l3lab/ntp-mathlib huggingface dataset:
private theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
    (hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  by_cases h : s ≤[i] 0
  · exact ⟨i, hi₁, Set.Subset.refl _, h, hi₂⟩
  push_neg at hn
  set k := Nat.find hn
  have hk₂ : s ≤[i \ ⋃ l < k, restrictNonposSeq s i l] 0 := Nat.find_spec hn
  have hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), restrictNonposSeq s i l) :=
    MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  refine' ⟨i \ ⋃ l < k, restrictNonposSeq s i l, hi₁.diff hmeas, Set.diff_subset _ _, hk₂, _⟩
  rw [of_diff hmeas hi₁, s.of_disjoint_iUnion_nat]
  · have h₁ : ∀ l < k, 0 ≤ s (restrictNonposSeq s i l) := by
      intro l hl
      refine' le_of_lt (measure_of_restrictNonposSeq h _ _)
      refine' mt (restrict_le_zero_subset _ (hi₁.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl)
      exact
        MeasurableSet.iUnion fun _ =>
          MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
    suffices 0 ≤ ∑' l : ℕ, s (⋃ _ : l < k, restrictNonposSeq s i l) by
      rw [sub_neg]
      exact lt_of_lt_of_le hi₂ this
    refine' tsum_nonneg _
    intro l; by_cases h : l < k
    · convert h₁ _ h
      ext x
      rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
      

Proof state:
case h.e'_4.h.e'_7.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ l < k, 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : l < k
x : α
⊢ x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i l → l < k

Next tactic:
exact fun _ => h
private theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
    (hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  by_cases h : s ≤[i] 0
  · exact ⟨i, hi₁, Set.Subset.refl _, h, hi₂⟩
  push_neg at hn
  set k := Nat.find hn
  have hk₂ : s ≤[i \ ⋃ l < k, restrictNonposSeq s i l] 0 := Nat.find_spec hn
  have hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), restrictNonposSeq s i l) :=
    MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  refine' ⟨i \ ⋃ l < k, restrictNonposSeq s i l, hi₁.diff hmeas, Set.diff_subset _ _, hk₂, _⟩
  rw [of_diff hmeas hi₁, s.of_disjoint_iUnion_nat]
  · have h₁ : ∀ l < k, 0 ≤ s (restrictNonposSeq s i l) := by
      intro l hl
      refine' le_of_lt (measure_of_restrictNonposSeq h _ _)
      refine' mt (restrict_le_zero_subset _ (hi₁.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl)
      exact
        MeasurableSet.iUnion fun _ =>
          MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
    suffices 0 ≤ ∑' l : ℕ, s (⋃ _ : l < k, restrictNonposSeq s i l) by
      rw [sub_neg]
      exact lt_of_lt_of_le hi₂ this
    refine' tsum_nonneg _
    intro l; by_cases h : l < k
    · convert h₁ _ h
      ext x
      rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
      exact fun _ => h
    · 

Proof state:
case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ l < k, 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : ¬l < k
⊢ 0 ≤ ↑s (⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)

Next tactic:
convert le_of_eq s.empty.symm
private theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
    (hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  by_cases h : s ≤[i] 0
  · exact ⟨i, hi₁, Set.Subset.refl _, h, hi₂⟩
  push_neg at hn
  set k := Nat.find hn
  have hk₂ : s ≤[i \ ⋃ l < k, restrictNonposSeq s i l] 0 := Nat.find_spec hn
  have hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), restrictNonposSeq s i l) :=
    MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  refine' ⟨i \ ⋃ l < k, restrictNonposSeq s i l, hi₁.diff hmeas, Set.diff_subset _ _, hk₂, _⟩
  rw [of_diff hmeas hi₁, s.of_disjoint_iUnion_nat]
  · have h₁ : ∀ l < k, 0 ≤ s (restrictNonposSeq s i l) := by
      intro l hl
      refine' le_of_lt (measure_of_restrictNonposSeq h _ _)
      refine' mt (restrict_le_zero_subset _ (hi₁.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl)
      exact
        MeasurableSet.iUnion fun _ =>
          MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
    suffices 0 ≤ ∑' l : ℕ, s (⋃ _ : l < k, restrictNonposSeq s i l) by
      rw [sub_neg]
      exact lt_of_lt_of_le hi₂ this
    refine' tsum_nonneg _
    intro l; by_cases h : l < k
    · convert h₁ _ h
      ext x
      rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
      exact fun _ => h
    · convert le_of_eq s.empty.symm
      

Proof state:
case h.e'_4.h.e'_7
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ l < k, 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : ¬l < k
⊢ ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l = ∅

Next tactic:
ext
private theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
    (hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  by_cases h : s ≤[i] 0
  · exact ⟨i, hi₁, Set.Subset.refl _, h, hi₂⟩
  push_neg at hn
  set k := Nat.find hn
  have hk₂ : s ≤[i \ ⋃ l < k, restrictNonposSeq s i l] 0 := Nat.find_spec hn
  have hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), restrictNonposSeq s i l) :=
    MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  refine' ⟨i \ ⋃ l < k, restrictNonposSeq s i l, hi₁.diff hmeas, Set.diff_subset _ _, hk₂, _⟩
  rw [of_diff hmeas hi₁, s.of_disjoint_iUnion_nat]
  · have h₁ : ∀ l < k, 0 ≤ s (restrictNonposSeq s i l) := by
      intro l hl
      refine' le_of_lt (measure_of_restrictNonposSeq h _ _)
      refine' mt (restrict_le_zero_subset _ (hi₁.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl)
      exact
        MeasurableSet.iUnion fun _ =>
          MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
    suffices 0 ≤ ∑' l : ℕ, s (⋃ _ : l < k, restrictNonposSeq s i l) by
      rw [sub_neg]
      exact lt_of_lt_of_le hi₂ this
    refine' tsum_nonneg _
    intro l; by_cases h : l < k
    · convert h₁ _ h
      ext x
      rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
      exact fun _ => h
    · convert le_of_eq s.empty.symm
      ext; 

Proof state:
case h.e'_4.h.e'_7.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ l < k, 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : ¬l < k
x✝ : α
⊢ x✝ ∈ ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l ↔ x✝ ∈ ∅

Next tactic:
simp only [exists_prop, Set.mem_empty_iff_false, Set.mem_iUnion, not_and, iff_false_iff]
private theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
    (hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  by_cases h : s ≤[i] 0
  · exact ⟨i, hi₁, Set.Subset.refl _, h, hi₂⟩
  push_neg at hn
  set k := Nat.find hn
  have hk₂ : s ≤[i \ ⋃ l < k, restrictNonposSeq s i l] 0 := Nat.find_spec hn
  have hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), restrictNonposSeq s i l) :=
    MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  refine' ⟨i \ ⋃ l < k, restrictNonposSeq s i l, hi₁.diff hmeas, Set.diff_subset _ _, hk₂, _⟩
  rw [of_diff hmeas hi₁, s.of_disjoint_iUnion_nat]
  · have h₁ : ∀ l < k, 0 ≤ s (restrictNonposSeq s i l) := by
      intro l hl
      refine' le_of_lt (measure_of_restrictNonposSeq h _ _)
      refine' mt (restrict_le_zero_subset _ (hi₁.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl)
      exact
        MeasurableSet.iUnion fun _ =>
          MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
    suffices 0 ≤ ∑' l : ℕ, s (⋃ _ : l < k, restrictNonposSeq s i l) by
      rw [sub_neg]
      exact lt_of_lt_of_le hi₂ this
    refine' tsum_nonneg _
    intro l; by_cases h : l < k
    · convert h₁ _ h
      ext x
      rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
      exact fun _ => h
    · convert le_of_eq s.empty.symm
      ext; simp only [exists_prop, Set.mem_empty_iff_false, Set.mem_iUnion, not_and, iff_false_iff]
      

Proof state:
case h.e'_4.h.e'_7.h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ l < k, 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : ¬l < k
x✝ : α
⊢ l < Nat.find hn → x✝ ∉ MeasureTheory.SignedMeasure.restrictNonposSeq s i l

Next tactic:
exact fun h' => False.elim (h h')
private theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
    (hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  by_cases h : s ≤[i] 0
  · exact ⟨i, hi₁, Set.Subset.refl _, h, hi₂⟩
  push_neg at hn
  set k := Nat.find hn
  have hk₂ : s ≤[i \ ⋃ l < k, restrictNonposSeq s i l] 0 := Nat.find_spec hn
  have hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), restrictNonposSeq s i l) :=
    MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  refine' ⟨i \ ⋃ l < k, restrictNonposSeq s i l, hi₁.diff hmeas, Set.diff_subset _ _, hk₂, _⟩
  rw [of_diff hmeas hi₁, s.of_disjoint_iUnion_nat]
  · have h₁ : ∀ l < k, 0 ≤ s (restrictNonposSeq s i l) := by
      intro l hl
      refine' le_of_lt (measure_of_restrictNonposSeq h _ _)
      refine' mt (restrict_le_zero_subset _ (hi₁.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl)
      exact
        MeasurableSet.iUnion fun _ =>
          MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
    suffices 0 ≤ ∑' l : ℕ, s (⋃ _ : l < k, restrictNonposSeq s i l) by
      rw [sub_neg]
      exact lt_of_lt_of_le hi₂ this
    refine' tsum_nonneg _
    intro l; by_cases h : l < k
    · convert h₁ _ h
      ext x
      rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
      exact fun _ => h
    · convert le_of_eq s.empty.symm
      ext; simp only [exists_prop, Set.mem_empty_iff_false, Set.mem_iUnion, not_and, iff_false_iff]
      exact fun h' => False.elim (h h')
  · 

Proof state:
case neg.hf₁
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ∀ (i_1 : ℕ), MeasurableSet (⋃ (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)

Next tactic:
intro
private theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
    (hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  by_cases h : s ≤[i] 0
  · exact ⟨i, hi₁, Set.Subset.refl _, h, hi₂⟩
  push_neg at hn
  set k := Nat.find hn
  have hk₂ : s ≤[i \ ⋃ l < k, restrictNonposSeq s i l] 0 := Nat.find_spec hn
  have hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), restrictNonposSeq s i l) :=
    MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  refine' ⟨i \ ⋃ l < k, restrictNonposSeq s i l, hi₁.diff hmeas, Set.diff_subset _ _, hk₂, _⟩
  rw [of_diff hmeas hi₁, s.of_disjoint_iUnion_nat]
  · have h₁ : ∀ l < k, 0 ≤ s (restrictNonposSeq s i l) := by
      intro l hl
      refine' le_of_lt (measure_of_restrictNonposSeq h _ _)
      refine' mt (restrict_le_zero_subset _ (hi₁.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl)
      exact
        MeasurableSet.iUnion fun _ =>
          MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
    suffices 0 ≤ ∑' l : ℕ, s (⋃ _ : l < k, restrictNonposSeq s i l) by
      rw [sub_neg]
      exact lt_of_lt_of_le hi₂ this
    refine' tsum_nonneg _
    intro l; by_cases h : l < k
    · convert h₁ _ h
      ext x
      rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
      exact fun _ => h
    · convert le_of_eq s.empty.symm
      ext; simp only [exists_prop, Set.mem_empty_iff_false, Set.mem_iUnion, not_and, iff_false_iff]
      exact fun h' => False.elim (h h')
  · intro; 

Proof state:
case neg.hf₁
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
i✝ : ℕ
⊢ MeasurableSet (⋃ (_ : i✝ < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i✝)

Next tactic:
exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
private theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
    (hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  by_cases h : s ≤[i] 0
  · exact ⟨i, hi₁, Set.Subset.refl _, h, hi₂⟩
  push_neg at hn
  set k := Nat.find hn
  have hk₂ : s ≤[i \ ⋃ l < k, restrictNonposSeq s i l] 0 := Nat.find_spec hn
  have hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), restrictNonposSeq s i l) :=
    MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  refine' ⟨i \ ⋃ l < k, restrictNonposSeq s i l, hi₁.diff hmeas, Set.diff_subset _ _, hk₂, _⟩
  rw [of_diff hmeas hi₁, s.of_disjoint_iUnion_nat]
  · have h₁ : ∀ l < k, 0 ≤ s (restrictNonposSeq s i l) := by
      intro l hl
      refine' le_of_lt (measure_of_restrictNonposSeq h _ _)
      refine' mt (restrict_le_zero_subset _ (hi₁.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl)
      exact
        MeasurableSet.iUnion fun _ =>
          MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
    suffices 0 ≤ ∑' l : ℕ, s (⋃ _ : l < k, restrictNonposSeq s i l) by
      rw [sub_neg]
      exact lt_of_lt_of_le hi₂ this
    refine' tsum_nonneg _
    intro l; by_cases h : l < k
    · convert h₁ _ h
      ext x
      rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
      exact fun _ => h
    · convert le_of_eq s.empty.symm
      ext; simp only [exists_prop, Set.mem_empty_iff_false, Set.mem_iUnion, not_and, iff_false_iff]
      exact fun h' => False.elim (h h')
  · intro; exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  · 

Proof state:
case neg.hf₂
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ Pairwise (Disjoint on fun l => ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)

Next tactic:
intro a b hab
private theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
    (hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  by_cases h : s ≤[i] 0
  · exact ⟨i, hi₁, Set.Subset.refl _, h, hi₂⟩
  push_neg at hn
  set k := Nat.find hn
  have hk₂ : s ≤[i \ ⋃ l < k, restrictNonposSeq s i l] 0 := Nat.find_spec hn
  have hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), restrictNonposSeq s i l) :=
    MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  refine' ⟨i \ ⋃ l < k, restrictNonposSeq s i l, hi₁.diff hmeas, Set.diff_subset _ _, hk₂, _⟩
  rw [of_diff hmeas hi₁, s.of_disjoint_iUnion_nat]
  · have h₁ : ∀ l < k, 0 ≤ s (restrictNonposSeq s i l) := by
      intro l hl
      refine' le_of_lt (measure_of_restrictNonposSeq h _ _)
      refine' mt (restrict_le_zero_subset _ (hi₁.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl)
      exact
        MeasurableSet.iUnion fun _ =>
          MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
    suffices 0 ≤ ∑' l : ℕ, s (⋃ _ : l < k, restrictNonposSeq s i l) by
      rw [sub_neg]
      exact lt_of_lt_of_le hi₂ this
    refine' tsum_nonneg _
    intro l; by_cases h : l < k
    · convert h₁ _ h
      ext x
      rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
      exact fun _ => h
    · convert le_of_eq s.empty.symm
      ext; simp only [exists_prop, Set.mem_empty_iff_false, Set.mem_iUnion, not_and, iff_false_iff]
      exact fun h' => False.elim (h h')
  · intro; exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  · intro a b hab
    

Proof state:
case neg.hf₂
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬restrict s i ≤ restrict 0 i
hn :
  ∃ n,
    restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
      restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
k : ℕ := Nat.find hn
hk₂ :
  restrict s (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
    restrict 0 (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
a b : ℕ
hab : a ≠ b
⊢ (Disjoint on fun l => ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) a b

Next tactic:
refine' Set.disjoint_iUnion_left.mpr fun _ => _