awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : MeasurableSpace E
inst_3 : BorelSpace E
inst_4 : FiniteDimensional ℝ E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
s : Set E
x : E
h :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
t_bound : t ⊆ closedBall 0 1
A :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
B :
  Tendsto
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞))
    (𝓝[>] 0)
    (𝓝
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x 1) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + u) : ℝ≥0∞)))
C :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
          ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞) *
        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
          ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
⊢ EventuallyEq (β := ℝ≥0∞) (𝓝[>] 0)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
          ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞) *
        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
          ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞))
    fun (r : ℝ) ↦
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
      ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞)
have h₁ : ∀ (a : ℝ),
  Membership.mem (γ := Set ℝ) (Ioi 0) a →
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + a • t)) : ℝ≥0∞)
          ((μ : (a : Set E) → ℝ≥0∞) (closedBall x a) : ℝ≥0∞) *
        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x a) : ℝ≥0∞)
          ((μ : (a : Set E) → ℝ≥0∞) ({x} + a • u) : ℝ≥0∞) =
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + a • t)) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + a • u) : ℝ≥0∞)
Tactic: filter_upwards [self_mem_nhdsWithin]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : MeasurableSpace E
inst_3 : BorelSpace E
inst_4 : FiniteDimensional ℝ E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
s : Set E
x : E
h :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
t_bound : t ⊆ closedBall 0 1
A :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
B :
  Tendsto
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞))
    (𝓝[>] 0)
    (𝓝
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x 1) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + u) : ℝ≥0∞)))
C :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
          ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞) *
        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
          ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
r : ℝ
rpos : 0 < r
⊢ HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞) *
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
        (((μ : (a : Set E) → ℝ≥0∞) (closedBall x r))⁻¹ : ℝ≥0∞) *
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞)
have simp : HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
      (((μ : (a : Set E) → ℝ≥0∞) (closedBall x r))⁻¹ : ℝ≥0∞) *
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
      (((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u))⁻¹ : ℝ≥0∞) =
  HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
      (((μ : (a : Set E) → ℝ≥0∞) (closedBall x r))⁻¹ : ℝ≥0∞) *
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
      (((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u))⁻¹ : ℝ≥0∞)
Tactic: simp only [div_eq_mul_inv]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : MeasurableSpace E
inst_3 : BorelSpace E
inst_4 : FiniteDimensional ℝ E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
s : Set E
x : E
h :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
t_bound : t ⊆ closedBall 0 1
A :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
B :
  Tendsto
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞))
    (𝓝[>] 0)
    (𝓝
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x 1) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + u) : ℝ≥0∞)))
C :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
          ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞) *
        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
          ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
r : ℝ
rpos : 0 < r
⊢ HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
        (((μ : (a : Set E) → ℝ≥0∞) (closedBall x r))⁻¹ : ℝ≥0∞) *
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞) =
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
      ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞)
have rw : 1 *
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
      ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞) =
  HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
    ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞)
Tactic: rw [ENNReal.mul_inv_cancel (measure_closedBall_pos μ x rpos).ne'
          measure_closedBall_lt_top.ne,
        one_mul]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : MeasurableSpace E
inst_3 : BorelSpace E
inst_4 : FiniteDimensional ℝ E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
s : Set E
x : E
h :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
t_bound : t ⊆ closedBall 0 1
A :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
B :
  Tendsto
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞))
    (𝓝[>] 0)
    (𝓝
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x 1) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + u) : ℝ≥0∞)))
C :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
          ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞) *
        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
          ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
r : ℝ
rpos : 0 < r
rw :
  1 *
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞) =
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
      ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞)
⊢ HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
        (((μ : (a : Set E) → ℝ≥0∞) (closedBall x r))⁻¹ : ℝ≥0∞) *
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞) =
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
      ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞)
have rw₁ : HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
    ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞) =
  HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
    ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞)
Tactic: rw [ENNReal.mul_inv_cancel (measure_closedBall_pos μ x rpos).ne'
          measure_closedBall_lt_top.ne,
        one_mul]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : MeasurableSpace E
inst_3 : BorelSpace E
inst_4 : FiniteDimensional ℝ E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
s : Set E
x : E
h :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
t_bound : t ⊆ closedBall 0 1
A :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
B :
  Tendsto
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞))
    (𝓝[>] 0)
    (𝓝
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x 1) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + u) : ℝ≥0∞)))
C :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
          ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞) *
        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
          ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
r : ℝ
rpos : 0 < r
⊢ HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞) *
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞) =
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
      ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞)
have calc : HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
      ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞) *
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
      ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞) =
  HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞)
      (((μ : (a : Set E) → ℝ≥0∞) (closedBall x r))⁻¹ : ℝ≥0∞) *
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
      ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞)
Tactic: calc μ (s ∩ ({x} + r • t)) / μ (closedBall x r) * (μ (closedBall x r) / μ ({x} + r • u))
    _ = μ (closedBall x r) * (μ (closedBall x r))⁻¹ *
        (μ (s ∩ ({x} + r • t)) / μ ({x} + r • u)) := by simp only [div_eq_mul_inv]; ring
    _ = μ (s ∩ ({x} + r • t)) / μ ({x} + r • u) := by
      rw [ENNReal.mul_inv_cancel (measure_closedBall_pos μ x rpos).ne'
          measure_closedBall_lt_top.ne,
        one_mul]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : MeasurableSpace E
inst_3 : BorelSpace E
inst_4 : FiniteDimensional ℝ E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
s : Set E
x : E
h :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
⊢ Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
have ht' : let t' := R⁻¹ • t;
t' = R⁻¹ • t
Tactic: set t' := R⁻¹ • t with ht'
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : MeasurableSpace E
inst_3 : BorelSpace E
inst_4 : FiniteDimensional ℝ E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
s : Set E
x : E
h :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := R⁻¹ • t
ht' : t' = R⁻¹ • t
⊢ Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t)) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
have hu' : let u' := R⁻¹ • u;
u' = R⁻¹ • u
Tactic: set u' := R⁻¹ • u with hu'
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : MeasurableSpace E
inst_3 : BorelSpace E
inst_4 : FiniteDimensional ℝ E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
s : Set E
x : E
h :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := R⁻¹ • t
ht' : t' = R⁻¹ • t
u' : Set E := R⁻¹ • u
hu' : u' = R⁻¹ • u
⊢ Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t')) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u') : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
have h'u₁ : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u' : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: apply tendsto_addHaar_inter_smul_zero_of_density_zero_aux1 μ s x h t' u'
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : MeasurableSpace E
inst_3 : BorelSpace E
inst_4 : FiniteDimensional ℝ E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
s : Set E
x : E
h :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := R⁻¹ • t
ht' : t' = R⁻¹ • t
u' : Set E := R⁻¹ • u
hu' : u' = R⁻¹ • u
h'u₁ : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u' : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ ({x} + r • t')) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) ({x} + r • u') : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
have t_bound₁ : t' ⊆ closedBall 0 1
Tactic: apply tendsto_addHaar_inter_smul_zero_of_density_zero_aux1 μ s x h t' u'
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : MeasurableSpace E
inst_3 : BorelSpace E
inst_4 : FiniteDimensional ℝ E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
s : Set E
x : E
h :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := R⁻¹ • t
ht' : t' = R⁻¹ • t
u' : Set E := R⁻¹ • u
hu' : u' = R⁻¹ • u
⊢ t' ⊆ closedBall 0 1
have t_bound₁ : HSMul.hSMul (β := Set E) R⁻¹ (closedBall 0 R) = closedBall 0 1
Tactic: refine (smul_set_mono t_bound).trans_eq ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : MeasurableSpace E
inst_3 : BorelSpace E
inst_4 : FiniteDimensional ℝ E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
s : Set E
x : E
h :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := R⁻¹ • t
ht' : t' = R⁻¹ • t
u' : Set E := R⁻¹ • u
hu' : u' = R⁻¹ • u
⊢ HSMul.hSMul (β := Set E) R⁻¹ (closedBall 0 R) = closedBall 0 1
have t_bound₁ : closedBall (HSMul.hSMul (β := E) R⁻¹ 0) (‖R⁻¹‖ * R) = closedBall 0 1
Tactic: rw [smul_closedBall _ _ Rpos.le, smul_zero, Real.norm_of_nonneg (inv_nonneg.2 Rpos.le),
        inv_mul_cancel₀ Rpos.ne']
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : MeasurableSpace E
inst_3 : BorelSpace E
inst_4 : FiniteDimensional ℝ E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
s : Set E
x : E
h :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := R⁻¹ • t
ht' : t' = R⁻¹ • t
u' : Set E := R⁻¹ • u
hu' : u' = R⁻¹ • u
t_bound₁ : closedBall (HSMul.hSMul (β := E) R⁻¹ 0) (‖R⁻¹‖ * R) = closedBall 0 1
⊢ HSMul.hSMul (β := Set E) R⁻¹ (closedBall 0 R) = closedBall 0 1
have t_bound₂ : Eq.{u_1 + 1} (α := Set E) (closedBall 0 (‖R⁻¹‖ * R)) (closedBall 0 1)
Tactic: rw [smul_closedBall _ _ Rpos.le, smul_zero, Real.norm_of_nonneg (inv_nonneg.2 Rpos.le),
        inv_mul_cancel₀ Rpos.ne']
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : MeasurableSpace E
inst_3 : BorelSpace E
inst_4 : FiniteDimensional ℝ E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
s : Set E
x : E
h :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := R⁻¹ • t
ht' : t' = R⁻¹ • t
u' : Set E := R⁻¹ • u
hu' : u' = R⁻¹ • u
t_bound₁ : closedBall (HSMul.hSMul (β := E) R⁻¹ 0) (‖R⁻¹‖ * R) = closedBall 0 1
t_bound₂ : Eq.{u_1 + 1} (α := Set E) (closedBall 0 (‖R⁻¹‖ * R)) (closedBall 0 1)
⊢ HSMul.hSMul (β := Set E) R⁻¹ (closedBall 0 R) = closedBall 0 1
have t_bound₃ : Eq.{u_1 + 1} (α := Set E) (closedBall 0 (R⁻¹ * R)) (closedBall 0 1)
Tactic: rw [smul_closedBall _ _ Rpos.le, smul_zero, Real.norm_of_nonneg (inv_nonneg.2 Rpos.le),
        inv_mul_cancel₀ Rpos.ne']
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : MeasurableSpace E
inst_3 : BorelSpace E
inst_4 : FiniteDimensional ℝ E
μ : Measure E
inst_5 : μ.IsAddHaarMeasure
s : Set E
x : E
h :
  Tendsto (β := ℝ≥0∞)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) (s ∩ closedBall x r) : ℝ≥0∞)
        ((μ : (a : Set E) → ℝ≥0∞) (closedBall x r) : ℝ≥0∞))
    (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : Ne (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := R⁻¹ • t
ht' : t' = R⁻¹ • t
u' : Set E := R⁻¹ • u
hu' : u' = R⁻¹ • u
t_bound₁ : closedBall (HSMul.hSMul (β := E) R⁻¹ 0) (‖R⁻¹‖ * R) = closedBall 0 1
t_bound₂ : Eq.{u_1 + 1} (α := Set E) (closedBall 0 (‖R⁻¹‖ * R)) (closedBall 0 1)
t_bound₃ : Eq.{u_1 + 1} (α := Set E) (closedBall 0 (R⁻¹ * R)) (closedBall 0 1)
⊢ HSMul.hSMul (β := Set E) R⁻¹ (closedBall 0 R) = closedBall 0 1
have t_bound₄ : Eq.{u_1 + 1} (α := Set E) (closedBall 0 1) (closedBall 0 1)
Tactic: rw [smul_closedBall _ _ Rpos.le, smul_zero, Real.norm_of_nonneg (inv_nonneg.2 Rpos.le),
        inv_mul_cancel₀ Rpos.ne']