l3lab/ntp-mathlib huggingface dataset:
private theorem restrictNonposSeq_disjoint' {n m : ℕ} (h : n < m) :
    restrictNonposSeq s i n ∩ restrictNonposSeq s i m = ∅ := by
  rw [Set.eq_empty_iff_forall_not_mem]
  rintro x ⟨hx₁, hx₂⟩
  

Proof state:
case intro
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
n m : ℕ
h : n < m
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i m
⊢ False

Next tactic:
cases m
private theorem restrictNonposSeq_disjoint' {n m : ℕ} (h : n < m) :
    restrictNonposSeq s i n ∩ restrictNonposSeq s i m = ∅ := by
  rw [Set.eq_empty_iff_forall_not_mem]
  rintro x ⟨hx₁, hx₂⟩
  cases m; · 

Proof state:
case intro.zero
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
n : ℕ
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
h : n < Nat.zero
hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero
⊢ False

Next tactic:
rw [Nat.zero_eq] at h
private theorem restrictNonposSeq_disjoint' {n m : ℕ} (h : n < m) :
    restrictNonposSeq s i n ∩ restrictNonposSeq s i m = ∅ := by
  rw [Set.eq_empty_iff_forall_not_mem]
  rintro x ⟨hx₁, hx₂⟩
  cases m; · rw [Nat.zero_eq] at h; 

Proof state:
case intro.zero
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
n : ℕ
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
h : n < 0
hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero
⊢ False

Next tactic:
linarith
private theorem restrictNonposSeq_disjoint' {n m : ℕ} (h : n < m) :
    restrictNonposSeq s i n ∩ restrictNonposSeq s i m = ∅ := by
  rw [Set.eq_empty_iff_forall_not_mem]
  rintro x ⟨hx₁, hx₂⟩
  cases m; · rw [Nat.zero_eq] at h; linarith
  · 

Proof state:
case intro.succ
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
n : ℕ
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
n✝ : ℕ
h : n < Nat.succ n✝
hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n✝)
⊢ False

Next tactic:
rw [restrictNonposSeq] at hx₂
private theorem restrictNonposSeq_disjoint' {n m : ℕ} (h : n < m) :
    restrictNonposSeq s i n ∩ restrictNonposSeq s i m = ∅ := by
  rw [Set.eq_empty_iff_forall_not_mem]
  rintro x ⟨hx₁, hx₂⟩
  cases m; · rw [Nat.zero_eq] at h; linarith
  · rw [restrictNonposSeq] at hx₂
    

Proof state:
case intro.succ
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
n : ℕ
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
n✝ : ℕ
h : n < Nat.succ n✝
hx₂ :
  x ∈
    MeasureTheory.SignedMeasure.someExistsOneDivLT s
      (i \
        ⋃ k,
          ⋃ (H : k ≤ n✝),
            let_fun this := (_ : k < Nat.succ n✝);
            MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ False

Next tactic:
exact
      (someExistsOneDivLT_subset hx₂).2
        (Set.mem_iUnion.2 ⟨n, Set.mem_iUnion.2 ⟨Nat.lt_succ_iff.mp h, hx₁⟩⟩)
private theorem restrictNonposSeq_disjoint : Pairwise (Disjoint on restrictNonposSeq s i) := by
  

Proof state:
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
⊢ Pairwise (Disjoint on MeasureTheory.SignedMeasure.restrictNonposSeq s i)

Next tactic:
intro n m h
private theorem restrictNonposSeq_disjoint : Pairwise (Disjoint on restrictNonposSeq s i) := by
  intro n m h
  

Proof state:
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
n m : ℕ
h : n ≠ m
⊢ (Disjoint on MeasureTheory.SignedMeasure.restrictNonposSeq s i) n m

Next tactic:
rw [Function.onFun, Set.disjoint_iff_inter_eq_empty]
private theorem restrictNonposSeq_disjoint : Pairwise (Disjoint on restrictNonposSeq s i) := by
  intro n m h
  rw [Function.onFun, Set.disjoint_iff_inter_eq_empty]
  

Proof state:
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
n m : ℕ
h : n ≠ m
⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅

Next tactic:
rcases lt_or_gt_of_ne h with (h | h)
private theorem restrictNonposSeq_disjoint : Pairwise (Disjoint on restrictNonposSeq s i) := by
  intro n m h
  rw [Function.onFun, Set.disjoint_iff_inter_eq_empty]
  rcases lt_or_gt_of_ne h with (h | h)
  · 

Proof state:
case inl
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
n m : ℕ
h✝ : n ≠ m
h : n < m
⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅

Next tactic:
rw [restrictNonposSeq_disjoint' h]
private theorem restrictNonposSeq_disjoint : Pairwise (Disjoint on restrictNonposSeq s i) := by
  intro n m h
  rw [Function.onFun, Set.disjoint_iff_inter_eq_empty]
  rcases lt_or_gt_of_ne h with (h | h)
  · rw [restrictNonposSeq_disjoint' h]
  · 

Proof state:
case inr
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
n m : ℕ
h✝ : n ≠ m
h : n > m
⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅

Next tactic:
rw [Set.inter_comm, restrictNonposSeq_disjoint' 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
  

Proof state:
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
hn :
  ¬∀ (n : 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)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0

Next tactic:
by_cases h : s ≤[i] 0
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
  · 

Proof state:
case pos
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
hn :
  ¬∀ (n : 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)
h : restrict s i ≤ restrict 0 i
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0

Next tactic:
exact ⟨i, hi₁, Set.Subset.refl _, h, hi₂⟩
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⟩
  

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
hn :
  ¬∀ (n : 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)
h : ¬restrict s i ≤ restrict 0 i
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0

Next tactic:
push_neg at hn
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
  

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)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0

Next tactic:
set k := Nat.find hn
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
  

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
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0

Next tactic:
have hk₂ : s ≤[i \ ⋃ l < k, restrictNonposSeq s i l] 0 := Nat.find_spec hn
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
  

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)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0

Next tactic:
have hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), restrictNonposSeq s i l) :=
    MeasurableSet.iUnion fun _ => 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 _
  

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)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0

Next tactic:
refine' ⟨i \ ⋃ l < k, restrictNonposSeq s i l, hi₁.diff hmeas, Set.diff_subset _ _, hk₂, _⟩
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, _⟩
  

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)
⊢ ↑s (i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) < 0

Next tactic:
rw [of_diff hmeas hi₁, s.of_disjoint_iUnion_nat]
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]
  · 

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)
⊢ ↑s i - ∑' (i_1 : ℕ), ↑s (⋃ (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) < 0

Next tactic:
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 _
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
      

Proof state:
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)
⊢ ∀ l < k, 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)

Next tactic:
intro l hl
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
      

Proof state:
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)
l : ℕ
hl : l < k
⊢ 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)

Next tactic:
refine' le_of_lt (measure_of_restrictNonposSeq 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 _ _)
      

Proof state:
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)
l : ℕ
hl : l < k
⊢ ¬restrict s (i \ ⋃ k, ⋃ (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)

Next tactic:
refine' mt (restrict_le_zero_subset _ (hi₁.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl)