l3lab/ntp-mathlib huggingface dataset:
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := 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
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i n)

Next tactic:
cases n with
  | zero =>
    rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
    rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
    exact lt_trans Nat.one_div_pos_of_nat h
  | succ n =>
    rw [restrictNonposSeq_succ]
    have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 := by
      refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn
      convert measurable_of_not_restrict_le_zero _ hn using 3
      exact funext fun x => by rw [Nat.lt_succ_iff]
    rcases someExistsOneDivLT_spec h₁ with ⟨_, _, h⟩
    exact lt_trans Nat.one_div_pos_of_nat h
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  

Proof state:
case 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
hi₂ : ¬restrict s i ≤ restrict 0 i
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero)

Next tactic:
| zero =>
    rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
    rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
    exact lt_trans Nat.one_div_pos_of_nat h
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  | zero =>
    

Proof state:
case 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
hi₂ : ¬restrict s i ≤ restrict 0 i
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero)

Next tactic:
rw [restrictNonposSeq]
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  | zero =>
    rw [restrictNonposSeq]; 

Proof state:
case 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
hi₂ : ¬restrict s i ≤ restrict 0 i
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅))

Next tactic:
rw [← @Set.diff_empty _ i] at hi₂
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  | zero =>
    rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
    

Proof state:
case 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
hi₂ : ¬restrict s (i \ ∅) ≤ restrict 0 (i \ ∅)
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅))

Next tactic:
rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  | zero =>
    rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
    rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
    

Proof state:
case zero.intro.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
hi₂ : ¬restrict s (i \ ∅) ≤ restrict 0 (i \ ∅)
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
left✝¹ : MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅) ⊆ i \ ∅
left✝ : MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅))
h :
  1 / (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ∅)) + 1) <
    ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅))
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅))

Next tactic:
exact lt_trans Nat.one_div_pos_of_nat h
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  | zero =>
    rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
    rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
    exact lt_trans Nat.one_div_pos_of_nat h
  

Proof state:
case 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
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n))

Next tactic:
| succ n =>
    rw [restrictNonposSeq_succ]
    have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 := by
      refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn
      convert measurable_of_not_restrict_le_zero _ hn using 3
      exact funext fun x => by rw [Nat.lt_succ_iff]
    rcases someExistsOneDivLT_spec h₁ with ⟨_, _, h⟩
    exact lt_trans Nat.one_div_pos_of_nat h
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  | zero =>
    rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
    rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
    exact lt_trans Nat.one_div_pos_of_nat h
  | succ n =>
    

Proof state:
case 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
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n))

Next tactic:
rw [restrictNonposSeq_succ]
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  | zero =>
    rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
    rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
    exact lt_trans Nat.one_div_pos_of_nat h
  | succ n =>
    rw [restrictNonposSeq_succ]
    

Proof state:
case 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
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 <
    ↑s
      (MeasureTheory.SignedMeasure.someExistsOneDivLT s
        (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k))

Next tactic:
have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 := by
      refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn
      convert measurable_of_not_restrict_le_zero _ hn using 3
      exact funext fun x => by rw [Nat.lt_succ_iff]
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  | zero =>
    rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
    rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
    exact lt_trans Nat.one_div_pos_of_nat h
  | succ n =>
    rw [restrictNonposSeq_succ]
    have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 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
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ ¬restrict s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)

Next tactic:
refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  | zero =>
    rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
    rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
    exact lt_trans Nat.one_div_pos_of_nat h
  | succ n =>
    rw [restrictNonposSeq_succ]
    have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 := by
      refine' mt (restrict_le_zero_subset _ _ (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
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k ⊆
    i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k

Next tactic:
simp [Nat.lt_succ_iff]
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  | zero =>
    rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
    rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
    exact lt_trans Nat.one_div_pos_of_nat h
  | succ n =>
    rw [restrictNonposSeq_succ]
    have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 := by
      refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; 

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
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k ⊆
    i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k

Next tactic:
rfl
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  | zero =>
    rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
    rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
    exact lt_trans Nat.one_div_pos_of_nat h
  | succ n =>
    rw [restrictNonposSeq_succ]
    have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 := by
      refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn
      

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
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ MeasurableSet (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)

Next tactic:
convert measurable_of_not_restrict_le_zero _ hn using 3
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  | zero =>
    rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
    rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
    exact lt_trans Nat.one_div_pos_of_nat h
  | succ n =>
    rw [restrictNonposSeq_succ]
    have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 := by
      refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn
      convert measurable_of_not_restrict_le_zero _ hn using 3
      

Proof state:
case h.e'_3.h.e'_4.h.e'_3
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
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ (fun k => ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) = fun k =>
    ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k

Next tactic:
exact funext fun x => by rw [Nat.lt_succ_iff]
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  | zero =>
    rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
    rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
    exact lt_trans Nat.one_div_pos_of_nat h
  | succ n =>
    rw [restrictNonposSeq_succ]
    have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 := by
      refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn
      convert measurable_of_not_restrict_le_zero _ hn using 3
      exact funext fun x => 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
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
x : ℕ
⊢ ⋃ (_ : x ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i x =
    ⋃ (_ : x < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i x

Next tactic:
rw [Nat.lt_succ_iff]
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  | zero =>
    rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
    rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
    exact lt_trans Nat.one_div_pos_of_nat h
  | succ n =>
    rw [restrictNonposSeq_succ]
    have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 := by
      refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn
      convert measurable_of_not_restrict_le_zero _ hn using 3
      exact funext fun x => by rw [Nat.lt_succ_iff]
    

Proof state:
case 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
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
h₁ :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 0 <
    ↑s
      (MeasureTheory.SignedMeasure.someExistsOneDivLT s
        (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k))

Next tactic:
rcases someExistsOneDivLT_spec h₁ with ⟨_, _, h⟩
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) := by
  cases n with
  | zero =>
    rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂
    rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩
    exact lt_trans Nat.one_div_pos_of_nat h
  | succ n =>
    rw [restrictNonposSeq_succ]
    have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 := by
      refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn
      convert measurable_of_not_restrict_le_zero _ hn using 3
      exact funext fun x => by rw [Nat.lt_succ_iff]
    rcases someExistsOneDivLT_spec h₁ with ⟨_, _, h⟩
    

Proof state:
case succ.intro.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
hi₂ : ¬restrict s i ≤ restrict 0 i
n : ℕ
hn :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
h₁ :
  ¬restrict s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      restrict 0 (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
left✝¹ :
  MeasureTheory.SignedMeasure.someExistsOneDivLT s
      (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊆
    i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k
left✝ :
  MeasurableSet
    (MeasureTheory.SignedMeasure.someExistsOneDivLT s
      (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k))
h :
  1 /
      (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s
            (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +
        1) <
    ↑s
      (MeasureTheory.SignedMeasure.someExistsOneDivLT s
        (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k))
⊢ 0 <
    ↑s
      (MeasureTheory.SignedMeasure.someExistsOneDivLT s
        (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k))

Next tactic:
exact lt_trans Nat.one_div_pos_of_nat h
private theorem restrictNonposSeq_measurableSet (n : ℕ) :
    MeasurableSet (restrictNonposSeq s i n) := 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
n : ℕ
⊢ MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i n)

Next tactic:
cases n
private theorem restrictNonposSeq_measurableSet (n : ℕ) :
    MeasurableSet (restrictNonposSeq s i n) := by
  cases n <;>
    · 

Proof state:
case 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
⊢ MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero)

Next tactic:
rw [restrictNonposSeq]
private theorem restrictNonposSeq_measurableSet (n : ℕ) :
    MeasurableSet (restrictNonposSeq s i n) := by
  cases n <;>
    · rw [restrictNonposSeq]
      

Proof state:
case 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
⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅))

Next tactic:
exact someExistsOneDivLT_measurableSet
private theorem restrictNonposSeq_measurableSet (n : ℕ) :
    MeasurableSet (restrictNonposSeq s i n) := by
  cases n <;>
    · 

Proof state:
case 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
⊢ MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n✝))

Next tactic:
rw [restrictNonposSeq]
private theorem restrictNonposSeq_measurableSet (n : ℕ) :
    MeasurableSet (restrictNonposSeq s i n) := by
  cases n <;>
    · rw [restrictNonposSeq]
      

Proof state:
case 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
⊢ MeasurableSet
    (MeasureTheory.SignedMeasure.someExistsOneDivLT s
      (i \
        ⋃ k,
          ⋃ (H : k ≤ n✝),
            let_fun this := (_ : k < Nat.succ n✝);
            MeasureTheory.SignedMeasure.restrictNonposSeq s i k))

Next tactic:
exact someExistsOneDivLT_measurableSet
private theorem restrictNonposSeq_disjoint' {n m : ℕ} (h : n < m) :
    restrictNonposSeq s i n ∩ restrictNonposSeq s i m = ∅ := 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
n m : ℕ
h : n < m
⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅

Next tactic:
rw [Set.eq_empty_iff_forall_not_mem]
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]
  

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
⊢ ∀ (x : α),
    x ∉ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m

Next tactic:
rintro x ⟨hx₁, hx₂⟩