l3lab/ntp-mathlib huggingface dataset:
/-- A measurable set of negative measure has a negative subset of negative measure. -/
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
  by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
  by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
  swap; · 

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
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
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:
exact exists_subset_restrict_nonpos' hi₁ hi hn
/-- A measurable set of negative measure has a negative subset of negative measure. -/
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
  by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
  by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
  swap; · exact exists_subset_restrict_nonpos' hi₁ hi hn
  

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
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
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:
set A := i \ ⋃ l, restrictNonposSeq s i l with hA
/-- A measurable set of negative measure has a negative subset of negative measure. -/
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
  by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
  by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
  swap; · exact exists_subset_restrict_nonpos' hi₁ hi hn
  set A := i \ ⋃ l, restrictNonposSeq s i l with hA
  

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
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
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)
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0

Next tactic:
set bdd : ℕ → ℕ := fun n => findExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k)
/-- A measurable set of negative measure has a negative subset of negative measure. -/
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
  by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
  by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
  swap; · exact exists_subset_restrict_nonpos' hi₁ hi hn
  set A := i \ ⋃ l, restrictNonposSeq s i l with hA
  set bdd : ℕ → ℕ := fun n => findExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k)
  

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
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
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)
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0

Next tactic:
have hn' : ∀ n : ℕ, ¬s ≤[i \ ⋃ l ≤ n, restrictNonposSeq s i l] 0 := by
    intro n
    convert hn (n + 1) using 5 <;>
      · ext l
        simp only [exists_prop, Set.mem_iUnion, and_congr_left_iff]
        exact fun _ => Nat.lt_succ_iff.symm
/-- A measurable set of negative measure has a negative subset of negative measure. -/
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
  by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
  by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
  swap; · exact exists_subset_restrict_nonpos' hi₁ hi hn
  set A := i \ ⋃ l, restrictNonposSeq s i l with hA
  set bdd : ℕ → ℕ := fun n => findExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k)
  have hn' : ∀ n : ℕ, ¬s ≤[i \ ⋃ l ≤ n, restrictNonposSeq s i l] 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
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
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)
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ ∀ (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)

Next tactic:
intro n
/-- A measurable set of negative measure has a negative subset of negative measure. -/
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
  by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
  by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
  swap; · exact exists_subset_restrict_nonpos' hi₁ hi hn
  set A := i \ ⋃ l, restrictNonposSeq s i l with hA
  set bdd : ℕ → ℕ := fun n => findExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k)
  have hn' : ∀ n : ℕ, ¬s ≤[i \ ⋃ l ≤ n, restrictNonposSeq s i l] 0 := by
    intro n
    

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
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
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)
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
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)

Next tactic:
convert hn (n + 1) using 5
/-- A measurable set of negative measure has a negative subset of negative measure. -/
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
  by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
  by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
  swap; · exact exists_subset_restrict_nonpos' hi₁ hi hn
  set A := i \ ⋃ l, restrictNonposSeq s i l with hA
  set bdd : ℕ → ℕ := fun n => findExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k)
  have hn' : ∀ n : ℕ, ¬s ≤[i \ ⋃ l ≤ n, restrictNonposSeq s i l] 0 := by
    intro n
    convert hn (n + 1) using 5 <;>
      · 

Proof state:
case h.e'_1.h.e'_3.h.e'_7.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
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
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)
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
n : ℕ
⊢ (fun l => ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) = fun l =>
    ⋃ (_ : l < n + 1), MeasureTheory.SignedMeasure.restrictNonposSeq s i l

Next tactic:
ext l
/-- A measurable set of negative measure has a negative subset of negative measure. -/
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
  by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
  by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
  swap; · exact exists_subset_restrict_nonpos' hi₁ hi hn
  set A := i \ ⋃ l, restrictNonposSeq s i l with hA
  set bdd : ℕ → ℕ := fun n => findExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k)
  have hn' : ∀ n : ℕ, ¬s ≤[i \ ⋃ l ≤ n, restrictNonposSeq s i l] 0 := by
    intro n
    convert hn (n + 1) using 5 <;>
      · ext l
        

Proof state:
case h.e'_1.h.e'_3.h.e'_7.h.e'_4.h.e'_3.h.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
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
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)
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
n l : ℕ
x✝ : α
⊢ x✝ ∈ ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l ↔
    x✝ ∈ ⋃ (_ : l < n + 1), MeasureTheory.SignedMeasure.restrictNonposSeq s i l

Next tactic:
simp only [exists_prop, Set.mem_iUnion, and_congr_left_iff]
/-- A measurable set of negative measure has a negative subset of negative measure. -/
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
  by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
  by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
  swap; · exact exists_subset_restrict_nonpos' hi₁ hi hn
  set A := i \ ⋃ l, restrictNonposSeq s i l with hA
  set bdd : ℕ → ℕ := fun n => findExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k)
  have hn' : ∀ n : ℕ, ¬s ≤[i \ ⋃ l ≤ n, restrictNonposSeq s i l] 0 := by
    intro n
    convert hn (n + 1) using 5 <;>
      · ext l
        simp only [exists_prop, Set.mem_iUnion, and_congr_left_iff]
        

Proof state:
case h.e'_1.h.e'_3.h.e'_7.h.e'_4.h.e'_3.h.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
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
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)
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
n l : ℕ
x✝ : α
⊢ x✝ ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i l → (l ≤ n ↔ l < n + 1)

Next tactic:
exact fun _ => Nat.lt_succ_iff.symm
/-- A measurable set of negative measure has a negative subset of negative measure. -/
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
  by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
  by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
  swap; · exact exists_subset_restrict_nonpos' hi₁ hi hn
  set A := i \ ⋃ l, restrictNonposSeq s i l with hA
  set bdd : ℕ → ℕ := fun n => findExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k)
  have hn' : ∀ n : ℕ, ¬s ≤[i \ ⋃ l ≤ n, restrictNonposSeq s i l] 0 := by
    intro n
    convert hn (n + 1) using 5 <;>
      · 

Proof state:
case h.e'_1.h.e'_4.h.e'_7.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
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
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)
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
n : ℕ
⊢ (fun l => ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) = fun l =>
    ⋃ (_ : l < n + 1), MeasureTheory.SignedMeasure.restrictNonposSeq s i l

Next tactic:
ext l
/-- A measurable set of negative measure has a negative subset of negative measure. -/
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
  by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
  by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
  swap; · exact exists_subset_restrict_nonpos' hi₁ hi hn
  set A := i \ ⋃ l, restrictNonposSeq s i l with hA
  set bdd : ℕ → ℕ := fun n => findExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k)
  have hn' : ∀ n : ℕ, ¬s ≤[i \ ⋃ l ≤ n, restrictNonposSeq s i l] 0 := by
    intro n
    convert hn (n + 1) using 5 <;>
      · ext l
        

Proof state:
case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3.h.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
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
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)
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
n l : ℕ
x✝ : α
⊢ x✝ ∈ ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l ↔
    x✝ ∈ ⋃ (_ : l < n + 1), MeasureTheory.SignedMeasure.restrictNonposSeq s i l

Next tactic:
simp only [exists_prop, Set.mem_iUnion, and_congr_left_iff]
/-- A measurable set of negative measure has a negative subset of negative measure. -/
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
  by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
  by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
  swap; · exact exists_subset_restrict_nonpos' hi₁ hi hn
  set A := i \ ⋃ l, restrictNonposSeq s i l with hA
  set bdd : ℕ → ℕ := fun n => findExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k)
  have hn' : ∀ n : ℕ, ¬s ≤[i \ ⋃ l ≤ n, restrictNonposSeq s i l] 0 := by
    intro n
    convert hn (n + 1) using 5 <;>
      · ext l
        simp only [exists_prop, Set.mem_iUnion, and_congr_left_iff]
        

Proof state:
case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3.h.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
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
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)
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
n l : ℕ
x✝ : α
⊢ x✝ ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i l → (l ≤ n ↔ l < n + 1)

Next tactic:
exact fun _ => Nat.lt_succ_iff.symm
/-- A measurable set of negative measure has a negative subset of negative measure. -/
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
  by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
  by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
  swap; · exact exists_subset_restrict_nonpos' hi₁ hi hn
  set A := i \ ⋃ l, restrictNonposSeq s i l with hA
  set bdd : ℕ → ℕ := fun n => findExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k)
  have hn' : ∀ n : ℕ, ¬s ≤[i \ ⋃ l ≤ n, restrictNonposSeq s i l] 0 := by
    intro n
    convert hn (n + 1) using 5 <;>
      · ext l
        simp only [exists_prop, Set.mem_iUnion, and_congr_left_iff]
        exact fun _ => Nat.lt_succ_iff.symm
  

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
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
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)
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
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:
have h₁ : s i = s A + ∑' l, s (restrictNonposSeq s i l) := by
    rw [hA, ← s.of_disjoint_iUnion_nat, add_comm, of_add_of_diff]
    exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
    exacts [hi₁, Set.iUnion_subset fun _ => restrictNonposSeq_subset _, fun _ =>
      restrictNonposSeq_measurableSet _, restrictNonposSeq_disjoint]
/-- A measurable set of negative measure has a negative subset of negative measure. -/
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
  have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
  by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
  by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
  swap; · exact exists_subset_restrict_nonpos' hi₁ hi hn
  set A := i \ ⋃ l, restrictNonposSeq s i l with hA
  set bdd : ℕ → ℕ := fun n => findExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k)
  have hn' : ∀ n : ℕ, ¬s ≤[i \ ⋃ l ≤ n, restrictNonposSeq s i l] 0 := by
    intro n
    convert hn (n + 1) using 5 <;>
      · ext l
        simp only [exists_prop, Set.mem_iUnion, and_congr_left_iff]
        exact fun _ => Nat.lt_succ_iff.symm
  have h₁ : s i = s A + ∑' l, 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
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
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)
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
  fun n =>
    MeasureTheory.SignedMeasure.findExistsOneDivLT s
      (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
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)
⊢ ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)

Next tactic:
rw [hA, ← s.of_disjoint_iUnion_nat, add_comm, of_add_of_diff]