awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
y : ℝ
hy : ¬y = 0
⊢ ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) =
    if r * y = 0 then 2 * (↑r : ℂ) else (↑y⁻¹ : ℂ) * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I)
have neg : ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) = (↑y : ℂ)⁻¹ * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I)
Tactic: simp only [mul_eq_zero, hr.ne', hy, or_self, ↓reduceIte, ofReal_inv]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
y : ℝ
hy : ¬y = 0
⊢ ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) = (↑y : ℂ)⁻¹ * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I)
have neg : ∀
  (h :
    Eq (α := ℂ)
      (∫ (x : ℝ) in -r..r,
        HSMul.hSMul (α := ℝ) (β := ℂ) ((fun (x : ℝ) ↦ y) x)
          (((fun (x : ℝ) ↦ cexp ((↑x : ℂ) * I)) ∘ fun (x : ℝ) ↦ y * x) x))
      (∫ (x : ℝ) in (fun (x : ℝ) ↦ y * x) (-r)..(fun (x : ℝ) ↦ y * x) r, (fun (x : ℝ) ↦ cexp ((↑x : ℂ) * I)) x)),
  ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) = (↑y : ℂ)⁻¹ * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I)
Tactic: have h := intervalIntegral.integral_deriv_smul_comp (E := ℂ) (a := -r) (b := r)
      (f := fun x ↦ y * x) (f' := fun _ ↦ y) (g := fun x ↦ cexp (x * I)) ?_ (by fun_prop)
      (by fun_prop)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
y : ℝ
hy : ¬y = 0
neg :
  ∀
    (h :
      Eq (α := ℂ)
        (∫ (x : ℝ) in -r..r,
          HSMul.hSMul (α := ℝ) (β := ℂ) ((fun (x : ℝ) ↦ y) x)
            (((fun (x : ℝ) ↦ cexp ((↑x : ℂ) * I)) ∘ fun (x : ℝ) ↦ y * x) x))
        (∫ (x : ℝ) in (fun (x : ℝ) ↦ y * x) (-r)..(fun (x : ℝ) ↦ y * x) r, (fun (x : ℝ) ↦ cexp ((↑x : ℂ) * I)) x)),
    ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) = (↑y : ℂ)⁻¹ * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I)
⊢ ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) = (↑y : ℂ)⁻¹ * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I)
have neg₁ : ∀ x ∈ Set.uIcc (-r) r, HasDerivAt (F := ℝ) (fun (x : ℝ) ↦ y * x) ((fun (x : ℝ) ↦ y) x) x
Tactic: have h := intervalIntegral.integral_deriv_smul_comp (E := ℂ) (a := -r) (b := r)
      (f := fun x ↦ y * x) (f' := fun _ ↦ y) (g := fun x ↦ cexp (x * I)) ?_ (by fun_prop)
      (by fun_prop)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
y : ℝ
hy : ¬y = 0
x : ℝ
hx : x ∈ Set.uIcc (-r) r
⊢ HasDerivAt (F := ℝ) (fun (x : ℝ) ↦ y * x) ((fun (x : ℝ) ↦ y) x) x
have neg : HasDerivAt (fun (x : ℝ) ↦ x * y) y x
Tactic: simp_rw [mul_comm y]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (hypothesis)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
y : ℝ
hy : ¬y = 0
h :
  Eq (α := ℂ)
    (∫ (x : ℝ) in -r..r,
      HSMul.hSMul (α := ℝ) (β := ℂ) ((fun (x : ℝ) ↦ y) x)
        (((fun (x : ℝ) ↦ cexp ((↑x : ℂ) * I)) ∘ fun (x : ℝ) ↦ y * x) x))
    (∫ (x : ℝ) in (fun (x : ℝ) ↦ y * x) (-r)..(fun (x : ℝ) ↦ y * x) r, (fun (x : ℝ) ↦ cexp ((↑x : ℂ) * I)) x)
⊢ ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) = (↑y : ℂ)⁻¹ * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I)
have h₁ : (↑y : ℂ) * ∫ (x : ℝ) in -r..r, cexp ((↑y : ℂ) * (↑x : ℂ) * I) = ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I)
Tactic: simp only [Function.comp_apply, ofReal_mul, real_smul, intervalIntegral.integral_const_mul,
      mul_neg] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
y : ℝ
hy : ¬y = 0
h : (↑y : ℂ) * ∫ (x : ℝ) in -r..r, cexp ((↑y : ℂ) * (↑x : ℂ) * I) = ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I)
⊢ ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) = (↑y : ℂ)⁻¹ * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I)
have neg : ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) =
  (↑y : ℂ)⁻¹ * (↑y : ℂ) * ∫ (x : ℝ) in -r..r, cexp ((↑y : ℂ) * (↑x : ℂ) * I)
Tactic: rw [← h, ← mul_assoc]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
y : ℝ
hy : ¬y = 0
h : (↑y : ℂ) * ∫ (x : ℝ) in -r..r, cexp ((↑y : ℂ) * (↑x : ℂ) * I) = ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I)
⊢ ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) =
    (↑y : ℂ)⁻¹ * (↑y : ℂ) * ∫ (x : ℝ) in -r..r, cexp ((↑y : ℂ) * (↑x : ℂ) * I)
have neg : ∫ (x : ℝ) in -r..r, cexp ((↑(x * y) : ℂ) * I) = (↑(y⁻¹ * y) : ℂ) * ∫ (x : ℝ) in -r..r, cexp ((↑(y * x) : ℂ) * I)
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
⊢ Eq (α := ℂ)
    (∫ (y : ℝ), if r * y = 0 then 2 * (↑r : ℂ) else (↑y⁻¹ : ℂ) * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I) ∂μ)
    (∫ (x : ℝ), 2 * (↑r : ℂ) * (↑(sinc (r * x)) : ℂ) ∂μ)
have e_f : ∀ (y : ℝ),
  (if r * y = 0 then 2 * (↑r : ℂ) else (↑y⁻¹ : ℂ) * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I)) =
    2 * (↑r : ℂ) * (↑(sinc (r * y)) : ℂ)
Tactic: congr with y
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
y : ℝ
⊢ (if r * y = 0 then 2 * (↑r : ℂ) else (↑y⁻¹ : ℂ) * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I)) =
    2 * (↑r : ℂ) * (↑(sinc (r * y)) : ℂ)
have e_f : (if r * y = 0 then 2 * (↑r : ℂ) else (↑y⁻¹ : ℂ) * (2 * (↑(y * r) : ℂ) * (↑(sinc (y * r)) : ℂ))) =
  2 * (↑r : ℂ) * (↑(sinc (r * y)) : ℂ)
Tactic: rw [integral_exp_mul_I_eq_sinc]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
y : ℝ
⊢ (if r * y = 0 then 2 * (↑r : ℂ) else (↑y⁻¹ : ℂ) * (2 * (↑(y * r) : ℂ) * (↑(sinc (y * r)) : ℂ))) =
    2 * (↑r : ℂ) * (↑(sinc (r * y)) : ℂ)
have pos : ∀ (hry : r * y = 0), 2 * (↑r : ℂ) = 2 * (↑r : ℂ) * (↑(sinc (r * y)) : ℂ)
Tactic: split_ifs with hry
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
y : ℝ
pos : ∀ (hry : r * y = 0), 2 * (↑r : ℂ) = 2 * (↑r : ℂ) * (↑(sinc (r * y)) : ℂ)
⊢ (if r * y = 0 then 2 * (↑r : ℂ) else (↑y⁻¹ : ℂ) * (2 * (↑(y * r) : ℂ) * (↑(sinc (y * r)) : ℂ))) =
    2 * (↑r : ℂ) * (↑(sinc (r * y)) : ℂ)
have neg : ∀ (hry : ¬r * y = 0), (↑y⁻¹ : ℂ) * (2 * (↑(y * r) : ℂ) * (↑(sinc (y * r)) : ℂ)) = 2 * (↑r : ℂ) * (↑(sinc (r * y)) : ℂ)
Tactic: split_ifs with hry
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (hypothesis)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
y : ℝ
hry : ¬r * y = 0
⊢ (↑y⁻¹ : ℂ) * (2 * (↑(y * r) : ℂ) * (↑(sinc (y * r)) : ℂ)) = 2 * (↑r : ℂ) * (↑(sinc (r * y)) : ℂ)
have hy : y ≠ 0
Tactic: have hy : y ≠ 0 := fun hy ↦ hry (by simp [hy])
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
y : ℝ
hry : ¬r * y = 0
hy : y ≠ 0
⊢ (↑y⁻¹ : ℂ) * (2 * (↑(y * r) : ℂ) * (↑(sinc (y * r)) : ℂ)) = 2 * (↑r : ℂ) * (↑(sinc (r * y)) : ℂ)
have neg : y⁻¹ * (2 * (y * r) * sinc (y * r)) = 2 * r * sinc (r * y)
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
⊢ ∫ (x : ℝ), 2 * (↑r : ℂ) * (↑(sinc (r * x)) : ℂ) ∂μ = 2 * (↑r : ℂ) * (↑(∫ (x : ℝ), sinc (r * x) ∂μ) : ℂ)
have norm_cast : ∫ (x : ℝ), 2 * r * sinc (r * x) ∂μ = 2 * r * ∫ (x : ℝ), sinc (r * x) ∂μ
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
⊢ ∫ (x : ℝ), 2 * r * sinc (r * x) ∂μ = 2 * r * ∫ (x : ℝ), sinc (r * x) ∂μ
have rw : Eq (α := ℝ) (∫ (x : ℝ), 2 * r * sinc (r * x) ∂μ) (∫ (a : ℝ), 2 * r * sinc (r * a) ∂μ)
Tactic: rw [← integral_const_mul]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
⊢ ∫ (t : ℝ) in -r..r, charFun μ t = 2 * (↑r : ℂ) * (↑(∫ (x : ℝ), sinc (r * x) ∂μ) : ℂ)
have calc : Eq (α := ℂ) (∫ (t : ℝ) in -r..r, charFun μ t) (∫ (x : ℝ) in -r..r, ∫ (y : ℝ), cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
Tactic: calc ∫ t in -r..r, charFun μ t
  _ = ∫ x in -r..r, ∫ y, cexp (x * y * I) ∂μ := by simp_rw [charFun_apply_real]
  _ = ∫ y, ∫ x in -r..r, cexp (x * y * I) ∂volume ∂μ := by
    rw [intervalIntegral_integral_swap h_int]
  _ = ∫ y, if r * y = 0 then 2 * (r : ℂ)
      else y⁻¹ * ∫ x in -(y * r)..y * r, cexp (x * I) ∂volume ∂μ := by
    congr with y
    by_cases hy : y = 0
    · simp [hy, two_mul]
    simp only [mul_eq_zero, hr.ne', hy, or_self, ↓reduceIte, ofReal_inv]
    have h := intervalIntegral.integral_deriv_smul_comp (E := ℂ) (a := -r) (b := r)
      (f := fun x ↦ y * x) (f' := fun _ ↦ y) (g := fun x ↦ cexp (x * I)) ?_ (by fun_prop)
      (by fun_prop)
    swap
    · intro x hx
      simp_rw [mul_comm y]
      exact hasDerivAt_mul_const _
    simp only [Function.comp_apply, ofReal_mul, real_smul, intervalIntegral.integral_const_mul,
      mul_neg] at h
    rw [← h, ← mul_assoc]
    norm_cast
    simp [mul_comm _ y, mul_inv_cancel₀ hy]
  _ = ∫ x, 2 * (r : ℂ) * sinc (r * x) ∂μ := by
    congr with y
    rw [integral_exp_mul_I_eq_sinc]
    split_ifs with hry
    · simp [hry]
    have hy : y ≠ 0 := fun hy ↦ hry (by simp [hy])
    norm_cast
    field_simp
  _ = 2 * r * ∫ x, sinc (r * x) ∂μ := by
    norm_cast
    rw [← integral_const_mul]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
calc : Eq (α := ℂ) (∫ (t : ℝ) in -r..r, charFun μ t) (∫ (x : ℝ) in -r..r, ∫ (y : ℝ), cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
⊢ ∫ (t : ℝ) in -r..r, charFun μ t = 2 * (↑r : ℂ) * (↑(∫ (x : ℝ), sinc (r * x) ∂μ) : ℂ)
have calc₁ : Eq (α := ℂ) (∫ (x : ℝ) in -r..r, ∫ (y : ℝ), cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
  (∫ (y : ℝ), ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
Tactic: calc ∫ t in -r..r, charFun μ t
  _ = ∫ x in -r..r, ∫ y, cexp (x * y * I) ∂μ := by simp_rw [charFun_apply_real]
  _ = ∫ y, ∫ x in -r..r, cexp (x * y * I) ∂volume ∂μ := by
    rw [intervalIntegral_integral_swap h_int]
  _ = ∫ y, if r * y = 0 then 2 * (r : ℂ)
      else y⁻¹ * ∫ x in -(y * r)..y * r, cexp (x * I) ∂volume ∂μ := by
    congr with y
    by_cases hy : y = 0
    · simp [hy, two_mul]
    simp only [mul_eq_zero, hr.ne', hy, or_self, ↓reduceIte, ofReal_inv]
    have h := intervalIntegral.integral_deriv_smul_comp (E := ℂ) (a := -r) (b := r)
      (f := fun x ↦ y * x) (f' := fun _ ↦ y) (g := fun x ↦ cexp (x * I)) ?_ (by fun_prop)
      (by fun_prop)
    swap
    · intro x hx
      simp_rw [mul_comm y]
      exact hasDerivAt_mul_const _
    simp only [Function.comp_apply, ofReal_mul, real_smul, intervalIntegral.integral_const_mul,
      mul_neg] at h
    rw [← h, ← mul_assoc]
    norm_cast
    simp [mul_comm _ y, mul_inv_cancel₀ hy]
  _ = ∫ x, 2 * (r : ℂ) * sinc (r * x) ∂μ := by
    congr with y
    rw [integral_exp_mul_I_eq_sinc]
    split_ifs with hry
    · simp [hry]
    have hy : y ≠ 0 := fun hy ↦ hry (by simp [hy])
    norm_cast
    field_simp
  _ = 2 * r * ∫ x, sinc (r * x) ∂μ := by
    norm_cast
    rw [← integral_const_mul]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
calc : Eq (α := ℂ) (∫ (t : ℝ) in -r..r, charFun μ t) (∫ (x : ℝ) in -r..r, ∫ (y : ℝ), cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
calc₁ :
  Eq (α := ℂ) (∫ (x : ℝ) in -r..r, ∫ (y : ℝ), cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
    (∫ (y : ℝ), ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
⊢ ∫ (t : ℝ) in -r..r, charFun μ t = 2 * (↑r : ℂ) * (↑(∫ (x : ℝ), sinc (r * x) ∂μ) : ℂ)
have calc₂ : Eq (α := ℂ) (∫ (y : ℝ), ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
  (∫ (y : ℝ), if r * y = 0 then 2 * (↑r : ℂ) else (↑y⁻¹ : ℂ) * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I) ∂μ)
Tactic: calc ∫ t in -r..r, charFun μ t
  _ = ∫ x in -r..r, ∫ y, cexp (x * y * I) ∂μ := by simp_rw [charFun_apply_real]
  _ = ∫ y, ∫ x in -r..r, cexp (x * y * I) ∂volume ∂μ := by
    rw [intervalIntegral_integral_swap h_int]
  _ = ∫ y, if r * y = 0 then 2 * (r : ℂ)
      else y⁻¹ * ∫ x in -(y * r)..y * r, cexp (x * I) ∂volume ∂μ := by
    congr with y
    by_cases hy : y = 0
    · simp [hy, two_mul]
    simp only [mul_eq_zero, hr.ne', hy, or_self, ↓reduceIte, ofReal_inv]
    have h := intervalIntegral.integral_deriv_smul_comp (E := ℂ) (a := -r) (b := r)
      (f := fun x ↦ y * x) (f' := fun _ ↦ y) (g := fun x ↦ cexp (x * I)) ?_ (by fun_prop)
      (by fun_prop)
    swap
    · intro x hx
      simp_rw [mul_comm y]
      exact hasDerivAt_mul_const _
    simp only [Function.comp_apply, ofReal_mul, real_smul, intervalIntegral.integral_const_mul,
      mul_neg] at h
    rw [← h, ← mul_assoc]
    norm_cast
    simp [mul_comm _ y, mul_inv_cancel₀ hy]
  _ = ∫ x, 2 * (r : ℂ) * sinc (r * x) ∂μ := by
    congr with y
    rw [integral_exp_mul_I_eq_sinc]
    split_ifs with hry
    · simp [hry]
    have hy : y ≠ 0 := fun hy ↦ hry (by simp [hy])
    norm_cast
    field_simp
  _ = 2 * r * ∫ x, sinc (r * x) ∂μ := by
    norm_cast
    rw [← integral_const_mul]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
calc : Eq (α := ℂ) (∫ (t : ℝ) in -r..r, charFun μ t) (∫ (x : ℝ) in -r..r, ∫ (y : ℝ), cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
calc₁ :
  Eq (α := ℂ) (∫ (x : ℝ) in -r..r, ∫ (y : ℝ), cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
    (∫ (y : ℝ), ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
calc₂ :
  Eq (α := ℂ) (∫ (y : ℝ), ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
    (∫ (y : ℝ), if r * y = 0 then 2 * (↑r : ℂ) else (↑y⁻¹ : ℂ) * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I) ∂μ)
⊢ ∫ (t : ℝ) in -r..r, charFun μ t = 2 * (↑r : ℂ) * (↑(∫ (x : ℝ), sinc (r * x) ∂μ) : ℂ)
have calc₃ : Eq (α := ℂ)
  (∫ (y : ℝ), if r * y = 0 then 2 * (↑r : ℂ) else (↑y⁻¹ : ℂ) * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I) ∂μ)
  (∫ (x : ℝ), 2 * (↑r : ℂ) * (↑(sinc (r * x)) : ℂ) ∂μ)
Tactic: calc ∫ t in -r..r, charFun μ t
  _ = ∫ x in -r..r, ∫ y, cexp (x * y * I) ∂μ := by simp_rw [charFun_apply_real]
  _ = ∫ y, ∫ x in -r..r, cexp (x * y * I) ∂volume ∂μ := by
    rw [intervalIntegral_integral_swap h_int]
  _ = ∫ y, if r * y = 0 then 2 * (r : ℂ)
      else y⁻¹ * ∫ x in -(y * r)..y * r, cexp (x * I) ∂volume ∂μ := by
    congr with y
    by_cases hy : y = 0
    · simp [hy, two_mul]
    simp only [mul_eq_zero, hr.ne', hy, or_self, ↓reduceIte, ofReal_inv]
    have h := intervalIntegral.integral_deriv_smul_comp (E := ℂ) (a := -r) (b := r)
      (f := fun x ↦ y * x) (f' := fun _ ↦ y) (g := fun x ↦ cexp (x * I)) ?_ (by fun_prop)
      (by fun_prop)
    swap
    · intro x hx
      simp_rw [mul_comm y]
      exact hasDerivAt_mul_const _
    simp only [Function.comp_apply, ofReal_mul, real_smul, intervalIntegral.integral_const_mul,
      mul_neg] at h
    rw [← h, ← mul_assoc]
    norm_cast
    simp [mul_comm _ y, mul_inv_cancel₀ hy]
  _ = ∫ x, 2 * (r : ℂ) * sinc (r * x) ∂μ := by
    congr with y
    rw [integral_exp_mul_I_eq_sinc]
    split_ifs with hry
    · simp [hry]
    have hy : y ≠ 0 := fun hy ↦ hry (by simp [hy])
    norm_cast
    field_simp
  _ = 2 * r * ∫ x, sinc (r * x) ∂μ := by
    norm_cast
    rw [← integral_const_mul]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsFiniteMeasure μ
hr : 0 < r
h_int :
  @Integrable ℂ _ _ _ Prod.instMeasurableSpace (Function.uncurry fun (x y : ℝ) ↦ cexp ((↑x : ℂ) * (↑y : ℂ) * I))
    (Measure.prod (Measure.restrict volume (Set.uIoc (-r) r)) μ)
calc : Eq (α := ℂ) (∫ (t : ℝ) in -r..r, charFun μ t) (∫ (x : ℝ) in -r..r, ∫ (y : ℝ), cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
calc₁ :
  Eq (α := ℂ) (∫ (x : ℝ) in -r..r, ∫ (y : ℝ), cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
    (∫ (y : ℝ), ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
calc₂ :
  Eq (α := ℂ) (∫ (y : ℝ), ∫ (x : ℝ) in -r..r, cexp ((↑x : ℂ) * (↑y : ℂ) * I) ∂μ)
    (∫ (y : ℝ), if r * y = 0 then 2 * (↑r : ℂ) else (↑y⁻¹ : ℂ) * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I) ∂μ)
calc₃ :
  Eq (α := ℂ)
    (∫ (y : ℝ), if r * y = 0 then 2 * (↑r : ℂ) else (↑y⁻¹ : ℂ) * ∫ (x : ℝ) in -(y * r)..y * r, cexp ((↑x : ℂ) * I) ∂μ)
    (∫ (x : ℝ), 2 * (↑r : ℂ) * (↑(sinc (r * x)) : ℂ) ∂μ)
⊢ ∫ (t : ℝ) in -r..r, charFun μ t = 2 * (↑r : ℂ) * (↑(∫ (x : ℝ), sinc (r * x) ∂μ) : ℂ)
have calc₄ : ∫ (x : ℝ), 2 * (↑r : ℂ) * (↑(sinc (r * x)) : ℂ) ∂μ = 2 * (↑r : ℂ) * (↑(∫ (x : ℝ), sinc (r * x) ∂μ) : ℂ)
Tactic: calc ∫ t in -r..r, charFun μ t
  _ = ∫ x in -r..r, ∫ y, cexp (x * y * I) ∂μ := by simp_rw [charFun_apply_real]
  _ = ∫ y, ∫ x in -r..r, cexp (x * y * I) ∂volume ∂μ := by
    rw [intervalIntegral_integral_swap h_int]
  _ = ∫ y, if r * y = 0 then 2 * (r : ℂ)
      else y⁻¹ * ∫ x in -(y * r)..y * r, cexp (x * I) ∂volume ∂μ := by
    congr with y
    by_cases hy : y = 0
    · simp [hy, two_mul]
    simp only [mul_eq_zero, hr.ne', hy, or_self, ↓reduceIte, ofReal_inv]
    have h := intervalIntegral.integral_deriv_smul_comp (E := ℂ) (a := -r) (b := r)
      (f := fun x ↦ y * x) (f' := fun _ ↦ y) (g := fun x ↦ cexp (x * I)) ?_ (by fun_prop)
      (by fun_prop)
    swap
    · intro x hx
      simp_rw [mul_comm y]
      exact hasDerivAt_mul_const _
    simp only [Function.comp_apply, ofReal_mul, real_smul, intervalIntegral.integral_const_mul,
      mul_neg] at h
    rw [← h, ← mul_assoc]
    norm_cast
    simp [mul_comm _ y, mul_inv_cancel₀ hy]
  _ = ∫ x, 2 * (r : ℂ) * sinc (r * x) ∂μ := by
    congr with y
    rw [integral_exp_mul_I_eq_sinc]
    split_ifs with hry
    · simp [hry]
    have hy : y ≠ 0 := fun hy ↦ hry (by simp [hy])
    norm_cast
    field_simp
  _ = 2 * r * ∫ x, sinc (r * x) ∂μ := by
    norm_cast
    rw [← integral_const_mul]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (hypothesis)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsProbabilityMeasure μ
hr : 0 < r
⊢ μ.real {x : ℝ | r < |x|} ≤ 2⁻¹ * r * Norm.norm (E := ℂ) (∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun μ t)
have integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (ε := ℝ) (fun (x : ℝ) ↦ sinc (r * x)) μ
Tactic: have integrable_sinc_const_mul (r : ℝ) : Integrable (fun x ↦ sinc (r * x)) μ :=
    (integrable_map_measure stronglyMeasurable_sinc.aestronglyMeasurable (by fun_prop)).mp
      integrable_sinc
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (ε := ℝ) (fun (x : ℝ) ↦ sinc (r * x)) μ
⊢ μ.real {x : ℝ | r < |x|} = μ.real {x : ℝ | 2 < |2 * r⁻¹ * x|}
have e_s : ∀ (x : ℝ), Membership.mem (γ := Set ℝ) {x : ℝ | r < |x|} x ↔ Membership.mem (γ := Set ℝ) {x : ℝ | 2 < |2 * r⁻¹ * x|} x
Tactic: congr 1 with x
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (ε := ℝ) (fun (x : ℝ) ↦ sinc (r * x)) μ
x : ℝ
⊢ Membership.mem (γ := Set ℝ) {x : ℝ | r < |x|} x ↔ Membership.mem (γ := Set ℝ) {x : ℝ | 2 < |2 * r⁻¹ * x|} x
have e_s : r < |x| ↔ 2 < 2 * |r⁻¹| * |x|
Tactic: simp only [Set.mem_setOf_eq, abs_mul, Nat.abs_ofNat]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (ε := ℝ) (fun (x : ℝ) ↦ sinc (r * x)) μ
x : ℝ
⊢ r < |x| ↔ 2 < 2 * |r⁻¹| * |x|
have e_s : r < |x| ↔ 2 < 2 * r⁻¹ * |x|
Tactic: rw [abs_of_nonneg (a := r⁻¹) (by positivity), mul_assoc, ← inv_mul_lt_iff₀ (by positivity),
      inv_mul_cancel₀ (by positivity), lt_inv_mul_iff₀ (by positivity), mul_one]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.IntegralCharFun (goal)
Goal:
μ : Measure ℝ
r : ℝ
inst : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (ε := ℝ) (fun (x : ℝ) ↦ sinc (r * x)) μ
x : ℝ
e_s : r < |x| ↔ 2 < 2 * r⁻¹ * |x|
⊢ r < |x| ↔ 2 < 2 * |r⁻¹| * |x|
have e_s₁ : r < |x| ↔ 2 < 2 * (r⁻¹ * |x|)
Tactic: rw [abs_of_nonneg (a := r⁻¹) (by positivity), mul_assoc, ← inv_mul_lt_iff₀ (by positivity),
      inv_mul_cancel₀ (by positivity), lt_inv_mul_iff₀ (by positivity), mul_one]