awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (goal)
Goal:
a : ℂ
r : ℝ
⊢ (volume : (a : Set ℂ) → ℝ≥0∞) (Metric.closedBall a r) = HPow.hPow (β := ℕ) (ENNReal.ofReal r) 2 * (↑NNReal.pi : ℝ≥0∞)
have rw : (volume : (a : Set ℂ) → ℝ≥0∞) (Metric.ball a r) = HPow.hPow (β := ℕ) (ENNReal.ofReal r) 2 * (↑NNReal.pi : ℝ≥0∞)
Tactic: rw [addHaar_closedBall_eq_addHaar_ball, Complex.volume_ball]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls (goal)
Goal:
a : ℂ
r : ℝ
rw : (volume : (a : Set ℂ) → ℝ≥0∞) (Metric.ball a r) = HPow.hPow (β := ℕ) (ENNReal.ofReal r) 2 * (↑NNReal.pi : ℝ≥0∞)
⊢ (volume : (a : Set ℂ) → ℝ≥0∞) (Metric.closedBall a r) = HPow.hPow (β := ℕ) (ENNReal.ofReal r) 2 * (↑NNReal.pi : ℝ≥0∞)
have rw₁ : HPow.hPow (β := ℕ) (ENNReal.ofReal r) 2 * (↑NNReal.pi : ℝ≥0∞) =
  HPow.hPow (β := ℕ) (ENNReal.ofReal r) 2 * (↑NNReal.pi : ℝ≥0∞)
Tactic: rw [addHaar_closedBall_eq_addHaar_ball, Complex.volume_ball]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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))
⊢ IsTightMeasureSet (Set.range μ)
have refine : ∀ (z : E) (hz : ‖z‖ = 1),
  Tendsto (β := ℝ) (fun (r : ℝ) ↦ limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < ‖⟪z, x⟫‖}) atTop) atTop (𝓝 0)
Tactic: refine isTightMeasureSet_range_of_tendsto_limsup_measureReal_inner_of_norm_eq_one ℝ
    (fun z hz ↦ ?_) 1 (.of_forall fun _ ↦ by simp)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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 * r⁻¹) ≤ 2 * r⁻¹
have rw : 0 ≤ 2 * r⁻¹
Tactic: rw [neg_le_self_iff]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (hypothesis)
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
have hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
Tactic: have hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹ := by rw [neg_le_self_iff]; positivity
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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
⊢ 2⁻¹ * 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)‖
have grw : 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)‖
Tactic: grw [neg_mul, intervalIntegral.norm_integral_le_integral_norm hr']
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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
grw :
  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)‖
⊢ 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)‖
have grw₁ : 2⁻¹ * r * ∫ (x : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, ‖1 - charFun (μ n) (x • z)‖ ≤
  2⁻¹ * r * ∫ (x : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, ‖1 - charFun (μ n) (x • z)‖
Tactic: grw [neg_mul, intervalIntegral.norm_integral_le_integral_norm hr']
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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
⊢ 2⁻¹ * 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
have hbc : LE.le (α := ℝ) (∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, ‖1 - charFun (μ n) (t • z)‖) (∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 2)
Tactic: gcongr
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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
⊢ LE.le (α := ℝ) (∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, ‖1 - charFun (μ n) (t • z)‖) (∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 2)
have hbc : LE.le (α := ℝ)
  (@integral _ _ _ _ Real.measurableSpace (Measure.restrict volume (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹))) fun (x : ℝ) ↦
    ‖1 - charFun (μ n) (x • z)‖)
  (@integral _ _ _ _ Real.measurableSpace (Measure.restrict volume (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹))) fun (x : ℝ) ↦ 2)
Tactic: rw [intervalIntegral.integral_of_le hr', intervalIntegral.integral_of_le hr']
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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
⊢ @Integrable ℝ _ _ _ Real.measurableSpace (fun (x : ℝ) ↦ 2) (Measure.restrict volume (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹)))
have funProp : enorm (E := ℝ) 2 ≠ ∞
Tactic: fun_prop
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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
⊢ LE.le (α := ℝ)
    (@integral _ _ _ _ Real.measurableSpace (Measure.restrict volume (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹))) fun (x : ℝ) ↦
      ‖1 - charFun (μ n) (x • z)‖)
    (@integral _ _ _ _ Real.measurableSpace (Measure.restrict volume (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹))) fun (x : ℝ) ↦ 2)
have hbc : (ae (F := Measure ℝ) (Measure.restrict volume (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹)))).EventuallyLE (β := ℝ) 0 fun (x : ℝ) ↦
  ‖1 - charFun (μ n) (x • z)‖
Tactic: refine integral_mono_of_nonneg ?_ (by fun_prop) ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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
hbc :
  (ae (F := Measure ℝ) (Measure.restrict volume (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹)))).EventuallyLE (β := ℝ) 0 fun (x : ℝ) ↦
    ‖1 - charFun (μ n) (x • z)‖
⊢ LE.le (α := ℝ)
    (@integral _ _ _ _ Real.measurableSpace (Measure.restrict volume (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹))) fun (x : ℝ) ↦
      ‖1 - charFun (μ n) (x • z)‖)
    (@integral _ _ _ _ Real.measurableSpace (Measure.restrict volume (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹))) fun (x : ℝ) ↦ 2)
have hbc₁ : (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: refine integral_mono_of_nonneg ?_ (by fun_prop) ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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
⊢ 2⁻¹ * r * ∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 2 ≤ 4
have simp : 2⁻¹ * r * ((2 * r⁻¹ + 2 * r⁻¹) * 2) ≤ 4
Tactic: simp only [intervalIntegral.integral_const, sub_neg_eq_add, smul_eq_mul]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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
⊢ 2⁻¹ * r * ((2 * r⁻¹ + 2 * r⁻¹) * 2) ≤ 4
have field_simp : 2 * HAdd.hAdd (α := ℝ) 1 1 ≤ 4
Tactic: field_simp
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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
⊢ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
have 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)‖
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
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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)‖
⊢ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
have 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
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
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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
⊢ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)) ≤ 4
have calc₂ : 2⁻¹ * r * ∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 2 ≤ 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
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (hypothesis)
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
⊢ Tendsto (β := ℝ) (fun (r : ℝ) ↦ limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < ‖⟪z, x⟫‖}) atTop) atTop (𝓝 0)
have 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
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
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (hypothesis)
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
⊢ Tendsto (β := ℝ) (fun (r : ℝ) ↦ limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < ‖⟪z, x⟫‖}) atTop) atTop (𝓝 0)
have 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))
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
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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)
have refine_1 : IsCoboundedUnder (fun (x1 x2 : ℝ) ↦ x1 ≤ x2) atTop fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}
Tactic: refine limsup_le_limsup (.of_forall fun n ↦ h_le n r hr) ?_ ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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⟫|}
⊢ 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)
have 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))
Tactic: refine limsup_le_limsup (.of_forall fun n ↦ h_le n r hr) ?_ ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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))
have 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
Tactic: refine ⟨4, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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
⊢ ∀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
have 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
Tactic: simp only [eventually_map, eventually_atTop, ge_iff_le]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyConvergence (goal)
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))
have 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)))
Tactic: refine ((Tendsto.norm ?_).const_mul _).limsup_eq