awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
⊢ closedBall (f (ε n) ⋯) (ε n / 4) ⊆ closure s ∩ closedBall x (ε n)

Tactic:
subset_inter ((hf₁ _ <| hε' n).trans interior_subset_closure) <| hf₀ _ <| hε' n

Premises used: HasSubset.Subset.trans, Set.subset_inter, interior_subset_closure
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
⊢ closedBall (f (ε n) ⋯) (ε n / 4) ⊆ closure s ∩ closedBall x (ε n)

Tactic:
exact subset_inter ((hf₁ _ <| hε' n).trans interior_subset_closure) <| hf₀ _ <| hε' n

Premises used: HasSubset.Subset.trans, Set.subset_inter, interior_subset_closure
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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 :
  ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι)) =
    HDiv.hDiv (α := ENNReal) (β := ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall (f (ε n) ⋯) (ε n / 4)) : ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x (ε n)) : ENNReal)
⊢ 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

Tactic:
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

Premises used: ENNReal.ofReal, Fintype.card, HDiv.hDiv, Inter.inter, Inv.inv, closure, HPow.hPow, MeasureTheory.MeasureSpace.volume, Metric.closedBall, OfNat.ofNat, DFunLike.coe
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
⊢ 0 < ε n

Tactic:
have := hε' n
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
this : 0 < ε n
refine_2 : ε n ≠ 0
refine_2₁ : 0 < (2 * ε n) ^ Fintype.card ι
refine_2₂ : 0 ≤ ε n
refine_2₃ : 0 ≤ ε n / 4
⊢ ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι)) =
    HDiv.hDiv (α := ENNReal) (β := ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall (f (ε n) ⋯) (ε n / 4)) : ENNReal)
      (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x (ε n)) : ENNReal)

Tactic:
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]

Premises used: Real, div_right_comm, ENNReal.ofReal_div_of_pos, two_ne_zero', div_pow, one_div, mul_div_mul_left, Real.volume_pi_closedBall, div_self
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
refine_2₃ : ∀ (this : 0 < ε n), 0 ≤ ε n / 4
⊢ 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

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]

Premises used: Real, div_right_comm, ENNReal.ofReal, ENNReal.ofReal_div_of_pos, Fintype.card, HDiv.hDiv, Inter.inter, two_ne_zero', Inv.inv, closure, HPow.hPow, MeasureTheory.MeasureSpace.volume, div_pow, one_div, mul_div_mul_left, Metric.closedBall, OfNat.ofNat, DFunLike.coe, Real.volume_pi_closedBall, div_self
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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)
⊢ 0 < ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι))

Tactic:
positivity
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
this : 0 < ε n
⊢ ε n ≠ 0

Tactic:
positivity
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
this : 0 < ε n
⊢ 0 < (2 * ε n) ^ Fintype.card ι

Tactic:
positivity
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
this : 0 < ε n
⊢ 0 ≤ ε n

Tactic:
positivity
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
this : 0 < ε n
⊢ 0 ≤ ε n / 4

Tactic:
positivity
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
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)
⊢ ¬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
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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)
obtain : ∀ (ε : ℕ → ℝ) (hε' : ∀ (n : ℕ), 0 < ε n) (hε₀ : Tendsto ε atTop (𝓝[>] 0)), False
⊢ False

Tactic:
obtain ⟨ε, -, hε', hε₀⟩ := exists_seq_strictAnti_tendsto_nhdsWithin (0 : ℝ)

Premises used: exists_seq_strictAnti_tendsto_nhdsWithin, OfNat.ofNat
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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

Tactic:
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' ?_)

Premises used: Iff.mpr, lt_irrefl, Filter.Frequently.of_forall, ENNReal.ofReal, tendsto_const_nhds, HSub.hSub, Fintype.card, lt_of_le_of_lt', Filter.Eventually.mono, Inv.inv, HPow.hPow, ENNReal.sub_lt_self, Filter.Tendsto.eventually_lt, Filter.not_eventually, Filter.Tendsto.comp, one_ne_zero, OfNat.ofNat, ENNReal.one_ne_top
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
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))
⊢ False

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' ?_)

Premises used: Iff.mpr, lt_irrefl, Filter.Frequently.of_forall, ENNReal.ofReal, tendsto_const_nhds, HSub.hSub, Fintype.card, lt_of_le_of_lt', Filter.Eventually.mono, Inv.inv, HPow.hPow, ENNReal.sub_lt_self, Filter.Tendsto.eventually_lt, Filter.not_eventually, Filter.Tendsto.comp, one_ne_zero, OfNat.ofNat, ENNReal.one_ne_top
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
hab : closure s ∩ closedBall x (ε n) ⊆ closedBall x (ε n) \ closedBall (f (ε n) ⋯) (ε n / 4)
⊢ 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)

Tactic:
gcongr
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
hab : closure s ∩ closedBall x (ε n) ⊆ (closedBall (f (ε n) ⋯) (ε n / 4))ᶜ ∩ closedBall x (ε n)
⊢ closure s ∩ closedBall x (ε n) ⊆ closedBall x (ε n) \ closedBall (f (ε n) ⋯) (ε n / 4)

Tactic:
rw [diff_eq_compl_inter]

Premises used: Set.diff_eq_compl_inter
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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)

Tactic:
inter_subset_inter_left _ ?_

Premises used: Set.inter_subset_inter_left
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
hab : closure s ⊆ (closedBall (f (ε n) ⋯) (ε n / 4))ᶜ
⊢ closure s ∩ closedBall x (ε n) ⊆ (closedBall (f (ε n) ⋯) (ε n / 4))ᶜ ∩ closedBall x (ε n)

Tactic:
refine inter_subset_inter_left _ ?_

Premises used: Set.inter_subset_inter_left
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
hab : closedBall (f (ε n) ⋯) (ε n / 4) ⊆ interior sᶜ
⊢ closure s ⊆ (closedBall (f (ε n) ⋯) (ε n / 4))ᶜ

Tactic:
rw [subset_compl_comm, ← interior_compl]

Premises used: Set.subset_compl_comm, interior_compl
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
⊢ closedBall (f (ε n) ⋯) (ε n / 4) ⊆ interior sᶜ

Tactic:
hf₁ _ _
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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
⊢ closedBall (f (ε n) ⋯) (ε n / 4) ⊆ interior sᶜ

Tactic:
exact hf₁ _ _
-- Module: Mathlib.MeasureTheory.Order.UpperLower
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))

Tactic:
?_