awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.Intersectivity (goal)
Goal:
α : Type u_2
inst : MeasurableSpace α
μ : Measure α
inst_1 : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ (μ : (a : Set α) → ℝ≥0∞) (s n)
M : (f : α → ℝ) → Set α := fun (f : α → ℝ) ↦ {x : α | eLpNormEssSup f μ < (↑‖f x‖₊ : ℝ≥0∞)}
N : Set α := ⋃ (u : Finset ℕ), M (indicator (⋂ n ∈ u, s n) 1)
hN₀ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) N : ℝ≥0∞) (0 : ℝ≥0∞)
hN₁ :
  ∀ (u : Finset ℕ),
    ((⋂ n ∈ u, s n) \ N).Nonempty → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
f : (n : ℕ) → α → ℝ≥0∞ :=
  fun (n : ℕ) ↦
    HSMul.hSMul (α := ℝ≥0∞) (β := (x : α) → ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1)
hfapp :
  ∀ (n : ℕ) (a : α), f n a = HMul.hMul (α := ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a)
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
n : ℕ
⊢ r ≤ HMul.hMul (α := ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ x ∈ Finset.range (n + 1), (μ : (a : Set α) → ℝ≥0∞) (s x))
have rw : (n + 1) • r ≤ ∑ x ∈ Finset.range (n + 1), (μ : (a : Set α) → ℝ≥0∞) (s x)
Tactic: rw [← ENNReal.div_eq_inv_mul, ENNReal.le_div_iff_mul_le (by simp) (by simp), ← nsmul_eq_mul']
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.Intersectivity (hypothesis)
Goal:
α : Type u_2
inst : MeasurableSpace α
μ : Measure α
inst_1 : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ (μ : (a : Set α) → ℝ≥0∞) (s n)
M : (f : α → ℝ) → Set α := fun (f : α → ℝ) ↦ {x : α | eLpNormEssSup f μ < (↑‖f x‖₊ : ℝ≥0∞)}
N : Set α := ⋃ (u : Finset ℕ), M (indicator (⋂ n ∈ u, s n) 1)
hN₀ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) N : ℝ≥0∞) (0 : ℝ≥0∞)
hN₁ :
  ∀ (u : Finset ℕ),
    ((⋂ n ∈ u, s n) \ N).Nonempty → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
f : (n : ℕ) → α → ℝ≥0∞ :=
  fun (n : ℕ) ↦
    HSMul.hSMul (α := ℝ≥0∞) (β := (x : α) → ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1)
hfapp :
  ∀ (n : ℕ) (a : α), f n a = HMul.hMul (α := ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a)
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
⊢ ∃ (t : Set ℕ),
    t.Infinite ∧
      ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
have hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
Tactic: have hrf n : r ≤ ∫⁻ a, f n a ∂μ := by
    simp_rw [hfapp]
    rw [lintegral_const_mul _ <| Finset.measurable_fun_sum _
        fun _ _ ↦ measurable_one.indicator <| hs _,
      lintegral_finsetSum _ fun _ _ ↦ measurable_one.indicator (hs _)]
    simp only [lintegral_indicator_one (hs _)]
    rw [← ENNReal.div_eq_inv_mul, ENNReal.le_div_iff_mul_le (by simp) (by simp), ← nsmul_eq_mul']
    simpa using Finset.card_nsmul_le_sum (Finset.range (n + 1)) _ _ fun _ _ ↦ hr _
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.Intersectivity (hypothesis)
Goal:
α : Type u_2
inst : MeasurableSpace α
μ : Measure α
inst_1 : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ (μ : (a : Set α) → ℝ≥0∞) (s n)
M : (f : α → ℝ) → Set α := fun (f : α → ℝ) ↦ {x : α | eLpNormEssSup f μ < (↑‖f x‖₊ : ℝ≥0∞)}
N : Set α := ⋃ (u : Finset ℕ), M (indicator (⋂ n ∈ u, s n) 1)
hN₀ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) N : ℝ≥0∞) (0 : ℝ≥0∞)
hN₁ :
  ∀ (u : Finset ℕ),
    ((⋂ n ∈ u, s n) \ N).Nonempty → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
f : (n : ℕ) → α → ℝ≥0∞ :=
  fun (n : ℕ) ↦
    HSMul.hSMul (α := ℝ≥0∞) (β := (x : α) → ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1)
hfapp :
  ∀ (n : ℕ) (a : α), f n a = HMul.hMul (α := ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a)
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
⊢ ∃ (t : Set ℕ),
    t.Infinite ∧
      ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
have hμ : μ ≠ 0
Tactic: have hμ : μ ≠ 0 := by rintro rfl; exact hr₀ <| le_bot_iff.1 <| hr 0
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.Intersectivity (goal)
Goal:
α : Type u_2
inst : MeasurableSpace α
μ : Measure α
inst_1 : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ (μ : (a : Set α) → ℝ≥0∞) (s n)
M : (f : α → ℝ) → Set α := fun (f : α → ℝ) ↦ {x : α | eLpNormEssSup f μ < (↑‖f x‖₊ : ℝ≥0∞)}
N : Set α := ⋃ (u : Finset ℕ), M (indicator (⋂ n ∈ u, s n) 1)
hN₀ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) N : ℝ≥0∞) (0 : ℝ≥0∞)
hN₁ :
  ∀ (u : Finset ℕ),
    ((⋂ n ∈ u, s n) \ N).Nonempty → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
f : (n : ℕ) → α → ℝ≥0∞ :=
  fun (n : ℕ) ↦
    HSMul.hSMul (α := ℝ≥0∞) (β := (x : α) → ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1)
hfapp :
  ∀ (n : ℕ) (a : α), f n a = HMul.hMul (α := ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a)
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
⊢ ∫⁻ (x : α), limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop ∂μ ≤ (μ : (a : Set α) → ℝ≥0∞) univ
have rw : ∫⁻ (x : α), limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop ∂μ ≤ ∫⁻ (x : α), 1 ∂μ
Tactic: rw [← lintegral_one]
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.Intersectivity (hypothesis)
Goal:
α : Type u_2
inst : MeasurableSpace α
μ : Measure α
inst_1 : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ (μ : (a : Set α) → ℝ≥0∞) (s n)
M : (f : α → ℝ) → Set α := fun (f : α → ℝ) ↦ {x : α | eLpNormEssSup f μ < (↑‖f x‖₊ : ℝ≥0∞)}
N : Set α := ⋃ (u : Finset ℕ), M (indicator (⋂ n ∈ u, s n) 1)
hN₀ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) N : ℝ≥0∞) (0 : ℝ≥0∞)
hN₁ :
  ∀ (u : Finset ℕ),
    ((⋂ n ∈ u, s n) \ N).Nonempty → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
f : (n : ℕ) → α → ℝ≥0∞ :=
  fun (n : ℕ) ↦
    HSMul.hSMul (α := ℝ≥0∞) (β := (x : α) → ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1)
hfapp :
  ∀ (n : ℕ) (a : α), f n a = HMul.hMul (α := ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a)
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
⊢ ∃ (t : Set ℕ),
    t.Infinite ∧
      ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
have this : ∫⁻ (x : α), limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop ∂μ ≤ (μ : (a : Set α) → ℝ≥0∞) univ
Tactic: have : ∫⁻ x, limsup (f · x) atTop ∂μ ≤ μ univ := by
    rw [← lintegral_one]
    exact lintegral_mono fun a ↦ limsup_le_of_le ⟨0, fun R _ ↦ bot_le⟩ <|
      Eventually.of_forall fun n ↦ hf₁ _ _
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.Intersectivity (goal)
Goal:
α : Type u_2
inst : MeasurableSpace α
μ : Measure α
inst_1 : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ (μ : (a : Set α) → ℝ≥0∞) (s n)
M : (f : α → ℝ) → Set α := fun (f : α → ℝ) ↦ {x : α | eLpNormEssSup f μ < (↑‖f x‖₊ : ℝ≥0∞)}
N : Set α := ⋃ (u : Finset ℕ), M (indicator (⋂ n ∈ u, s n) 1)
hN₀ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) N : ℝ≥0∞) (0 : ℝ≥0∞)
hN₁ :
  ∀ (u : Finset ℕ),
    ((⋂ n ∈ u, s n) \ N).Nonempty → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
f : (n : ℕ) → α → ℝ≥0∞ :=
  fun (n : ℕ) ↦
    HSMul.hSMul (α := ℝ≥0∞) (β := (x : α) → ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1)
hfapp :
  ∀ (n : ℕ) (a : α), f n a = HMul.hMul (α := ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a)
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop ∂μ ≤ (μ : (a : Set α) → ℝ≥0∞) univ
⊢ ∃ (t : Set ℕ),
    t.Infinite ∧
      ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
have obtain : ∀ (x : α) (hxN : x ∉ N)
  (hx : ⨍⁻ (a : α), limsup (fun (x : ℕ) ↦ f x a) atTop ∂μ ≤ limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop),
  ∃ (t : Set ℕ),
    t.Infinite ∧
      ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
Tactic: obtain ⟨x, hxN, hx⟩ := exists_notMem_null_laverage_le hμ
    (ne_top_of_le_ne_top (by finiteness) this) hN₀
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.Intersectivity (goal)
Goal:
α : Type u_2
inst : MeasurableSpace α
μ : Measure α
inst_1 : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ (μ : (a : Set α) → ℝ≥0∞) (s n)
M : (f : α → ℝ) → Set α := fun (f : α → ℝ) ↦ {x : α | eLpNormEssSup f μ < (↑‖f x‖₊ : ℝ≥0∞)}
N : Set α := ⋃ (u : Finset ℕ), M (indicator (⋂ n ∈ u, s n) 1)
hN₀ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) N : ℝ≥0∞) (0 : ℝ≥0∞)
hN₁ :
  ∀ (u : Finset ℕ),
    ((⋂ n ∈ u, s n) \ N).Nonempty → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
f : (n : ℕ) → α → ℝ≥0∞ :=
  fun (n : ℕ) ↦
    HSMul.hSMul (α := ℝ≥0∞) (β := (x : α) → ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1)
hfapp :
  ∀ (n : ℕ) (a : α), f n a = HMul.hMul (α := ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a)
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop ∂μ ≤ (μ : (a : Set α) → ℝ≥0∞) univ
x : α
hxN : x ∉ N
hx : ⨍⁻ (a : α), limsup (fun (x : ℕ) ↦ f x a) atTop ∂μ ≤ limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop
⊢ ∃ (t : Set ℕ),
    t.Infinite ∧
      ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
have refine_3 : ∀ (hx : HDiv.hDiv (β := ℝ≥0∞) r ((μ : (a : Set α) → ℝ≥0∞) univ : ℝ≥0∞) ≤ limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop),
  ∃ (t : Set ℕ),
    t.Infinite ∧
      ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
Tactic: replace hx : r / μ univ ≤ limsup (f · x) atTop :=
    calc
      _ ≤ limsup (⨍⁻ x, f · x ∂μ) atTop := le_limsup_of_le ⟨1, eventually_map.2 ?_⟩ fun b hb ↦ ?_
      _ ≤ ⨍⁻ x, limsup (f · x) atTop ∂μ := limsup_lintegral_le 1 hf (ae_of_all _ <| hf₁ ·) (by simp)
      _ ≤ limsup (f · x) atTop := hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.Intersectivity (goal)
Goal:
α : Type u_2
inst : MeasurableSpace α
μ : Measure α
inst_1 : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ (μ : (a : Set α) → ℝ≥0∞) (s n)
M : (f : α → ℝ) → Set α := fun (f : α → ℝ) ↦ {x : α | eLpNormEssSup f μ < (↑‖f x‖₊ : ℝ≥0∞)}
N : Set α := ⋃ (u : Finset ℕ), M (indicator (⋂ n ∈ u, s n) 1)
hN₀ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) N : ℝ≥0∞) (0 : ℝ≥0∞)
hN₁ :
  ∀ (u : Finset ℕ),
    ((⋂ n ∈ u, s n) \ N).Nonempty → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
f : (n : ℕ) → α → ℝ≥0∞ :=
  fun (n : ℕ) ↦
    HSMul.hSMul (α := ℝ≥0∞) (β := (x : α) → ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1)
hfapp :
  ∀ (n : ℕ) (a : α), f n a = HMul.hMul (α := ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a)
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop ∂μ ≤ (μ : (a : Set α) → ℝ≥0∞) univ
x : α
hxN : x ∉ N
hx : ⨍⁻ (a : α), limsup (fun (x : ℕ) ↦ f x a) atTop ∂μ ≤ limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop
refine_3 :
  ∀ (hx : HDiv.hDiv (β := ℝ≥0∞) r ((μ : (a : Set α) → ℝ≥0∞) univ : ℝ≥0∞) ≤ limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop),
    ∃ (t : Set ℕ),
      t.Infinite ∧
        ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
⊢ ∃ (t : Set ℕ),
    t.Infinite ∧
      ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
have refine_1 : ∀ᶠ (a : ℕ) in atTop, (fun (x1 x2 : ℝ≥0∞) ↦ x1 ≤ x2) (⨍⁻ (x : α), f a x ∂μ) 1
Tactic: replace hx : r / μ univ ≤ limsup (f · x) atTop :=
    calc
      _ ≤ limsup (⨍⁻ x, f · x ∂μ) atTop := le_limsup_of_le ⟨1, eventually_map.2 ?_⟩ fun b hb ↦ ?_
      _ ≤ ⨍⁻ x, limsup (f · x) atTop ∂μ := limsup_lintegral_le 1 hf (ae_of_all _ <| hf₁ ·) (by simp)
      _ ≤ limsup (f · x) atTop := hx
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.Intersectivity (goal)
Goal:
α : Type u_2
inst : MeasurableSpace α
μ : Measure α
inst_1 : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ (μ : (a : Set α) → ℝ≥0∞) (s n)
M : (f : α → ℝ) → Set α := fun (f : α → ℝ) ↦ {x : α | eLpNormEssSup f μ < (↑‖f x‖₊ : ℝ≥0∞)}
N : Set α := ⋃ (u : Finset ℕ), M (indicator (⋂ n ∈ u, s n) 1)
hN₀ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) N : ℝ≥0∞) (0 : ℝ≥0∞)
hN₁ :
  ∀ (u : Finset ℕ),
    ((⋂ n ∈ u, s n) \ N).Nonempty → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
f : (n : ℕ) → α → ℝ≥0∞ :=
  fun (n : ℕ) ↦
    HSMul.hSMul (α := ℝ≥0∞) (β := (x : α) → ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1)
hfapp :
  ∀ (n : ℕ) (a : α), f n a = HMul.hMul (α := ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a)
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop ∂μ ≤ (μ : (a : Set α) → ℝ≥0∞) univ
x : α
hxN : x ∉ N
hx : ⨍⁻ (a : α), limsup (fun (x : ℕ) ↦ f x a) atTop ∂μ ≤ limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop
refine_3 :
  ∀ (hx : HDiv.hDiv (β := ℝ≥0∞) r ((μ : (a : Set α) → ℝ≥0∞) univ : ℝ≥0∞) ≤ limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop),
    ∃ (t : Set ℕ),
      t.Infinite ∧
        ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
refine_1 : ∀ᶠ (a : ℕ) in atTop, (fun (x1 x2 : ℝ≥0∞) ↦ x1 ≤ x2) (⨍⁻ (x : α), f a x ∂μ) 1
⊢ ∃ (t : Set ℕ),
    t.Infinite ∧
      ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
have refine_2 : ∀ (b : ℝ≥0∞) (hb : ∀ᶠ (n : ℕ) in atTop, ⨍⁻ (x : α), f n x ∂μ ≤ b),
  HDiv.hDiv (β := ℝ≥0∞) r ((μ : (a : Set α) → ℝ≥0∞) univ : ℝ≥0∞) ≤ b
Tactic: replace hx : r / μ univ ≤ limsup (f · x) atTop :=
    calc
      _ ≤ limsup (⨍⁻ x, f · x ∂μ) atTop := le_limsup_of_le ⟨1, eventually_map.2 ?_⟩ fun b hb ↦ ?_
      _ ≤ ⨍⁻ x, limsup (f · x) atTop ∂μ := limsup_lintegral_le 1 hf (ae_of_all _ <| hf₁ ·) (by simp)
      _ ≤ limsup (f · x) atTop := hx
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.Intersectivity (goal)
Goal:
α : Type u_2
inst : MeasurableSpace α
μ : Measure α
inst_1 : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ (μ : (a : Set α) → ℝ≥0∞) (s n)
M : (f : α → ℝ) → Set α := fun (f : α → ℝ) ↦ {x : α | eLpNormEssSup f μ < (↑‖f x‖₊ : ℝ≥0∞)}
N : Set α := ⋃ (u : Finset ℕ), M (indicator (⋂ n ∈ u, s n) 1)
hN₀ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) N : ℝ≥0∞) (0 : ℝ≥0∞)
hN₁ :
  ∀ (u : Finset ℕ),
    ((⋂ n ∈ u, s n) \ N).Nonempty → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
f : (n : ℕ) → α → ℝ≥0∞ :=
  fun (n : ℕ) ↦
    HSMul.hSMul (α := ℝ≥0∞) (β := (x : α) → ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1)
hfapp :
  ∀ (n : ℕ) (a : α), f n a = HMul.hMul (α := ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a)
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop ∂μ ≤ (μ : (a : Set α) → ℝ≥0∞) univ
x : α
hxN : x ∉ N
hx : HDiv.hDiv (β := ℝ≥0∞) r ((μ : (a : Set α) → ℝ≥0∞) univ : ℝ≥0∞) ≤ limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop
⊢ ∃ (t : Set ℕ),
    t.Infinite ∧
      ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
have refine_3 : ∀ (hxs : Set.Finite (α := ℕ) {n : ℕ | x ∈ s n}), False
Tactic: refine ⟨{n | x ∈ s n}, fun hxs ↦ ?_, fun u hux hu ↦ ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.Intersectivity (goal)
Goal:
α : Type u_2
inst : MeasurableSpace α
μ : Measure α
inst_1 : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ (μ : (a : Set α) → ℝ≥0∞) (s n)
M : (f : α → ℝ) → Set α := fun (f : α → ℝ) ↦ {x : α | eLpNormEssSup f μ < (↑‖f x‖₊ : ℝ≥0∞)}
N : Set α := ⋃ (u : Finset ℕ), M (indicator (⋂ n ∈ u, s n) 1)
hN₀ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) N : ℝ≥0∞) (0 : ℝ≥0∞)
hN₁ :
  ∀ (u : Finset ℕ),
    ((⋂ n ∈ u, s n) \ N).Nonempty → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
f : (n : ℕ) → α → ℝ≥0∞ :=
  fun (n : ℕ) ↦
    HSMul.hSMul (α := ℝ≥0∞) (β := (x : α) → ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1)
hfapp :
  ∀ (n : ℕ) (a : α), f n a = HMul.hMul (α := ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a)
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop ∂μ ≤ (μ : (a : Set α) → ℝ≥0∞) univ
x : α
hxN : x ∉ N
hx : HDiv.hDiv (β := ℝ≥0∞) r ((μ : (a : Set α) → ℝ≥0∞) univ : ℝ≥0∞) ≤ limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop
refine_3 : ∀ (hxs : Set.Finite (α := ℕ) {n : ℕ | x ∈ s n}), False
⊢ ∃ (t : Set ℕ),
    t.Infinite ∧
      ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
have refine_3₁ : ∀ (u : Set ℕ) (hux : u ⊆ {n : ℕ | x ∈ s n}) (hu : u.Finite),
  LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
Tactic: refine ⟨{n | x ∈ s n}, fun hxs ↦ ?_, fun u hux hu ↦ ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.Intersectivity (goal)
Goal:
α : Type u_2
inst : MeasurableSpace α
μ : Measure α
inst_1 : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ (μ : (a : Set α) → ℝ≥0∞) (s n)
M : (f : α → ℝ) → Set α := fun (f : α → ℝ) ↦ {x : α | eLpNormEssSup f μ < (↑‖f x‖₊ : ℝ≥0∞)}
N : Set α := ⋃ (u : Finset ℕ), M (indicator (⋂ n ∈ u, s n) 1)
hN₀ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) N : ℝ≥0∞) (0 : ℝ≥0∞)
hN₁ :
  ∀ (u : Finset ℕ),
    ((⋂ n ∈ u, s n) \ N).Nonempty → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
f : (n : ℕ) → α → ℝ≥0∞ :=
  fun (n : ℕ) ↦
    HSMul.hSMul (α := ℝ≥0∞) (β := (x : α) → ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1)
hfapp :
  ∀ (n : ℕ) (a : α), f n a = HMul.hMul (α := ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a)
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop ∂μ ≤ (μ : (a : Set α) → ℝ≥0∞) univ
x : α
hxN : x ∉ N
hx : HDiv.hDiv (β := ℝ≥0∞) r ((μ : (a : Set α) → ℝ≥0∞) univ : ℝ≥0∞) ≤ limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop
hxs : Set.Finite (α := ℕ) {n : ℕ | x ∈ s n}
⊢ False
have refine_3 : Tendsto (β := ℝ≥0∞)
  (fun (n : ℕ) ↦ HMul.hMul (α := ℝ≥0∞) (↑n.succ : ℝ≥0∞)⁻¹ (↑hxs.toFinset (s := {n : ℕ | x ∈ s n}).card : ℝ≥0∞)) atTop
  (nhds ⊥)
Tactic: refine ENNReal.div_ne_zero.2 ⟨hr₀, by finiteness⟩ <| eq_bot_mono hx <|
        Tendsto.limsup_eq <| tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
        (h := fun n ↦ (n.succ : ℝ≥0∞)⁻¹ * hxs.toFinset.card) ?_ bot_le fun n ↦ mul_le_mul_right ?_ _
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.Intersectivity (goal)
Goal:
α : Type u_2
inst : MeasurableSpace α
μ : Measure α
inst_1 : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ (μ : (a : Set α) → ℝ≥0∞) (s n)
M : (f : α → ℝ) → Set α := fun (f : α → ℝ) ↦ {x : α | eLpNormEssSup f μ < (↑‖f x‖₊ : ℝ≥0∞)}
N : Set α := ⋃ (u : Finset ℕ), M (indicator (⋂ n ∈ u, s n) 1)
hN₀ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) N : ℝ≥0∞) (0 : ℝ≥0∞)
hN₁ :
  ∀ (u : Finset ℕ),
    ((⋂ n ∈ u, s n) \ N).Nonempty → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
f : (n : ℕ) → α → ℝ≥0∞ :=
  fun (n : ℕ) ↦
    HSMul.hSMul (α := ℝ≥0∞) (β := (x : α) → ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1)
hfapp :
  ∀ (n : ℕ) (a : α), f n a = HMul.hMul (α := ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a)
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop ∂μ ≤ (μ : (a : Set α) → ℝ≥0∞) univ
x : α
hxN : x ∉ N
hx : HDiv.hDiv (β := ℝ≥0∞) r ((μ : (a : Set α) → ℝ≥0∞) univ : ℝ≥0∞) ≤ limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop
hxs : Set.Finite (α := ℕ) {n : ℕ | x ∈ s n}
refine_3 :
  Tendsto (β := ℝ≥0∞)
    (fun (n : ℕ) ↦ HMul.hMul (α := ℝ≥0∞) (↑n.succ : ℝ≥0∞)⁻¹ (↑hxs.toFinset (s := {n : ℕ | x ∈ s n}).card : ℝ≥0∞)) atTop
    (nhds ⊥)
⊢ False
have refine_3₁ : ∀ (n : ℕ),
  LE.le (α := (fun (a : ℕ) ↦ ℝ≥0∞) n) ((∑ k ∈ Finset.range (n + 1), (s k).indicator 1) x)
    (↑hxs.toFinset (s := {n : ℕ | x ∈ s n}).card : ℝ≥0∞)
Tactic: refine ENNReal.div_ne_zero.2 ⟨hr₀, by finiteness⟩ <| eq_bot_mono hx <|
        Tendsto.limsup_eq <| tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
        (h := fun n ↦ (n.succ : ℝ≥0∞)⁻¹ * hxs.toFinset.card) ?_ bot_le fun n ↦ mul_le_mul_right ?_ _
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.Intersectivity (goal)
Goal:
α : Type u_2
inst : MeasurableSpace α
μ : Measure α
inst_1 : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ (μ : (a : Set α) → ℝ≥0∞) (s n)
M : (f : α → ℝ) → Set α := fun (f : α → ℝ) ↦ {x : α | eLpNormEssSup f μ < (↑‖f x‖₊ : ℝ≥0∞)}
N : Set α := ⋃ (u : Finset ℕ), M (indicator (⋂ n ∈ u, s n) 1)
hN₀ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) N : ℝ≥0∞) (0 : ℝ≥0∞)
hN₁ :
  ∀ (u : Finset ℕ),
    ((⋂ n ∈ u, s n) \ N).Nonempty → LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
f : (n : ℕ) → α → ℝ≥0∞ :=
  fun (n : ℕ) ↦
    HSMul.hSMul (α := ℝ≥0∞) (β := (x : α) → ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1)
hfapp :
  ∀ (n : ℕ) (a : α), f n a = HMul.hMul (α := ℝ≥0∞) (↑(n + 1) : ℝ≥0∞)⁻¹ (∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a)
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop ∂μ ≤ (μ : (a : Set α) → ℝ≥0∞) univ
x : α
hxN : x ∉ N
hx : HDiv.hDiv (β := ℝ≥0∞) r ((μ : (a : Set α) → ℝ≥0∞) univ : ℝ≥0∞) ≤ limsup (fun (x_1 : ℕ) ↦ f x_1 x) atTop
u : Set ℕ
hux : u ⊆ {n : ℕ | x ∈ s n}
hu : u.Finite
⊢ LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)
have refine_3 : LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ hu.toFinset (s := u), s n) : ℝ≥0∞)
Tactic: simp_rw [← hu.mem_toFinset]