awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Function.Intersectivity
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 ≤ ∫⁻ (x : α), 1 ∂μ

Tactic:
exact lintegral_mono fun a ↦ limsup_le_of_le ⟨0, fun R _ ↦ bot_le⟩ <|
      Eventually.of_forall fun n ↦ hf₁ _ _

Premises used: Filter.Eventually.of_forall, Filter.limsup_le_of_le, MeasureTheory.lintegral_mono, Exists.intro, OfNat.ofNat, bot_le
-- Module: Mathlib.MeasureTheory.Function.Intersectivity
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

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 _ _
-- Module: Mathlib.MeasureTheory.Function.Intersectivity
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
⊢ Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) univ : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
finiteness
-- Module: Mathlib.MeasureTheory.Function.Intersectivity
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
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∞)
⊢ ∃ (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

Premises used: ne_top_of_le_ne_top, MeasureTheory.exists_notMem_null_laverage_le
-- Module: Mathlib.MeasureTheory.Function.Intersectivity
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
⊢ ∫⁻ (a : α), 1 a ∂HSMul.hSMul (α := ℝ≥0∞) (((μ : (a : Set α) → ℝ≥0∞) univ)⁻¹ : ℝ≥0∞) μ ≠ ∞

Tactic:
simp
-- Module: Mathlib.MeasureTheory.Function.Intersectivity
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
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
⊢ ∃ (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
-- Module: Mathlib.MeasureTheory.Function.Intersectivity
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∞)

Tactic:
⟨{n | x ∈ s n}, fun hxs ↦ ?_, fun u hux hu ↦ ?_⟩

Premises used: setOf, Membership.mem, And.intro, Exists.intro
-- Module: Mathlib.MeasureTheory.Function.Intersectivity
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
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∞)
⊢ ∃ (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:
refine ⟨{n | x ∈ s n}, fun hxs ↦ ?_, fun u hux hu ↦ ?_⟩
-- Module: Mathlib.MeasureTheory.Function.Intersectivity
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}
⊢ Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) univ : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
finiteness
-- Module: Mathlib.MeasureTheory.Function.Intersectivity
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 ⊥)
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∞)
⊢ False

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 ?_ _

Premises used: Filter.Tendsto.limsup_eq, Iff.mpr, tendsto_of_tendsto_of_tendsto_of_le_of_le, tendsto_const_nhds, mul_le_mul_right, Inv.inv, And.intro, eq_bot_mono, ENNReal.div_ne_zero
-- Module: Mathlib.MeasureTheory.Function.Intersectivity
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}
⊢ 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:
simpa using ENNReal.Tendsto.mul_const (ENNReal.tendsto_inv_nat_nhds_zero.comp <|
          tendsto_add_atTop_nat 1) (.inr <| ENNReal.natCast_ne_top _)

Premises used: ENNReal.Tendsto.mul_const, ENNReal.natCast_ne_top, ENNReal.tendsto_inv_nat_nhds_zero, Filter.tendsto_add_atTop_nat, Filter.Tendsto.comp, OfNat.ofNat, Or.inr
-- Module: Mathlib.MeasureTheory.Function.Intersectivity
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}
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:
simpa only [Finset.sum_apply, indicator_apply, Pi.one_apply, Finset.sum_boole, Nat.cast_le]
          using Finset.card_le_card fun m hm ↦ hxs.mem_toFinset.2 (Finset.mem_filter.1 hm).2

Premises used: Iff.mpr, Set.indicator_apply, Set.Finite.mem_toFinset, Finset.card_le_card, Pi.one_apply, Finset.sum_apply, And.right, Iff.mp, Finset.mem_filter, Nat.cast_le, Finset.sum_boole
-- Module: Mathlib.MeasureTheory.Function.Intersectivity
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}
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:
classical
        simpa only [Finset.sum_apply, indicator_apply, Pi.one_apply, Finset.sum_boole, Nat.cast_le]
          using Finset.card_le_card fun m hm ↦ hxs.mem_toFinset.2 (Finset.mem_filter.1 hm).2

Premises used: Iff.mpr, Set.indicator_apply, Set.Finite.mem_toFinset, Finset.card_le_card, Pi.one_apply, Finset.sum_apply, And.right, Iff.mp, Finset.mem_filter, Nat.cast_le, Finset.sum_boole
-- Module: Mathlib.MeasureTheory.Function.Intersectivity
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
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∞)
⊢ LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋂ n ∈ u, s n) : ℝ≥0∞)

Tactic:
simp_rw [← hu.mem_toFinset]

Premises used: Set.Finite.mem_toFinset
-- Module: Mathlib.MeasureTheory.Function.Intersectivity
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 ∈ hu.toFinset (s := u), s n) : ℝ≥0∞)

Tactic:
hN₁ _ ⟨x, mem_iInter₂.2 fun n hn ↦ hux <| hu.mem_toFinset.1 hn, hxN⟩

Premises used: Iff.mpr, Set.Finite.mem_toFinset, Set.mem_iInter₂, And.intro, Exists.intro, Iff.mp