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; · 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
    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]
  have h₂ : s A ≤ s i := by
    rw [h₁]
    apply le_add_of_nonneg_right
    exact tsum_nonneg fun n => le_of_lt (measure_of_restrictNonposSeq h _ (hn n))
  have h₃' : Summable fun n => (1 / (bdd n + 1) : ℝ) := by
    have : Summable fun l => s (restrictNonposSeq s i l) :=
      HasSum.summable
        (s.m_iUnion (fun _ => restrictNonposSeq_measurableSet _) restrictNonposSeq_disjoint)
    refine' .of_nonneg_of_le (fun n => _) (fun n => _)
        (this.comp_injective Nat.succ_injective)
    · exact le_of_lt Nat.one_div_pos_of_nat
    · 

Proof state:
case refine'_2
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)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
this : Summable fun l => ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
n : ℕ
⊢ 1 / (↑(bdd n) + 1) ≤ ((fun l => ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)) ∘ Nat.succ) n

Next tactic:
exact le_of_lt (restrictNonposSeq_lt n (hn' 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
    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
    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]
  have h₂ : s A ≤ s i := by
    rw [h₁]
    apply le_add_of_nonneg_right
    exact tsum_nonneg fun n => le_of_lt (measure_of_restrictNonposSeq h _ (hn n))
  have h₃' : Summable fun n => (1 / (bdd n + 1) : ℝ) := by
    have : Summable fun l => s (restrictNonposSeq s i l) :=
      HasSum.summable
        (s.m_iUnion (fun _ => restrictNonposSeq_measurableSet _) restrictNonposSeq_disjoint)
    refine' .of_nonneg_of_le (fun n => _) (fun n => _)
        (this.comp_injective Nat.succ_injective)
    · exact le_of_lt Nat.one_div_pos_of_nat
    · exact le_of_lt (restrictNonposSeq_lt n (hn' n))
  have h₃ : Tendsto (fun n => (bdd n : ℝ) + 1) atTop atTop := 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)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
⊢ Tendsto (fun n => ↑(bdd n) + 1) atTop atTop

Next tactic:
simp only [one_div] at h₃'
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := 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
⊢ BddBelow (measureOfNegatives s)

Next tactic:
simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  

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
⊢ ∃ x, ∀ x_1 ∈ measureOfNegatives s, x ≤ x_1

Next tactic:
by_contra! h
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! 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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
⊢ False

Next tactic:
have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! h
  have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
h' : ∀ (n : ℕ), ∃ y ∈ measureOfNegatives s, y < -↑n
⊢ False

Next tactic:
choose f hf using h'
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! h
  have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
  choose f hf using 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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
⊢ False

Next tactic:
have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by
    intro n
    rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
    exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! h
  have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
  choose f hf using h'
  have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
⊢ ∀ (n : ℕ), ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n

Next tactic:
intro n
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! h
  have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
  choose f hf using h'
  have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := 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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
n : ℕ
⊢ ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n

Next tactic:
rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! h
  have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
  choose f hf using h'
  have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by
    intro n
    rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
    

Proof state:
case intro.intro.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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
n : ℕ
hlt : f n < -↑n
B : Set α
hB₂ : ↑s B = f n
hB₁ : MeasurableSet B
hBr : restrict s B ≤ restrict 0 B
⊢ ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n

Next tactic:
exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! h
  have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
  choose f hf using h'
  have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by
    intro n
    rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
    exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩
  

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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
hf' : ∀ (n : ℕ), ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n
⊢ False

Next tactic:
choose B hmeas hr h_lt using hf'
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! h
  have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
  choose f hf using h'
  have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by
    intro n
    rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
    exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩
  choose B hmeas hr h_lt using hf'
  

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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
⊢ False

Next tactic:
set A := ⋃ n, B n with hA
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! h
  have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
  choose f hf using h'
  have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by
    intro n
    rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
    exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩
  choose B hmeas hr h_lt using hf'
  set A := ⋃ n, B n with hA
  

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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
⊢ False

Next tactic:
have hfalse : ∀ n : ℕ, s A ≤ -n := by
    intro n
    refine' le_trans _ (le_of_lt (h_lt _))
    rw [hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
      of_union Set.disjoint_sdiff_left _ (hmeas n)]
    · refine' add_le_of_nonpos_left _
      have : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hmeas hr
      refine' nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
      exact MeasurableSet.iUnion hmeas
    · exact (MeasurableSet.iUnion hmeas).diff (hmeas n)
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! h
  have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
  choose f hf using h'
  have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by
    intro n
    rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
    exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩
  choose B hmeas hr h_lt using hf'
  set A := ⋃ n, B n with hA
  have hfalse : ∀ n : ℕ, s A ≤ -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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
⊢ ∀ (n : ℕ), ↑s A ≤ -↑n

Next tactic:
intro n
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! h
  have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
  choose f hf using h'
  have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by
    intro n
    rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
    exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩
  choose B hmeas hr h_lt using hf'
  set A := ⋃ n, B n with hA
  have hfalse : ∀ n : ℕ, s A ≤ -n := 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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
n : ℕ
⊢ ↑s A ≤ -↑n

Next tactic:
refine' le_trans _ (le_of_lt (h_lt _))
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! h
  have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
  choose f hf using h'
  have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by
    intro n
    rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
    exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩
  choose B hmeas hr h_lt using hf'
  set A := ⋃ n, B n with hA
  have hfalse : ∀ n : ℕ, s A ≤ -n := by
    intro n
    refine' le_trans _ (le_of_lt (h_lt _))
    

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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
n : ℕ
⊢ ↑s A ≤ ↑s (B n)

Next tactic:
rw [hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
      of_union Set.disjoint_sdiff_left _ (hmeas n)]
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! h
  have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
  choose f hf using h'
  have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by
    intro n
    rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
    exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩
  choose B hmeas hr h_lt using hf'
  set A := ⋃ n, B n with hA
  have hfalse : ∀ n : ℕ, s A ≤ -n := by
    intro n
    refine' le_trans _ (le_of_lt (h_lt _))
    rw [hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
      of_union Set.disjoint_sdiff_left _ (hmeas 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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
n : ℕ
⊢ ↑s ((⋃ i, B i) \ B n) + ↑s (B n) ≤ ↑s (B n)

Next tactic:
refine' add_le_of_nonpos_left _
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! h
  have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
  choose f hf using h'
  have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by
    intro n
    rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
    exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩
  choose B hmeas hr h_lt using hf'
  set A := ⋃ n, B n with hA
  have hfalse : ∀ n : ℕ, s A ≤ -n := by
    intro n
    refine' le_trans _ (le_of_lt (h_lt _))
    rw [hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
      of_union Set.disjoint_sdiff_left _ (hmeas n)]
    · refine' add_le_of_nonpos_left _
      

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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
n : ℕ
⊢ ↑s ((⋃ i, B i) \ B n) ≤ 0

Next tactic:
have : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hmeas hr
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! h
  have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
  choose f hf using h'
  have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by
    intro n
    rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
    exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩
  choose B hmeas hr h_lt using hf'
  set A := ⋃ n, B n with hA
  have hfalse : ∀ n : ℕ, s A ≤ -n := by
    intro n
    refine' le_trans _ (le_of_lt (h_lt _))
    rw [hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
      of_union Set.disjoint_sdiff_left _ (hmeas n)]
    · refine' add_le_of_nonpos_left _
      have : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hmeas hr
      

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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
n : ℕ
this : restrict s A ≤ restrict 0 A
⊢ ↑s ((⋃ i, B i) \ B n) ≤ 0

Next tactic:
refine' nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by
  simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
  by_contra! h
  have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
  choose f hf using h'
  have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by
    intro n
    rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
    exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩
  choose B hmeas hr h_lt using hf'
  set A := ⋃ n, B n with hA
  have hfalse : ∀ n : ℕ, s A ≤ -n := by
    intro n
    refine' le_trans _ (le_of_lt (h_lt _))
    rw [hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
      of_union Set.disjoint_sdiff_left _ (hmeas n)]
    · refine' add_le_of_nonpos_left _
      have : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hmeas hr
      refine' nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
      

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
h : ∀ (x : ℝ), ∃ x_1 ∈ measureOfNegatives s, x_1 < x
f : ℕ → ℝ
hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n
B : ℕ → Set α
hmeas : ∀ (n : ℕ), MeasurableSet (B n)
hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
n : ℕ
this : restrict s A ≤ restrict 0 A
⊢ MeasurableSet (⋃ i, B i)

Next tactic:
exact MeasurableSet.iUnion hmeas