x2 : ℝ) ↦ x1 ≤ x2) x 4
⊢ IsBoundedUnder (fun (x1 x2 : ℝ) ↦ x1 ≤ x2) atTop fun (n : ℕ) ↦
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))

Tactic:
refine ⟨4, ?_⟩
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
refine_2 :
  ∃ (a : ℕ),
    ∀ (b : ℕ), a ≤ b → 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ b) (t • z)) ≤ 4
⊢ ∀ᶠ (x : ℝ) in
    Filter.map (fun (n : ℕ) ↦ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)))
      atTop,
    (fun (x1 x2 : ℝ) ↦ x1 ≤ x2) x 4

Tactic:
simp only [eventually_map, eventually_atTop, ge_iff_le]

Premises used: ge_iff_le, Filter.eventually_map, Filter.eventually_atTop
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
⊢ ∃ (a : ℕ),
    ∀ (b : ℕ), a ≤ b → 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ b) (t • z)) ≤ 4

Tactic:
⟨0, fun n _ ↦ h_le_4 n r hr⟩

Premises used: Exists.intro, OfNat.ofNat
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
⊢ ∃ (a : ℕ),
    ∀ (b : ℕ), a ≤ b → 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ b) (t • z)) ≤ 4

Tactic:
exact ⟨0, fun n _ ↦ h_le_4 n r hr⟩
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
⊢ limsup (fun (n : ℕ) ↦ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)))
      atTop =
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))

Tactic:
((Tendsto.norm ?_).const_mul _).limsup_eq

Premises used: Filter.Tendsto.limsup_eq, Filter.Tendsto.norm, Filter.Tendsto.const_mul
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
refine :
  Tendsto (β := ℂ) (fun (n : ℕ) ↦ ∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) atTop
    (𝓝 (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z)))
⊢ limsup (fun (n : ℕ) ↦ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)))
      atTop =
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))

Tactic:
refine ((Tendsto.norm ?_).const_mul _).limsup_eq

Premises used: Filter.Tendsto.limsup_eq, Filter.Tendsto.norm, Filter.Tendsto.const_mul
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
simp :
  Tendsto (β := ℂ) (fun (n : ℕ) ↦ ∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1 - charFun (μ n) (t • z)) atTop
    (𝓝 (∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1 - f (t • z)))
⊢ Tendsto (β := ℂ) (fun (n : ℕ) ↦ ∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) atTop
    (𝓝 (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z)))

Tactic:
simp only [neg_mul]

Premises used: neg_mul
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
rw : 0 ≤ 2 * r⁻¹
⊢ -(2 * r⁻¹) ≤ 2 * r⁻¹

Tactic:
rw [neg_le_self_iff]

Premises used: neg_le_self_iff
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
⊢ 0 ≤ 2 * r⁻¹

Tactic:
positivity
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
⊢ -(2 * r⁻¹) ≤ 2 * r⁻¹

Tactic:
have hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹ := by rw [neg_le_self_iff]; positivity
-- Module: Mathlib.MeasureTheory.Measure.LevyConvergence
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : InnerProductSpace ℝ E
inst_2 : FiniteDimensional ℝ E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : ℕ → Measure E
inst_5 : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun (n : ℕ) ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ) (hr : 0 < r),
    (μ n).real {x : E | r < |⟪z, x⟫|} ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z))
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
simp_rw :
  Tendsto (β := ℂ)
    (fun (n : ℕ) ↦
      @integral _ _ _ _ Real.measurableSpace (Measure.restrict volume (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹))) fun (t : ℝ) ↦
        1 - charFun (μ n) (t • z))
    atTop
    (𝓝
      (@integral _ _ _ _ Real.measurableSpace (Measure.restrict volume (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹))) fun (t : ℝ) ↦
        1 - f (t • z)))
⊢ Tendsto (β := ℂ) (fun (n : ℕ) ↦ ∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1 - charFun (μ n) (t • z)) atTop
    (𝓝 (∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1 - f (t • z)))

Tactic:
simp_rw [intervalIntegral.integral_of_le hr']

Premises used: intervalIntegral.integral_of_le