awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
⊢ Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), s n) : ℝ≥0∞) (⨆ (n : ℕ), (m : (a : Set α) → ℝ≥0∞) (s n) : ℝ≥0∞)
have refine : Tendsto (β := ℝ≥0∞) (fun (k : ℕ) ↦ (m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s k)) atTop (𝓝 0)
Tactic: refine measure_iUnion_of_tendsto_zero m atTop ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
⊢ Tendsto (β := ℝ≥0∞) (fun (k : ℕ) ↦ (m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s k)) atTop (𝓝 0)
have refine : ∀ (n : ℕ),
  LE.le (α := (fun (a : ℕ) ↦ ℝ≥0∞) n) ((m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s n) : ℝ≥0∞)
    (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + n + 1) \ s (k + n)) : ℝ≥0∞)
Tactic: refine tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
n : ℕ
⊢ LE.le (α := (fun (a : ℕ) ↦ ℝ≥0∞) n) ((m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s n) : ℝ≥0∞)
    (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + n + 1) \ s (k + n)) : ℝ≥0∞)
have refine : (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
Tactic: refine (m.mono ?_).trans (measure_iUnion_le _)
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (hypothesis)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
n : ℕ
⊢ (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
have h' : Monotone s
Tactic: have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
n : ℕ
h' : Monotone s
⊢ (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
have simp : ∀ (i : ℕ), s i ⊆ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
Tactic: simp only [diff_subset_iff, iUnion_subset_iff]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (hypothesis)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
n : ℕ
h' : Monotone s
i : ℕ
x : α
hx : x ∈ s i
⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
have this : ∃ (i : ℕ), x ∈ s i
Tactic: have : ∃ i, x ∈ s i := by exists i
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
n : ℕ
h' : Monotone s
i : ℕ
x : α
hx : x ∈ s i
this : ∃ (i : ℕ), x ∈ s i
⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
have rcases : ∀ (j : ℕ) (hj : x ∈ s j) (hlt : ∀ m < j, x ∉ s m), x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
Tactic: rcases Nat.findX this with ⟨j, hj, hlt⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
n : ℕ
h' : Monotone s
x : α
this : ∃ (i : ℕ), x ∈ s i
j : ℕ
hj : x ∈ s j
hlt : ∀ m < j, x ∉ s m
⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
have inl : ∀ (hjn : j ≤ n), x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
Tactic: rcases le_or_gt j n with hjn | hnj
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
n : ℕ
h' : Monotone s
x : α
this : ∃ (i : ℕ), x ∈ s i
j : ℕ
hj : x ∈ s j
hlt : ∀ m < j, x ∉ s m
inl : ∀ (hjn : j ≤ n), x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
have inr : ∀ (hnj : n < j), x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
Tactic: rcases le_or_gt j n with hjn | hnj
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (hypothesis)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
n : ℕ
h' : Monotone s
x : α
this : ∃ (i : ℕ), x ∈ s i
j : ℕ
hj : x ∈ s j
hlt : ∀ m < j, x ∉ s m
hnj : n < j
⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
have this₁ : j - (n + 1) + n + 1 = j
Tactic: have : j - (n + 1) + n + 1 = j := by lia
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
n : ℕ
h' : Monotone s
x : α
this_1 : ∃ (i : ℕ), x ∈ s i
j : ℕ
hj : x ∈ s j
hlt : ∀ m < j, x ∉ s m
hnj : n < j
this : j - (n + 1) + n + 1 = j
⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
have inr : x ∈ s (j - (n + 1) + n + 1)
Tactic: refine Or.inr (mem_iUnion.2 ⟨j - (n + 1), ?_, hlt _ ?_⟩)
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
n : ℕ
h' : Monotone s
x : α
this_1 : ∃ (i : ℕ), x ∈ s i
j : ℕ
hj : x ∈ s j
hlt : ∀ m < j, x ∉ s m
hnj : n < j
this : j - (n + 1) + n + 1 = j
inr : x ∈ s (j - (n + 1) + n + 1)
⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
have inr₁ : j - (n + 1) + n < j
Tactic: refine Or.inr (mem_iUnion.2 ⟨j - (n + 1), ?_, hlt _ ?_⟩)
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
n : ℕ
h' : Monotone s
x : α
this_1 : ∃ (i : ℕ), x ∈ s i
j : ℕ
hj : x ∈ s j
hlt : ∀ m < j, x ∉ s m
hnj : n < j
this : j - (n + 1) + n + 1 = j
⊢ j - (n + 1) + n < j
have inr : (j - (n + 1) + n).succ ≤ j
Tactic: rw [← Nat.succ_le_iff, Nat.succ_eq_add_one, this]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
n : ℕ
h' : Monotone s
x : α
this_1 : ∃ (i : ℕ), x ∈ s i
j : ℕ
hj : x ∈ s j
hlt : ∀ m < j, x ∉ s m
hnj : n < j
this : j - (n + 1) + n + 1 = j
inr : (j - (n + 1) + n).succ ≤ j
inr₁ : j - (n + 1) + n + 1 ≤ j
⊢ j - (n + 1) + n < j
have inr₂ : j ≤ j
Tactic: rw [← Nat.succ_le_iff, Nat.succ_eq_add_one, this]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
⊢ Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), s n) : ℝ≥0∞) (⨆ (n : ℕ), (m : (a : Set α) → ℝ≥0∞) (s n) : ℝ≥0∞)
have refine : Tendsto (β := ℝ≥0∞) (fun (k : ℕ) ↦ (m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s k)) atTop (𝓝 0)
Tactic: classical
  refine measure_iUnion_of_tendsto_zero m atTop ?_
  refine tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => ?_
  refine (m.mono ?_).trans (measure_iUnion_le _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
  simp only [diff_subset_iff, iUnion_subset_iff]
  intro i x hx
  have : ∃ i, x ∈ s i := by exists i
  rcases Nat.findX this with ⟨j, hj, hlt⟩
  clear hx i
  rcases le_or_gt j n with hjn | hnj
  · exact Or.inl (h' hjn hj)
  have : j - (n + 1) + n + 1 = j := by lia
  refine Or.inr (mem_iUnion.2 ⟨j - (n + 1), ?_, hlt _ ?_⟩)
  · rwa [this]
  · rw [← Nat.succ_le_iff, Nat.succ_eq_add_one, this]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
refine : Tendsto (β := ℝ≥0∞) (fun (k : ℕ) ↦ (m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s k)) atTop (𝓝 0)
⊢ Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), s n) : ℝ≥0∞) (⨆ (n : ℕ), (m : (a : Set α) → ℝ≥0∞) (s n) : ℝ≥0∞)
have refine₁ : ∀ (n : ℕ),
  LE.le (α := (fun (a : ℕ) ↦ ℝ≥0∞) n) ((m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s n) : ℝ≥0∞)
    (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + n + 1) \ s (k + n)) : ℝ≥0∞)
Tactic: classical
  refine measure_iUnion_of_tendsto_zero m atTop ?_
  refine tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => ?_
  refine (m.mono ?_).trans (measure_iUnion_le _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
  simp only [diff_subset_iff, iUnion_subset_iff]
  intro i x hx
  have : ∃ i, x ∈ s i := by exists i
  rcases Nat.findX this with ⟨j, hj, hlt⟩
  clear hx i
  rcases le_or_gt j n with hjn | hnj
  · exact Or.inl (h' hjn hj)
  have : j - (n + 1) + n + 1 = j := by lia
  refine Or.inr (mem_iUnion.2 ⟨j - (n + 1), ?_, hlt _ ?_⟩)
  · rwa [this]
  · rw [← Nat.succ_le_iff, Nat.succ_eq_add_one, this]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
refine : Tendsto (β := ℝ≥0∞) (fun (k : ℕ) ↦ (m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s k)) atTop (𝓝 0)
refine₁ :
  ∀ (n : ℕ),
    LE.le (α := (fun (a : ℕ) ↦ ℝ≥0∞) n) ((m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s n) : ℝ≥0∞)
      (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + n + 1) \ s (k + n)) : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), s n) : ℝ≥0∞) (⨆ (n : ℕ), (m : (a : Set α) → ℝ≥0∞) (s n) : ℝ≥0∞)
have refine₂ : ∀ (n : ℕ), (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
Tactic: classical
  refine measure_iUnion_of_tendsto_zero m atTop ?_
  refine tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => ?_
  refine (m.mono ?_).trans (measure_iUnion_le _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
  simp only [diff_subset_iff, iUnion_subset_iff]
  intro i x hx
  have : ∃ i, x ∈ s i := by exists i
  rcases Nat.findX this with ⟨j, hj, hlt⟩
  clear hx i
  rcases le_or_gt j n with hjn | hnj
  · exact Or.inl (h' hjn hj)
  have : j - (n + 1) + n + 1 = j := by lia
  refine Or.inr (mem_iUnion.2 ⟨j - (n + 1), ?_, hlt _ ?_⟩)
  · rwa [this]
  · rw [← Nat.succ_le_iff, Nat.succ_eq_add_one, this]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
refine : Tendsto (β := ℝ≥0∞) (fun (k : ℕ) ↦ (m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s k)) atTop (𝓝 0)
refine₁ :
  ∀ (n : ℕ),
    LE.le (α := (fun (a : ℕ) ↦ ℝ≥0∞) n) ((m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s n) : ℝ≥0∞)
      (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + n + 1) \ s (k + n)) : ℝ≥0∞)
refine₂ : ∀ (n : ℕ), (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
⊢ Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), s n) : ℝ≥0∞) (⨆ (n : ℕ), (m : (a : Set α) → ℝ≥0∞) (s n) : ℝ≥0∞)
have refine₃ : ∀ (n : ℕ) (h' : Monotone s), (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
Tactic: classical
  refine measure_iUnion_of_tendsto_zero m atTop ?_
  refine tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => ?_
  refine (m.mono ?_).trans (measure_iUnion_le _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
  simp only [diff_subset_iff, iUnion_subset_iff]
  intro i x hx
  have : ∃ i, x ∈ s i := by exists i
  rcases Nat.findX this with ⟨j, hj, hlt⟩
  clear hx i
  rcases le_or_gt j n with hjn | hnj
  · exact Or.inl (h' hjn hj)
  have : j - (n + 1) + n + 1 = j := by lia
  refine Or.inr (mem_iUnion.2 ⟨j - (n + 1), ?_, hlt _ ?_⟩)
  · rwa [this]
  · rw [← Nat.succ_le_iff, Nat.succ_eq_add_one, this]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
refine : Tendsto (β := ℝ≥0∞) (fun (k : ℕ) ↦ (m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s k)) atTop (𝓝 0)
refine₁ :
  ∀ (n : ℕ),
    LE.le (α := (fun (a : ℕ) ↦ ℝ≥0∞) n) ((m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s n) : ℝ≥0∞)
      (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + n + 1) \ s (k + n)) : ℝ≥0∞)
refine₂ : ∀ (n : ℕ), (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₃ : ∀ (n : ℕ) (h' : Monotone s), (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
⊢ Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), s n) : ℝ≥0∞) (⨆ (n : ℕ), (m : (a : Set α) → ℝ≥0∞) (s n) : ℝ≥0∞)
have refine₄ : ∀ (n : ℕ) (h' : Monotone s) (i : ℕ), s i ⊆ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
Tactic: classical
  refine measure_iUnion_of_tendsto_zero m atTop ?_
  refine tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => ?_
  refine (m.mono ?_).trans (measure_iUnion_le _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
  simp only [diff_subset_iff, iUnion_subset_iff]
  intro i x hx
  have : ∃ i, x ∈ s i := by exists i
  rcases Nat.findX this with ⟨j, hj, hlt⟩
  clear hx i
  rcases le_or_gt j n with hjn | hnj
  · exact Or.inl (h' hjn hj)
  have : j - (n + 1) + n + 1 = j := by lia
  refine Or.inr (mem_iUnion.2 ⟨j - (n + 1), ?_, hlt _ ?_⟩)
  · rwa [this]
  · rw [← Nat.succ_le_iff, Nat.succ_eq_add_one, this]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
refine : Tendsto (β := ℝ≥0∞) (fun (k : ℕ) ↦ (m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s k)) atTop (𝓝 0)
refine₁ :
  ∀ (n : ℕ),
    LE.le (α := (fun (a : ℕ) ↦ ℝ≥0∞) n) ((m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s n) : ℝ≥0∞)
      (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + n + 1) \ s (k + n)) : ℝ≥0∞)
refine₂ : ∀ (n : ℕ), (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₃ : ∀ (n : ℕ) (h' : Monotone s), (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₄ : ∀ (n : ℕ) (h' : Monotone s) (i : ℕ), s i ⊆ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
⊢ Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), s n) : ℝ≥0∞) (⨆ (n : ℕ), (m : (a : Set α) → ℝ≥0∞) (s n) : ℝ≥0∞)
have refine₅ : ∀ (n : ℕ) (h' : Monotone s) (i : ℕ) ⦃x : α⦄ (hx : x ∈ s i), x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
Tactic: classical
  refine measure_iUnion_of_tendsto_zero m atTop ?_
  refine tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => ?_
  refine (m.mono ?_).trans (measure_iUnion_le _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
  simp only [diff_subset_iff, iUnion_subset_iff]
  intro i x hx
  have : ∃ i, x ∈ s i := by exists i
  rcases Nat.findX this with ⟨j, hj, hlt⟩
  clear hx i
  rcases le_or_gt j n with hjn | hnj
  · exact Or.inl (h' hjn hj)
  have : j - (n + 1) + n + 1 = j := by lia
  refine Or.inr (mem_iUnion.2 ⟨j - (n + 1), ?_, hlt _ ?_⟩)
  · rwa [this]
  · rw [← Nat.succ_le_iff, Nat.succ_eq_add_one, this]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
refine : Tendsto (β := ℝ≥0∞) (fun (k : ℕ) ↦ (m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s k)) atTop (𝓝 0)
refine₁ :
  ∀ (n : ℕ),
    LE.le (α := (fun (a : ℕ) ↦ ℝ≥0∞) n) ((m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s n) : ℝ≥0∞)
      (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + n + 1) \ s (k + n)) : ℝ≥0∞)
refine₂ : ∀ (n : ℕ), (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₃ : ∀ (n : ℕ) (h' : Monotone s), (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₄ : ∀ (n : ℕ) (h' : Monotone s) (i : ℕ), s i ⊆ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₅ : ∀ (n : ℕ) (h' : Monotone s) (i : ℕ) ⦃x : α⦄ (hx : x ∈ s i), x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
⊢ Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), s n) : ℝ≥0∞) (⨆ (n : ℕ), (m : (a : Set α) → ℝ≥0∞) (s n) : ℝ≥0∞)
have refine₆ : ∀ (n : ℕ) (h' : Monotone s) (i : ℕ) ⦃x : α⦄ (hx : x ∈ s i) (this : ∃ (i : ℕ), x ∈ s i),
  x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
Tactic: classical
  refine measure_iUnion_of_tendsto_zero m atTop ?_
  refine tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => ?_
  refine (m.mono ?_).trans (measure_iUnion_le _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
  simp only [diff_subset_iff, iUnion_subset_iff]
  intro i x hx
  have : ∃ i, x ∈ s i := by exists i
  rcases Nat.findX this with ⟨j, hj, hlt⟩
  clear hx i
  rcases le_or_gt j n with hjn | hnj
  · exact Or.inl (h' hjn hj)
  have : j - (n + 1) + n + 1 = j := by lia
  refine Or.inr (mem_iUnion.2 ⟨j - (n + 1), ?_, hlt _ ?_⟩)
  · rwa [this]
  · rw [← Nat.succ_le_iff, Nat.succ_eq_add_one, this]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
refine : Tendsto (β := ℝ≥0∞) (fun (k : ℕ) ↦ (m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s k)) atTop (𝓝 0)
refine₁ :
  ∀ (n : ℕ),
    LE.le (α := (fun (a : ℕ) ↦ ℝ≥0∞) n) ((m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s n) : ℝ≥0∞)
      (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + n + 1) \ s (k + n)) : ℝ≥0∞)
refine₂ : ∀ (n : ℕ), (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₃ : ∀ (n : ℕ) (h' : Monotone s), (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₄ : ∀ (n : ℕ) (h' : Monotone s) (i : ℕ), s i ⊆ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₅ : ∀ (n : ℕ) (h' : Monotone s) (i : ℕ) ⦃x : α⦄ (hx : x ∈ s i), x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₆ :
  ∀ (n : ℕ) (h' : Monotone s) (i : ℕ) ⦃x : α⦄ (hx : x ∈ s i) (this : ∃ (i : ℕ), x ∈ s i),
    x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
⊢ Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), s n) : ℝ≥0∞) (⨆ (n : ℕ), (m : (a : Set α) → ℝ≥0∞) (s n) : ℝ≥0∞)
have refine₇ : ∀ (n : ℕ) (h' : Monotone s) (i : ℕ) ⦃x : α⦄ (hx : x ∈ s i) (this : ∃ (i : ℕ), x ∈ s i) (j : ℕ) (hj : x ∈ s j)
  (hlt : ∀ m < j, x ∉ s m), x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
Tactic: classical
  refine measure_iUnion_of_tendsto_zero m atTop ?_
  refine tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => ?_
  refine (m.mono ?_).trans (measure_iUnion_le _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
  simp only [diff_subset_iff, iUnion_subset_iff]
  intro i x hx
  have : ∃ i, x ∈ s i := by exists i
  rcases Nat.findX this with ⟨j, hj, hlt⟩
  clear hx i
  rcases le_or_gt j n with hjn | hnj
  · exact Or.inl (h' hjn hj)
  have : j - (n + 1) + n + 1 = j := by lia
  refine Or.inr (mem_iUnion.2 ⟨j - (n + 1), ?_, hlt _ ?_⟩)
  · rwa [this]
  · rw [← Nat.succ_le_iff, Nat.succ_eq_add_one, this]
-- Lean 4 proof step from Mathlib.MeasureTheory.OuterMeasure.Basic (goal)
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
refine : Tendsto (β := ℝ≥0∞) (fun (k : ℕ) ↦ (m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s k)) atTop (𝓝 0)
refine₁ :
  ∀ (n : ℕ),
    LE.le (α := (fun (a : ℕ) ↦ ℝ≥0∞) n) ((m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s n) : ℝ≥0∞)
      (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + n + 1) \ s (k + n)) : ℝ≥0∞)
refine₂ : ∀ (n : ℕ), (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₃ : ∀ (n : ℕ) (h' : Monotone s), (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₄ : ∀ (n : ℕ) (h' : Monotone s) (i : ℕ), s i ⊆ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₅ : ∀ (n : ℕ) (h' : Monotone s) (i : ℕ) ⦃x : α⦄ (hx : x ∈ s i), x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₆ :
  ∀ (n : ℕ) (h' : Monotone s) (i : ℕ) ⦃x : α⦄ (hx : x ∈ s i) (this : ∃ (i : ℕ), x ∈ s i),
    x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₇ :
  ∀ (n : ℕ) (h' : Monotone s) (i : ℕ) ⦃x : α⦄ (hx : x ∈ s i) (this : ∃ (i : ℕ), x ∈ s i) (j : ℕ) (hj : x ∈ s j)
    (hlt : ∀ m < j, x ∉ s m), x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
⊢ Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), s n) : ℝ≥0∞) (⨆ (n : ℕ), (m : (a : Set α) → ℝ≥0∞) (s n) : ℝ≥0∞)
have refine₈ : ∀ (n : ℕ) (h' : Monotone s) ⦃x : α⦄ (this : ∃ (i : ℕ), x ∈ s i) (j : ℕ) (hj : x ∈ s j) (hlt : ∀ m < j, x ∉ s m),
  x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
Tactic: classical
  refine measure_iUnion_of_tendsto_zero m atTop ?_
  refine tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => ?_
  refine (m.mono ?_).trans (measure_iUnion_le _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
  simp only [diff_subset_iff, iUnion_subset_iff]
  intro i x hx
  have : ∃ i, x ∈ s i := by exists i
  rcases Nat.findX this with ⟨j, hj, hlt⟩
  clear hx i
  rcases le_or_gt j n with hjn | hnj
  · exact Or.inl (h' hjn hj)
  have : j - (n + 1) + n + 1 = j := by lia
  refine Or.inr (mem_iUnion.2 ⟨j - (n + 1), ?_, hlt _ ?_⟩)
  · rwa [this]
  · rw [← Nat.succ_le_iff, Nat.succ_eq_add_one, this]