l3lab/ntp-mathlib huggingface dataset:
/-- Alternative formulation of `measure_theory.signed_measure.exists_is_compl_positive_negative`
(the Hahn decomposition theorem) using set complements. -/
theorem exists_compl_positive_negative (s : SignedMeasure α) :
    ∃ i : Set α, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0 := by
  obtain ⟨f, _, hf₂, hf₁⟩ :=
    exists_seq_tendsto_sInf ⟨0, @zero_mem_measureOfNegatives _ _ s⟩ bddBelow_measureOfNegatives
  choose B hB using hf₁
  have hB₁ : ∀ n, MeasurableSet (B n) := fun n => (hB n).1.1
  have hB₂ : ∀ n, s ≤[B n] 0 := fun n => (hB n).1.2
  set A := ⋃ n, B n with hA
  have hA₁ : MeasurableSet A := MeasurableSet.iUnion hB₁
  have hA₂ : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hB₁ hB₂
  have hA₃ : s A = sInf s.measureOfNegatives := by
    apply le_antisymm
    · refine' le_of_tendsto_of_tendsto tendsto_const_nhds hf₂ (eventually_of_forall fun n => _)
      rw [← (hB n).2, hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
        of_union Set.disjoint_sdiff_left _ (hB₁ n)]
      · refine' add_le_of_nonpos_left _
        have : s ≤[A] 0 :=
          restrict_le_restrict_iUnion _ _ hB₁ fun m =>
            let ⟨_, h⟩ := (hB m).1
            h
        

Proof state:
case a
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
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
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)
/-- Alternative formulation of `measure_theory.signed_measure.exists_is_compl_positive_negative`
(the Hahn decomposition theorem) using set complements. -/
theorem exists_compl_positive_negative (s : SignedMeasure α) :
    ∃ i : Set α, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0 := by
  obtain ⟨f, _, hf₂, hf₁⟩ :=
    exists_seq_tendsto_sInf ⟨0, @zero_mem_measureOfNegatives _ _ s⟩ bddBelow_measureOfNegatives
  choose B hB using hf₁
  have hB₁ : ∀ n, MeasurableSet (B n) := fun n => (hB n).1.1
  have hB₂ : ∀ n, s ≤[B n] 0 := fun n => (hB n).1.2
  set A := ⋃ n, B n with hA
  have hA₁ : MeasurableSet A := MeasurableSet.iUnion hB₁
  have hA₂ : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hB₁ hB₂
  have hA₃ : s A = sInf s.measureOfNegatives := by
    apply le_antisymm
    · refine' le_of_tendsto_of_tendsto tendsto_const_nhds hf₂ (eventually_of_forall fun n => _)
      rw [← (hB n).2, hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
        of_union Set.disjoint_sdiff_left _ (hB₁ n)]
      · refine' add_le_of_nonpos_left _
        have : s ≤[A] 0 :=
          restrict_le_restrict_iUnion _ _ hB₁ fun m =>
            let ⟨_, h⟩ := (hB m).1
            h
        refine'
          nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
        

Proof state:
case a
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
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
n : ℕ
this : restrict s A ≤ restrict 0 A
⊢ MeasurableSet (⋃ i, B i)

Next tactic:
exact MeasurableSet.iUnion hB₁
/-- Alternative formulation of `measure_theory.signed_measure.exists_is_compl_positive_negative`
(the Hahn decomposition theorem) using set complements. -/
theorem exists_compl_positive_negative (s : SignedMeasure α) :
    ∃ i : Set α, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0 := by
  obtain ⟨f, _, hf₂, hf₁⟩ :=
    exists_seq_tendsto_sInf ⟨0, @zero_mem_measureOfNegatives _ _ s⟩ bddBelow_measureOfNegatives
  choose B hB using hf₁
  have hB₁ : ∀ n, MeasurableSet (B n) := fun n => (hB n).1.1
  have hB₂ : ∀ n, s ≤[B n] 0 := fun n => (hB n).1.2
  set A := ⋃ n, B n with hA
  have hA₁ : MeasurableSet A := MeasurableSet.iUnion hB₁
  have hA₂ : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hB₁ hB₂
  have hA₃ : s A = sInf s.measureOfNegatives := by
    apply le_antisymm
    · refine' le_of_tendsto_of_tendsto tendsto_const_nhds hf₂ (eventually_of_forall fun n => _)
      rw [← (hB n).2, hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
        of_union Set.disjoint_sdiff_left _ (hB₁ n)]
      · refine' add_le_of_nonpos_left _
        have : s ≤[A] 0 :=
          restrict_le_restrict_iUnion _ _ hB₁ fun m =>
            let ⟨_, h⟩ := (hB m).1
            h
        refine'
          nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
        exact MeasurableSet.iUnion hB₁
      · 

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
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
n : ℕ
⊢ MeasurableSet ((⋃ i, B i) \ B n)

Next tactic:
exact (MeasurableSet.iUnion hB₁).diff (hB₁ n)
/-- Alternative formulation of `measure_theory.signed_measure.exists_is_compl_positive_negative`
(the Hahn decomposition theorem) using set complements. -/
theorem exists_compl_positive_negative (s : SignedMeasure α) :
    ∃ i : Set α, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0 := by
  obtain ⟨f, _, hf₂, hf₁⟩ :=
    exists_seq_tendsto_sInf ⟨0, @zero_mem_measureOfNegatives _ _ s⟩ bddBelow_measureOfNegatives
  choose B hB using hf₁
  have hB₁ : ∀ n, MeasurableSet (B n) := fun n => (hB n).1.1
  have hB₂ : ∀ n, s ≤[B n] 0 := fun n => (hB n).1.2
  set A := ⋃ n, B n with hA
  have hA₁ : MeasurableSet A := MeasurableSet.iUnion hB₁
  have hA₂ : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hB₁ hB₂
  have hA₃ : s A = sInf s.measureOfNegatives := by
    apply le_antisymm
    · refine' le_of_tendsto_of_tendsto tendsto_const_nhds hf₂ (eventually_of_forall fun n => _)
      rw [← (hB n).2, hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
        of_union Set.disjoint_sdiff_left _ (hB₁ n)]
      · refine' add_le_of_nonpos_left _
        have : s ≤[A] 0 :=
          restrict_le_restrict_iUnion _ _ hB₁ fun m =>
            let ⟨_, h⟩ := (hB m).1
            h
        refine'
          nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
        exact MeasurableSet.iUnion hB₁
      · exact (MeasurableSet.iUnion hB₁).diff (hB₁ n)
    · 

Proof state:
case a
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
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
⊢ sInf (measureOfNegatives s) ≤ ↑s A

Next tactic:
exact csInf_le bddBelow_measureOfNegatives ⟨A, ⟨hA₁, hA₂⟩, rfl⟩
/-- Alternative formulation of `measure_theory.signed_measure.exists_is_compl_positive_negative`
(the Hahn decomposition theorem) using set complements. -/
theorem exists_compl_positive_negative (s : SignedMeasure α) :
    ∃ i : Set α, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0 := by
  obtain ⟨f, _, hf₂, hf₁⟩ :=
    exists_seq_tendsto_sInf ⟨0, @zero_mem_measureOfNegatives _ _ s⟩ bddBelow_measureOfNegatives
  choose B hB using hf₁
  have hB₁ : ∀ n, MeasurableSet (B n) := fun n => (hB n).1.1
  have hB₂ : ∀ n, s ≤[B n] 0 := fun n => (hB n).1.2
  set A := ⋃ n, B n with hA
  have hA₁ : MeasurableSet A := MeasurableSet.iUnion hB₁
  have hA₂ : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hB₁ hB₂
  have hA₃ : s A = sInf s.measureOfNegatives := by
    apply le_antisymm
    · refine' le_of_tendsto_of_tendsto tendsto_const_nhds hf₂ (eventually_of_forall fun n => _)
      rw [← (hB n).2, hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
        of_union Set.disjoint_sdiff_left _ (hB₁ n)]
      · refine' add_le_of_nonpos_left _
        have : s ≤[A] 0 :=
          restrict_le_restrict_iUnion _ _ hB₁ fun m =>
            let ⟨_, h⟩ := (hB m).1
            h
        refine'
          nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
        exact MeasurableSet.iUnion hB₁
      · exact (MeasurableSet.iUnion hB₁).diff (hB₁ n)
    · exact csInf_le bddBelow_measureOfNegatives ⟨A, ⟨hA₁, hA₂⟩, rfl⟩
  

Proof state:
case 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
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
⊢ ∃ i, MeasurableSet i ∧ restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ

Next tactic:
refine' ⟨Aᶜ, hA₁.compl, _, (compl_compl A).symm ▸ hA₂⟩
/-- Alternative formulation of `measure_theory.signed_measure.exists_is_compl_positive_negative`
(the Hahn decomposition theorem) using set complements. -/
theorem exists_compl_positive_negative (s : SignedMeasure α) :
    ∃ i : Set α, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0 := by
  obtain ⟨f, _, hf₂, hf₁⟩ :=
    exists_seq_tendsto_sInf ⟨0, @zero_mem_measureOfNegatives _ _ s⟩ bddBelow_measureOfNegatives
  choose B hB using hf₁
  have hB₁ : ∀ n, MeasurableSet (B n) := fun n => (hB n).1.1
  have hB₂ : ∀ n, s ≤[B n] 0 := fun n => (hB n).1.2
  set A := ⋃ n, B n with hA
  have hA₁ : MeasurableSet A := MeasurableSet.iUnion hB₁
  have hA₂ : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hB₁ hB₂
  have hA₃ : s A = sInf s.measureOfNegatives := by
    apply le_antisymm
    · refine' le_of_tendsto_of_tendsto tendsto_const_nhds hf₂ (eventually_of_forall fun n => _)
      rw [← (hB n).2, hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
        of_union Set.disjoint_sdiff_left _ (hB₁ n)]
      · refine' add_le_of_nonpos_left _
        have : s ≤[A] 0 :=
          restrict_le_restrict_iUnion _ _ hB₁ fun m =>
            let ⟨_, h⟩ := (hB m).1
            h
        refine'
          nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
        exact MeasurableSet.iUnion hB₁
      · exact (MeasurableSet.iUnion hB₁).diff (hB₁ n)
    · exact csInf_le bddBelow_measureOfNegatives ⟨A, ⟨hA₁, hA₂⟩, rfl⟩
  refine' ⟨Aᶜ, hA₁.compl, _, (compl_compl A).symm ▸ hA₂⟩
  

Proof state:
case 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
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
⊢ restrict 0 Aᶜ ≤ restrict s Aᶜ

Next tactic:
rw [restrict_le_restrict_iff _ _ hA₁.compl]
/-- Alternative formulation of `measure_theory.signed_measure.exists_is_compl_positive_negative`
(the Hahn decomposition theorem) using set complements. -/
theorem exists_compl_positive_negative (s : SignedMeasure α) :
    ∃ i : Set α, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0 := by
  obtain ⟨f, _, hf₂, hf₁⟩ :=
    exists_seq_tendsto_sInf ⟨0, @zero_mem_measureOfNegatives _ _ s⟩ bddBelow_measureOfNegatives
  choose B hB using hf₁
  have hB₁ : ∀ n, MeasurableSet (B n) := fun n => (hB n).1.1
  have hB₂ : ∀ n, s ≤[B n] 0 := fun n => (hB n).1.2
  set A := ⋃ n, B n with hA
  have hA₁ : MeasurableSet A := MeasurableSet.iUnion hB₁
  have hA₂ : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hB₁ hB₂
  have hA₃ : s A = sInf s.measureOfNegatives := by
    apply le_antisymm
    · refine' le_of_tendsto_of_tendsto tendsto_const_nhds hf₂ (eventually_of_forall fun n => _)
      rw [← (hB n).2, hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
        of_union Set.disjoint_sdiff_left _ (hB₁ n)]
      · refine' add_le_of_nonpos_left _
        have : s ≤[A] 0 :=
          restrict_le_restrict_iUnion _ _ hB₁ fun m =>
            let ⟨_, h⟩ := (hB m).1
            h
        refine'
          nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
        exact MeasurableSet.iUnion hB₁
      · exact (MeasurableSet.iUnion hB₁).diff (hB₁ n)
    · exact csInf_le bddBelow_measureOfNegatives ⟨A, ⟨hA₁, hA₂⟩, rfl⟩
  refine' ⟨Aᶜ, hA₁.compl, _, (compl_compl A).symm ▸ hA₂⟩
  rw [restrict_le_restrict_iff _ _ hA₁.compl]
  

Proof state:
case 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
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
⊢ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ Aᶜ → ↑0 j ≤ ↑s j

Next tactic:
intro C _ hC₁
/-- Alternative formulation of `measure_theory.signed_measure.exists_is_compl_positive_negative`
(the Hahn decomposition theorem) using set complements. -/
theorem exists_compl_positive_negative (s : SignedMeasure α) :
    ∃ i : Set α, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0 := by
  obtain ⟨f, _, hf₂, hf₁⟩ :=
    exists_seq_tendsto_sInf ⟨0, @zero_mem_measureOfNegatives _ _ s⟩ bddBelow_measureOfNegatives
  choose B hB using hf₁
  have hB₁ : ∀ n, MeasurableSet (B n) := fun n => (hB n).1.1
  have hB₂ : ∀ n, s ≤[B n] 0 := fun n => (hB n).1.2
  set A := ⋃ n, B n with hA
  have hA₁ : MeasurableSet A := MeasurableSet.iUnion hB₁
  have hA₂ : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hB₁ hB₂
  have hA₃ : s A = sInf s.measureOfNegatives := by
    apply le_antisymm
    · refine' le_of_tendsto_of_tendsto tendsto_const_nhds hf₂ (eventually_of_forall fun n => _)
      rw [← (hB n).2, hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
        of_union Set.disjoint_sdiff_left _ (hB₁ n)]
      · refine' add_le_of_nonpos_left _
        have : s ≤[A] 0 :=
          restrict_le_restrict_iUnion _ _ hB₁ fun m =>
            let ⟨_, h⟩ := (hB m).1
            h
        refine'
          nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
        exact MeasurableSet.iUnion hB₁
      · exact (MeasurableSet.iUnion hB₁).diff (hB₁ n)
    · exact csInf_le bddBelow_measureOfNegatives ⟨A, ⟨hA₁, hA₂⟩, rfl⟩
  refine' ⟨Aᶜ, hA₁.compl, _, (compl_compl A).symm ▸ hA₂⟩
  rw [restrict_le_restrict_iff _ _ hA₁.compl]
  intro C _ hC₁
  

Proof state:
case 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
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
C : Set α
a✝ : MeasurableSet C
hC₁ : C ⊆ Aᶜ
⊢ ↑0 C ≤ ↑s C

Next tactic:
by_contra! hC₂
/-- Alternative formulation of `measure_theory.signed_measure.exists_is_compl_positive_negative`
(the Hahn decomposition theorem) using set complements. -/
theorem exists_compl_positive_negative (s : SignedMeasure α) :
    ∃ i : Set α, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0 := by
  obtain ⟨f, _, hf₂, hf₁⟩ :=
    exists_seq_tendsto_sInf ⟨0, @zero_mem_measureOfNegatives _ _ s⟩ bddBelow_measureOfNegatives
  choose B hB using hf₁
  have hB₁ : ∀ n, MeasurableSet (B n) := fun n => (hB n).1.1
  have hB₂ : ∀ n, s ≤[B n] 0 := fun n => (hB n).1.2
  set A := ⋃ n, B n with hA
  have hA₁ : MeasurableSet A := MeasurableSet.iUnion hB₁
  have hA₂ : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hB₁ hB₂
  have hA₃ : s A = sInf s.measureOfNegatives := by
    apply le_antisymm
    · refine' le_of_tendsto_of_tendsto tendsto_const_nhds hf₂ (eventually_of_forall fun n => _)
      rw [← (hB n).2, hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
        of_union Set.disjoint_sdiff_left _ (hB₁ n)]
      · refine' add_le_of_nonpos_left _
        have : s ≤[A] 0 :=
          restrict_le_restrict_iUnion _ _ hB₁ fun m =>
            let ⟨_, h⟩ := (hB m).1
            h
        refine'
          nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
        exact MeasurableSet.iUnion hB₁
      · exact (MeasurableSet.iUnion hB₁).diff (hB₁ n)
    · exact csInf_le bddBelow_measureOfNegatives ⟨A, ⟨hA₁, hA₂⟩, rfl⟩
  refine' ⟨Aᶜ, hA₁.compl, _, (compl_compl A).symm ▸ hA₂⟩
  rw [restrict_le_restrict_iff _ _ hA₁.compl]
  intro C _ hC₁
  by_contra! hC₂
  

Proof state:
case 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
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
C : Set α
a✝ : MeasurableSet C
hC₁ : C ⊆ Aᶜ
hC₂ : ↑s C < ↑0 C
⊢ False

Next tactic:
rcases exists_subset_restrict_nonpos hC₂ with ⟨D, hD₁, hD, hD₂, hD₃⟩
/-- Alternative formulation of `measure_theory.signed_measure.exists_is_compl_positive_negative`
(the Hahn decomposition theorem) using set complements. -/
theorem exists_compl_positive_negative (s : SignedMeasure α) :
    ∃ i : Set α, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0 := by
  obtain ⟨f, _, hf₂, hf₁⟩ :=
    exists_seq_tendsto_sInf ⟨0, @zero_mem_measureOfNegatives _ _ s⟩ bddBelow_measureOfNegatives
  choose B hB using hf₁
  have hB₁ : ∀ n, MeasurableSet (B n) := fun n => (hB n).1.1
  have hB₂ : ∀ n, s ≤[B n] 0 := fun n => (hB n).1.2
  set A := ⋃ n, B n with hA
  have hA₁ : MeasurableSet A := MeasurableSet.iUnion hB₁
  have hA₂ : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hB₁ hB₂
  have hA₃ : s A = sInf s.measureOfNegatives := by
    apply le_antisymm
    · refine' le_of_tendsto_of_tendsto tendsto_const_nhds hf₂ (eventually_of_forall fun n => _)
      rw [← (hB n).2, hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n),
        of_union Set.disjoint_sdiff_left _ (hB₁ n)]
      · refine' add_le_of_nonpos_left _
        have : s ≤[A] 0 :=
          restrict_le_restrict_iUnion _ _ hB₁ fun m =>
            let ⟨_, h⟩ := (hB m).1
            h
        refine'
          nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this)
        exact MeasurableSet.iUnion hB₁
      · exact (MeasurableSet.iUnion hB₁).diff (hB₁ n)
    · exact csInf_le bddBelow_measureOfNegatives ⟨A, ⟨hA₁, hA₂⟩, rfl⟩
  refine' ⟨Aᶜ, hA₁.compl, _, (compl_compl A).symm ▸ hA₂⟩
  rw [restrict_le_restrict_iff _ _ hA₁.compl]
  intro C _ hC₁
  by_contra! hC₂
  rcases exists_subset_restrict_nonpos hC₂ with ⟨D, hD₁, hD, hD₂, hD₃⟩
  

Proof state:
case intro.intro.intro.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
s : SignedMeasure α
f : ℕ → ℝ
left✝ : Antitone f
hf₂ : Tendsto f atTop (nhds (sInf (measureOfNegatives s)))
B : ℕ → Set α
hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n
hB₁ : ∀ (n : ℕ), MeasurableSet (B n)
hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)
A : Set α := ⋃ n, B n
hA : A = ⋃ n, B n
hA₁ : MeasurableSet A
hA₂ : restrict s A ≤ restrict 0 A
hA₃ : ↑s A = sInf (measureOfNegatives s)
C : Set α
a✝ : MeasurableSet C
hC₁ : C ⊆ Aᶜ
hC₂ : ↑s C < ↑0 C
D : Set α
hD₁ : MeasurableSet D
hD : D ⊆ C
hD₂ : restrict s D ≤ restrict 0 D
hD₃ : ↑s D < 0
⊢ False

Next tactic:
have : s (A ∪ D) < sInf s.measureOfNegatives := by
    rw [← hA₃,
      of_union (Set.disjoint_of_subset_right (Set.Subset.trans hD hC₁) disjoint_compl_right) hA₁
        hD₁]
    linarith