awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 0)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
⊢ ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι)) ≤
    ((fun (r : ℝ) ↦
          HDiv.hDiv (α := ENNReal) (β := ENNReal)
            (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
            (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal)) ∘
        ε)
      n
have refine_2 : ∀ (this : 0 < ε n), ε n ≠ 0
Tactic: on_goal 2 =>
    calc
      ENNReal.ofReal (4⁻¹ ^ Fintype.card ι)
        = volume (closedBall (f (ε n) (hε' n)) (ε n / 4)) / volume (closedBall x (ε n)) := ?_
      _ ≤ volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n)) := by
        gcongr
        exact subset_inter ((hf₁ _ <| hε' n).trans interior_subset_closure) <| hf₀ _ <| hε' n
    have := hε' n
    rw [Real.volume_pi_closedBall, Real.volume_pi_closedBall, ← ENNReal.ofReal_div_of_pos,
      ← div_pow, mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 0)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
refine_2 : ∀ (this : 0 < ε n), ε n ≠ 0
⊢ ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι)) ≤
    ((fun (r : ℝ) ↦
          HDiv.hDiv (α := ENNReal) (β := ENNReal)
            (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
            (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal)) ∘
        ε)
      n
have refine_2₁ : ∀ (this : 0 < ε n), 0 < (2 * ε n) ^ Fintype.card ι
Tactic: on_goal 2 =>
    calc
      ENNReal.ofReal (4⁻¹ ^ Fintype.card ι)
        = volume (closedBall (f (ε n) (hε' n)) (ε n / 4)) / volume (closedBall x (ε n)) := ?_
      _ ≤ volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n)) := by
        gcongr
        exact subset_inter ((hf₁ _ <| hε' n).trans interior_subset_closure) <| hf₀ _ <| hε' n
    have := hε' n
    rw [Real.volume_pi_closedBall, Real.volume_pi_closedBall, ← ENNReal.ofReal_div_of_pos,
      ← div_pow, mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 0)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
refine_2 : ∀ (this : 0 < ε n), ε n ≠ 0
refine_2₁ : ∀ (this : 0 < ε n), 0 < (2 * ε n) ^ Fintype.card ι
⊢ ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι)) ≤
    ((fun (r : ℝ) ↦
          HDiv.hDiv (α := ENNReal) (β := ENNReal)
            (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
            (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal)) ∘
        ε)
      n
have refine_2₂ : ∀ (this : 0 < ε n), 0 ≤ ε n
Tactic: on_goal 2 =>
    calc
      ENNReal.ofReal (4⁻¹ ^ Fintype.card ι)
        = volume (closedBall (f (ε n) (hε' n)) (ε n / 4)) / volume (closedBall x (ε n)) := ?_
      _ ≤ volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n)) := by
        gcongr
        exact subset_inter ((hf₁ _ <| hε' n).trans interior_subset_closure) <| hf₀ _ <| hε' n
    have := hε' n
    rw [Real.volume_pi_closedBall, Real.volume_pi_closedBall, ← ENNReal.ofReal_div_of_pos,
      ← div_pow, mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 0)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
refine_2 : ∀ (this : 0 < ε n), ε n ≠ 0
refine_2₁ : ∀ (this : 0 < ε n), 0 < (2 * ε n) ^ Fintype.card ι
refine_2₂ : ∀ (this : 0 < ε n), 0 ≤ ε n
⊢ ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι)) ≤
    ((fun (r : ℝ) ↦
          HDiv.hDiv (α := ENNReal) (β := ENNReal)
            (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
            (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal)) ∘
        ε)
      n
have refine_2₃ : ∀ (this : 0 < ε n), 0 ≤ ε n / 4
Tactic: on_goal 2 =>
    calc
      ENNReal.ofReal (4⁻¹ ^ Fintype.card ι)
        = volume (closedBall (f (ε n) (hε' n)) (ε n / 4)) / volume (closedBall x (ε n)) := ?_
      _ ≤ volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n)) := by
        gcongr
        exact subset_inter ((hf₁ _ <| hε' n).trans interior_subset_closure) <| hf₀ _ <| hε' n
    have := hε' n
    rw [Real.volume_pi_closedBall, Real.volume_pi_closedBall, ← ENNReal.ofReal_div_of_pos,
      ← div_pow, mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
h : ∀ (δ : ℝ), 0 < δ → ∃ (y : ι → ℝ), closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior sᶜ
⊢ ¬Tendsto (β := ENNReal)
      (fun (r : ℝ) ↦
        HDiv.hDiv (α := ENNReal) (β := ENNReal)
          (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
          (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
      (𝓝[>] 0) (𝓝 1)
have choose : ∀ (f : (δ : ℝ) → 0 < δ → ι → ℝ) (hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ)
  (hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ),
  ¬Tendsto (β := ENNReal)
      (fun (r : ℝ) ↦
        HDiv.hDiv (α := ENNReal) (β := ENNReal)
          (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
          (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
      (𝓝[>] 0) (𝓝 1)
Tactic: choose f hf₀ hf₁ using h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 1)
⊢ False
have obtain : ∀ (ε : ℕ → ℝ) (hε' : ∀ (n : ℕ), 0 < ε n) (hε₀ : Tendsto ε atTop (𝓝[>] 0)), False
Tactic: obtain ⟨ε, -, hε', hε₀⟩ := exists_seq_strictAnti_tendsto_nhdsWithin (0 : ℝ)
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
⊢ False
have refine_1 : ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι)) ≠ 0
Tactic: refine not_eventually.2
      (Frequently.of_forall fun _ ↦ lt_irrefl <| 1 - ENNReal.ofReal (4⁻¹ ^ Fintype.card ι))
      ((Filter.Tendsto.eventually_lt tendsto_const_nhds (H.comp hε₀) <|
            ENNReal.sub_lt_self ENNReal.one_ne_top one_ne_zero ?_).mono
        fun n ↦ lt_of_le_of_lt' ?_)
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
refine_1 : ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι)) ≠ 0
⊢ False
have refine_2 : ∀ (n : ℕ),
  ((fun (r : ℝ) ↦
          HDiv.hDiv (α := ENNReal) (β := ENNReal)
            (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
            (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal)) ∘
        ε)
      n ≤
    1 - ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι))
Tactic: refine not_eventually.2
      (Frequently.of_forall fun _ ↦ lt_irrefl <| 1 - ENNReal.ofReal (4⁻¹ ^ Fintype.card ι))
      ((Filter.Tendsto.eventually_lt tendsto_const_nhds (H.comp hε₀) <|
            ENNReal.sub_lt_self ENNReal.one_ne_top one_ne_zero ?_).mono
        fun n ↦ lt_of_le_of_lt' ?_)
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
⊢ HDiv.hDiv (α := ENNReal) (β := ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x (ε n)) : ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x (ε n)) : ENNReal) ≤
    HDiv.hDiv (α := ENNReal) (β := ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal)
          (closedBall x (ε n) \ closedBall (f (ε n) ⋯) (ε n / 4)) :
        ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x (ε n)) : ENNReal)
have hab : closure s ∩ closedBall x (ε n) ⊆ closedBall x (ε n) \ closedBall (f (ε n) sorry) (ε n / 4)
Tactic: gcongr
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
⊢ closure s ∩ closedBall x (ε n) ⊆ closedBall x (ε n) \ closedBall (f (ε n) ⋯) (ε n / 4)
have hab : closure s ∩ closedBall x (ε n) ⊆ (closedBall (f (ε n) sorry) (ε n / 4))ᶜ ∩ closedBall x (ε n)
Tactic: rw [diff_eq_compl_inter]
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
⊢ closure s ∩ closedBall x (ε n) ⊆ (closedBall (f (ε n) ⋯) (ε n / 4))ᶜ ∩ closedBall x (ε n)
have hab : closure s ⊆ (closedBall (f (ε n) sorry) (ε n / 4))ᶜ
Tactic: refine inter_subset_inter_left _ ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
⊢ closure s ⊆ (closedBall (f (ε n) ⋯) (ε n / 4))ᶜ
have hab : closedBall (f (ε n) sorry) (ε n / 4) ⊆ interior sᶜ
Tactic: rw [subset_compl_comm, ← interior_compl]
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
⊢ ((fun (r : ℝ) ↦
          HDiv.hDiv (α := ENNReal) (β := ENNReal)
            (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
            (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal)) ∘
        ε)
      n ≤
    1 - ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι))
have refine_2 : HDiv.hDiv (α := ENNReal) (β := ENNReal)
    (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal)
        (closedBall x (ε n) \ closedBall (f (ε n) sorry) (ε n / 4)) :
      ENNReal)
    (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x (ε n)) : ENNReal) =
  1 - ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι))
Tactic: calc
      volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n))
        ≤ volume (closedBall x (ε n) \ closedBall (f (ε n) <| hε' n) (ε n / 4)) /
          volume (closedBall x (ε n)) := by
        gcongr
        rw [diff_eq_compl_inter]
        refine inter_subset_inter_left _ ?_
        rw [subset_compl_comm, ← interior_compl]
        exact hf₁ _ _
      _ = 1 - ENNReal.ofReal (4⁻¹ ^ Fintype.card ι) := ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (hypothesis)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
⊢ HDiv.hDiv (α := ENNReal) (β := ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal)
          (closedBall x (ε n) \ closedBall (f (ε n) ⋯) (ε n / 4)) :
        ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x (ε n)) : ENNReal) =
    1 - ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι))
have this : 0 < ε n
Tactic: have := hε' n
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
⊢ HDiv.hDiv (α := ENNReal) (β := ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal)
          (closedBall x (ε n) \ closedBall (f (ε n) ⋯) (ε n / 4)) :
        ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x (ε n)) : ENNReal) =
    1 - ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι))
have refine_2 : ε n ≠ 0
Tactic: rw [measure_diff (hf₀ _ _) _ ((Real.volume_pi_closedBall _ _).trans_ne ENNReal.ofReal_ne_top),
      Real.volume_pi_closedBall, Real.volume_pi_closedBall, ENNReal.sub_div fun _ _ ↦ _,
      ENNReal.div_self _ ENNReal.ofReal_ne_top, ← ENNReal.ofReal_div_of_pos, ← div_pow,
      mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
refine_2 : ε n ≠ 0
⊢ HDiv.hDiv (α := ENNReal) (β := ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal)
          (closedBall x (ε n) \ closedBall (f (ε n) ⋯) (ε n / 4)) :
        ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x (ε n)) : ENNReal) =
    1 - ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι))
have refine_2₁ : 0 < (2 * ε n) ^ Fintype.card ι
Tactic: rw [measure_diff (hf₀ _ _) _ ((Real.volume_pi_closedBall _ _).trans_ne ENNReal.ofReal_ne_top),
      Real.volume_pi_closedBall, Real.volume_pi_closedBall, ENNReal.sub_div fun _ _ ↦ _,
      ENNReal.div_self _ ENNReal.ofReal_ne_top, ← ENNReal.ofReal_div_of_pos, ← div_pow,
      mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.UpperLower (goal)
Goal:
ι : Type u_1
inst : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H :
  Tendsto (β := ENNReal)
    (fun (r : ℝ) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closure s ∩ closedBall x r) : ENNReal)
        (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x r) : ENNReal))
    (𝓝[>] 0) (𝓝 1)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
this : 0 < ε n
refine_2 : ε n ≠ 0
refine_2₁ : 0 < (2 * ε n) ^ Fintype.card ι
⊢ HDiv.hDiv (α := ENNReal) (β := ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal)
          (closedBall x (ε n) \ closedBall (f (ε n) ⋯) (ε n / 4)) :
        ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x (ε n)) : ENNReal) =
    1 - ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι))
have rw : ENNReal.ofReal ((2 * ε n) ^ Fintype.card ι) ≠ 0
Tactic: rw [measure_diff (hf₀ _ _) _ ((Real.volume_pi_closedBall _ _).trans_ne ENNReal.ofReal_ne_top),
      Real.volume_pi_closedBall, Real.volume_pi_closedBall, ENNReal.sub_div fun _ _ ↦ _,
      ENNReal.div_self _ ENNReal.ofReal_ne_top, ← ENNReal.ofReal_div_of_pos, ← div_pow,
      mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]