awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.WithDensityFinite (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst : SFinite μ
⊢ IsFiniteMeasure
    (if IsFiniteMeasure μ then μ else ⋯.choose (p := fun (ν : Measure α) ↦ IsFiniteMeasure ν ∧ μ ≪ ν ∧ ν ≪ μ))
have pos : IsFiniteMeasure μ → IsFiniteMeasure μ
Tactic: split_ifs
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.WithDensityFinite (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst : SFinite μ
pos : IsFiniteMeasure μ → IsFiniteMeasure μ
⊢ IsFiniteMeasure
    (if IsFiniteMeasure μ then μ else ⋯.choose (p := fun (ν : Measure α) ↦ IsFiniteMeasure ν ∧ μ ≪ ν ∧ ν ≪ μ))
have neg : ¬IsFiniteMeasure μ → IsFiniteMeasure (sorry.choose (p := fun (ν : Measure α) ↦ IsFiniteMeasure ν ∧ μ ≪ ν ∧ ν ≪ μ))
Tactic: split_ifs
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.WithDensityFinite (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst : SFinite μ
⊢ μ.toFinite = 0 ↔ μ = 0
have simp_rw : Eq (α := ℝ≥0∞) ((μ.toFinite : (a : Set α) → ℝ≥0∞) univ : ℝ≥0∞) (0 : ℝ≥0∞) ↔
  Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) univ : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: simp_rw [← Measure.measure_univ_eq_zero, toFinite_apply_eq_zero_iff]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.WithDensityFinite (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst : IsProbabilityMeasure μ
⊢ μ.toFinite = μ
have hc : IsFiniteMeasure μ
Tactic: rw [Measure.toFinite, Measure.toFiniteAux, if_pos, ProbabilityTheory.cond_univ]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.WithDensityFinite (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst : SFinite μ
inst_1 : NeZero μ
⊢ IsProbabilityMeasure μ.toFinite
have hcs : Ne (α := ℝ≥0∞) ((μ.toFiniteAux : (a : Set α) → ℝ≥0∞) univ : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: apply ProbabilityTheory.cond_isProbabilityMeasure
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.WithDensityFinite (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst : SFinite μ
⊢ μ.restrict μ.sigmaFiniteSetᶜ = HSMul.hSMul (α := ℝ≥0∞) ∞ (μ.toFinite.restrict μ.sigmaFiniteSetᶜ)
have rw : HSMul.hSMul (α := ℝ≥0∞) ∞ (μ.restrict (μ.sigmaFiniteSetWRT μ)ᶜ) =
  HSMul.hSMul (α := ℝ≥0∞) ∞ (μ.toFinite.restrict (μ.sigmaFiniteSetWRT μ)ᶜ)
Tactic: rw [Measure.sigmaFiniteSet,
    restrict_compl_sigmaFiniteSetWRT (Measure.AbsolutelyContinuous.refl μ)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.WithDensityFinite (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst : SFinite μ
⊢ HSMul.hSMul (α := ℝ≥0∞) ∞ (μ.restrict (μ.sigmaFiniteSetWRT μ)ᶜ) =
    HSMul.hSMul (α := ℝ≥0∞) ∞ (μ.toFinite.restrict (μ.sigmaFiniteSetWRT μ)ᶜ)
have h : ∀ (t : Set α) (ht : MeasurableSet t),
  Eq (α := ℝ≥0∞) ((HSMul.hSMul (α := ℝ≥0∞) ∞ (μ.restrict (μ.sigmaFiniteSetWRT μ)ᶜ) : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
    ((HSMul.hSMul (α := ℝ≥0∞) ∞ (μ.toFinite.restrict (μ.sigmaFiniteSetWRT μ)ᶜ) : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
Tactic: ext t ht
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.WithDensityFinite (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst : SFinite μ
t : Set α
ht : MeasurableSet t
⊢ HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞) ((μ.restrict (μ.sigmaFiniteSetWRT μ)ᶜ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞)
      ((μ.toFinite.restrict (μ.sigmaFiniteSetWRT μ)ᶜ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
have h : HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) =
  HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞) ((μ.toFinite : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞)
Tactic: rw [Measure.restrict_apply ht, Measure.restrict_apply ht]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.WithDensityFinite (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst : SFinite μ
t : Set α
ht : MeasurableSet t
⊢ HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞)
      ((μ.toFinite : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞)
have pos : ∀ (hμt : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) (0 : ℝ≥0∞)),
  HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞)
      ((μ.toFinite : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞)
Tactic: by_cases hμt : μ (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) = 0
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.WithDensityFinite (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst : SFinite μ
t : Set α
ht : MeasurableSet t
pos :
  ∀ (hμt : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) (0 : ℝ≥0∞)),
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞)
        ((μ.toFinite : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞)
⊢ HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞)
      ((μ.toFinite : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞)
have neg : ∀ (hμt : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) (0 : ℝ≥0∞)),
  HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞)
      ((μ.toFinite : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞)
Tactic: by_cases hμt : μ (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) = 0
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.WithDensityFinite (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst : SFinite μ
t : Set α
ht : MeasurableSet t
hμt : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞)
      ((μ.toFinite : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞)
have pos : HMul.hMul (α := ℝ≥0∞) ∞ 0 =
  HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞) ((μ.toFinite : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞)
Tactic: rw [hμt, toFinite_absolutelyContinuous μ hμt]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.WithDensityFinite (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst : SFinite μ
t : Set α
ht : MeasurableSet t
hμt : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) (0 : ℝ≥0∞)
pos :
  HMul.hMul (α := ℝ≥0∞) ∞ 0 =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞)
      ((μ.toFinite : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞)
⊢ HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞)
      ((μ.toFinite : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞)
have pos₁ : HMul.hMul (α := ℝ≥0∞) ∞ 0 = HMul.hMul (α := ℝ≥0∞) ∞ 0
Tactic: rw [hμt, toFinite_absolutelyContinuous μ hμt]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.WithDensityFinite (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst : SFinite μ
t : Set α
ht : MeasurableSet t
hμt : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞ : ℝ≥0∞)
      ((μ.toFinite : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞)
have neg : Ne (α := ℝ≥0∞) ((μ.toFinite : (a : Set α) → ℝ≥0∞) (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: rw [ENNReal.top_mul hμt, ENNReal.top_mul]
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.Lattice (goal)
Goal:
α : Type u_2
m : MeasurableSpace α
δ : Type u_3
inst : MeasurableSpace δ
inst_1 : SemilatticeSup α
inst_2 : MeasurableSup₂ α
f : ℕ → δ → α
n : ℕ
hf : ∀ k ≤ n, Measurable (f k)
⊢ Measurable ((range (n + 1)).sup' ⋯ f)
have refine : ∀ n_1 ∈ range (n + 1), Measurable (f n_1)
Tactic: refine Finset.measurable_sup' _ ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.Lattice (goal)
Goal:
α : Type u_2
m : MeasurableSpace α
δ : Type u_3
inst : MeasurableSpace δ
inst_1 : SemilatticeSup α
inst_2 : MeasurableSup₂ α
f : ℕ → δ → α
n : ℕ
hf : ∀ k ≤ n, Measurable (f k)
⊢ Measurable.{u_3, u_2} (β := α) fun (x : δ) ↦ (range (n + 1)).sup' ⋯ fun (k : ℕ) ↦ f k x
have h : (fun (x : δ) ↦ (range (n + 1)).sup' sorry fun (k : ℕ) ↦ f k x) = (range (n + 1)).sup' sorry f
Tactic: convert! Finset.measurable_range_sup' hf using 1
-- Lean 4 proof step from Mathlib.MeasureTheory.Order.Lattice (goal)
Goal:
α : Type u_2
m : MeasurableSpace α
δ : Type u_3
inst : MeasurableSpace δ
inst_1 : SemilatticeSup α
inst_2 : MeasurableSup₂ α
f : ℕ → δ → α
n : ℕ
hf : ∀ k ≤ n, Measurable (f k)
⊢ (fun (x : δ) ↦ (range (n + 1)).sup' ⋯ fun (k : ℕ) ↦ f k x) = (range (n + 1)).sup' ⋯ f
have h : ∀ (x : δ), ((range (n + 1)).sup' sorry fun (k : ℕ) ↦ f k x) = (range (n + 1)).sup' sorry f x
Tactic: ext x
-- 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 s
⊢ ¬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)
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 s),
  ¬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)
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 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)
⊢ 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 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)
⊢ False
have refine_1 : 0 < ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι))
Tactic: refine not_eventually.2
    (Frequently.of_forall fun _ ↦ lt_irrefl <| ENNReal.ofReal <| 4⁻¹ ^ Fintype.card ι)
    ((Filter.Tendsto.eventually_lt (H.comp hε₀) tendsto_const_nhds ?_).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 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)
refine_1 : 0 < ENNReal.ofReal (HPow.hPow (α := ℝ) 4⁻¹ (Fintype.card ι))
⊢ False
have refine_2 : ∀ (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
Tactic: refine not_eventually.2
    (Frequently.of_forall fun _ ↦ lt_irrefl <| ENNReal.ofReal <| 4⁻¹ ^ Fintype.card ι)
    ((Filter.Tendsto.eventually_lt (H.comp hε₀) tendsto_const_nhds ?_).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 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
⊢ 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) ≤
    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)
have hab : closedBall (f (ε n) sorry) (ε n / 4) ⊆ closure s ∩ closedBall x (ε n)
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 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 : 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) sorry) (ε n / 4)) : ENNReal)
    (((volume : Measure (ι → ℝ)) : (a : Set (ι → ℝ)) → ENNReal) (closedBall x (ε n)) : ENNReal)
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
-- 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 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)) =
    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)
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 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
⊢ 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)
have refine_2 : ε n ≠ 0
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]
-- 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
this : 0 < ε n
refine_2 : ε n ≠ 0
⊢ 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)
have refine_2₁ : 0 < (2 * ε n) ^ Fintype.card ι
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]
-- 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
this : 0 < ε n
refine_2 : ε n ≠ 0
refine_2₁ : 0 < (2 * ε n) ^ Fintype.card ι
⊢ 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)
have refine_2₂ : 0 ≤ ε n
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]
-- 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
this : 0 < ε n
refine_2 : ε n ≠ 0
refine_2₁ : 0 < (2 * ε n) ^ Fintype.card ι
refine_2₂ : 0 ≤ ε n
⊢ 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)
have refine_2₃ : 0 ≤ ε n / 4
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]