awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
r : ℝ
hr : 0 < r
h₁ : IsBoundedUnder (fun (x1 x2 : ℝ) ↦ x1 ≤ x2) atTop fun (n : ℕ) ↦ (μ n).real {x : E | r < ‖⟪z, x⟫‖}
⊢ LE.le (α := ℝ) 0 (limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < ‖⟪z, x⟫‖}) atTop)
have h₂ : ∀ (u : ℝ) (hu : ∀ᶠ (n : ℕ) in atTop, (μ n).real {x : E | r < ‖⟪z, x⟫‖} ≤ u), 0 ≤ u
Tactic: refine le_limsup_of_le ?_ fun u hu ↦ ?_
-- 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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
r : ℝ
hr : 0 < r
⊢ IsBoundedUnder (fun (x1 x2 : ℝ) ↦ x1 ≤ x2) atTop fun (n : ℕ) ↦ (μ n).real {x : E | r < ‖⟪z, x⟫‖}
have h₁ : ∀ᶠ (x : ℝ) in Filter.map (fun (n : ℕ) ↦ (μ n).real {x : E | r < ‖⟪z, x⟫‖}) 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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
r : ℝ
hr : 0 < r
⊢ ∀ᶠ (x : ℝ) in Filter.map (fun (n : ℕ) ↦ (μ n).real {x : E | r < ‖⟪z, x⟫‖}) atTop, (fun (x1 x2 : ℝ) ↦ x1 ≤ x2) x 4
have h₁ : ∃ (a : ℕ), ∀ (b : ℕ), a ≤ b → (μ b).real {x : E | r < ‖⟪z, x⟫‖} ≤ 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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
⊢ ∀ᶠ (b : ℝ) in atTop,
    LE.le (α := ℝ) (limsup (fun (n : ℕ) ↦ (μ n).real {x : E | b < ‖⟪z, x⟫‖}) atTop)
      ((fun (r : ℝ) ↦ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))) b)
have h₁ : ∀ (a : ℝ),
  Membership.mem (γ := Set ℝ)
    {x : ℝ |
      Membership.mem (γ := Set ℝ) {x : ℝ | (fun (x : ℝ) ↦ 0 < x) x} x →
        Membership.mem (γ := Set ℝ)
          {x : ℝ |
            (fun (b : ℝ) ↦
                LE.le (α := ℝ) (limsup (fun (n : ℕ) ↦ (μ n).real {x : E | b < ‖⟪z, x⟫‖}) atTop)
                  ((fun (r : ℝ) ↦ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))) b))
              x}
          x}
    a
Tactic: filter_upwards [eventually_gt_atTop 0] with r hr using h_limsup_le 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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
h₁ :
  ∀ (a : ℝ),
    Membership.mem (γ := Set ℝ)
      {x : ℝ |
        Membership.mem (γ := Set ℝ) {x : ℝ | (fun (x : ℝ) ↦ 0 < x) x} x →
          Membership.mem (γ := Set ℝ)
            {x : ℝ |
              (fun (b : ℝ) ↦
                  LE.le (α := ℝ) (limsup (fun (n : ℕ) ↦ (μ n).real {x : E | b < ‖⟪z, x⟫‖}) atTop)
                    ((fun (r : ℝ) ↦ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))) b))
                x}
            x}
      a
⊢ ∀ᶠ (b : ℝ) in atTop,
    LE.le (α := ℝ) (limsup (fun (n : ℕ) ↦ (μ n).real {x : E | b < ‖⟪z, x⟫‖}) atTop)
      ((fun (r : ℝ) ↦ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))) b)
have h₂ : ∀ (a : ℝ),
  0 < a →
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | a < ‖⟪z, x⟫‖}) atTop ≤
      2⁻¹ * a * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * a⁻¹..2 * a⁻¹, 1 - f (t • z))
Tactic: filter_upwards [eventually_gt_atTop 0] with r hr using h_limsup_le r hr
-- 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
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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
⊢ Tendsto (β := ℝ) (fun (r : ℝ) ↦ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))) atTop
    (𝓝 0)
have hf_tendsto : Tendsto f (𝓝 0) (𝓝 (f 0))
Tactic: have hf_tendsto := hf.tendsto
-- 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
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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
hf_tendsto : Tendsto f (𝓝 0) (𝓝 (f 0))
⊢ Tendsto (β := ℝ) (fun (r : ℝ) ↦ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))) atTop
    (𝓝 0)
have hf_tendsto₁ : ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : E⦄, dist x 0 < δ → dist (f x) (f 0) < ε
Tactic: rw [Metric.tendsto_nhds_nhds] at hf_tendsto
Cleared hypotheses: hf_tendsto
-- 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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
hf_tendsto : ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : E⦄, dist x 0 < δ → dist (f x) (f 0) < ε
⊢ Tendsto (β := ℝ) (fun (r : ℝ) ↦ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))) atTop
    (𝓝 0)
have refine_1 : ∀ ε > 0, ∃ (N : ℝ), ∀ n ≥ N, dist (2⁻¹ * n * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * n⁻¹..2 * n⁻¹, 1 - f (t • z))) 0 < ε
Tactic: rw [Metric.tendsto_atTop]
-- 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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
hf_tendsto : ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : E⦄, dist x 0 < δ → dist (f x) (f 0) < ε
ε : ℝ
hε : ε > 0
⊢ f 0 = 1
have symm : 1 = f 0
Tactic: symm
-- 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
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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
hf_tendsto : ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : E⦄, dist x 0 < δ → dist (f x) (f 0) < ε
ε : ℝ
hε : ε > 0
⊢ ∃ (N : ℝ), ∀ n ≥ N, dist (2⁻¹ * n * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * n⁻¹..2 * n⁻¹, 1 - f (t • z))) 0 < ε
have hf0 : f 0 = 1
Tactic: have hf0 : f 0 = 1 := by symm; simpa using h 0
-- 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
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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
hf_tendsto : ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : E⦄, dist x 0 < δ → dist (f x) (f 0) < ε
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
⊢ ∃ (N : ℝ), ∀ n ≥ N, dist (2⁻¹ * n * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * n⁻¹..2 * n⁻¹, 1 - f (t • z))) 0 < ε
have hf_tendsto₁ : ∀ (ε : ℝ), 0 < ε → ∃ (δ : ℝ), 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
Tactic: simp only [gt_iff_lt, dist_eq_norm_sub', zero_sub, norm_neg, hf0] at hf_tendsto
Cleared hypotheses: hf_tendsto
-- 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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ (δ : ℝ), 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
⊢ ∃ (N : ℝ), ∀ n ≥ N, dist (2⁻¹ * n * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * n⁻¹..2 * n⁻¹, 1 - f (t • z))) 0 < ε
have refine_1 : ∃ (N : ℝ), ∀ (n : ℝ), N ≤ n → 2⁻¹ * |n| * Norm.norm (E := ℂ) (∫ (t : ℝ) in -(2 * n⁻¹)..2 * n⁻¹, 1 - f (t • z)) < ε
Tactic: simp only [ge_iff_le, neg_mul, dist_zero_right, norm_mul, norm_inv,
    Real.norm_ofNat, Real.norm_eq_abs, abs_of_nonneg (norm_nonneg _)]
-- 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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ (δ : ℝ), 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
⊢ ∃ (N : ℝ), ∀ (n : ℝ), N ≤ n → 2⁻¹ * |n| * Norm.norm (E := ℂ) (∫ (t : ℝ) in -(2 * n⁻¹)..2 * n⁻¹, 1 - f (t • z)) < ε
have refine_1 : ∀ (δ : ℝ) (hδ : 0 < δ) (hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4),
  ∃ (N : ℝ), ∀ (n : ℝ), N ≤ n → 2⁻¹ * |n| * Norm.norm (E := ℂ) (∫ (t : ℝ) in -(2 * n⁻¹)..2 * n⁻¹, 1 - f (t • z)) < ε
Tactic: obtain ⟨δ, hδ, hδ_lt⟩ : ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4 :=
    hf_tendsto (ε / 4) (by 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
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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ (δ : ℝ), 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
δ : ℝ
hδ : 0 < δ
hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4
⊢ ∃ (N : ℝ), ∀ (n : ℝ), N ≤ n → 2⁻¹ * |n| * Norm.norm (E := ℂ) (∫ (t : ℝ) in -(2 * n⁻¹)..2 * n⁻¹, 1 - f (t • z)) < ε
have refine_1 : ∀ (r : ℝ) (hrδ : 4 * δ⁻¹ ≤ r), 2⁻¹ * |r| * Norm.norm (E := ℂ) (∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1 - f (t • z)) < ε
Tactic: refine ⟨4 * δ⁻¹, fun r hrδ ↦ ?_⟩
-- 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
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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ (δ : ℝ), 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
δ : ℝ
hδ : 0 < δ
hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4
r : ℝ
hrδ : 4 * δ⁻¹ ≤ r
⊢ 2⁻¹ * |r| * Norm.norm (E := ℂ) (∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1 - f (t • z)) < ε
have hr : 0 < r
Tactic: have hr : 0 < r := lt_of_lt_of_le (by positivity) hrδ
-- 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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ (δ : ℝ), 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
δ : ℝ
hδ : 0 < δ
hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4
r : ℝ
hrδ : 4 * δ⁻¹ ≤ 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
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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ (δ : ℝ), 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
δ : ℝ
hδ : 0 < δ
hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4
r : ℝ
hrδ : 4 * δ⁻¹ ≤ r
hr : 0 < r
⊢ 2⁻¹ * |r| * Norm.norm (E := ℂ) (∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1 - f (t • z)) < ε
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
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))
h_limsup_le :
  ∀ (r : ℝ) (hr : 0 < r),
    limsup (fun (n : ℕ) ↦ (μ n).real {x : E | r < |⟪z, x⟫|}) atTop ≤
      2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z))
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ (δ : ℝ), 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
δ : ℝ
hδ : 0 < δ
hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4
r : ℝ
hrδ : 4 * δ⁻¹ ≤ r
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
x : ℝ
hx : x ∈ Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹)
⊢ ‖1 - f (x • z)‖ ≤ ε / 4
have refine : ‖x • z‖ < δ
Tactic: refine (hδ_lt ?_).le