-- Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
hn : ¬∀ (n : ℕ), ¬s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
pos : ∀ (h : s ≤[i] 0), ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0
neg neg₁ : ∀ (h : ¬s ≤[i] 0), ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0
neg₂ neg₃ :
  ∀ (h : ¬s ≤[i] 0) (hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0),
    ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0
neg₄ :
  ∀ (h : ¬s ≤[i] 0) (hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0),
    let k := Nat.find (p := fun (n : ℕ) ↦ s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0) hn;
    ∀ (hk₂ : s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l] 0),
      ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0
neg₅ :
  ∀ (h : ¬s ≤[i] 0) (hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0),
    let k := Nat.find (p := fun (n : ℕ) ↦ s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0) hn;
    ∀ (hk₂ : s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l] 0)
      (hmeas : MeasurableSet.{u_1} (α := α) (⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l)),
      ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0
neg₆ :
  ∀ (h : ¬s ≤[i] 0) (hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0),
    let k := Nat.find (p := fun (n : ℕ) ↦ s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0) hn;
    ∀ (hk₂ : s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l] 0)
      (hmeas : MeasurableSet.{u_1} (α := α) (⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l)),
      (↑s : Set α → ℝ) (i \ ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l) < 0
neg₇ :
  ∀ (h : ¬s ≤[i] 0) (hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0),
    let k := Nat.find (p := fun (n : ℕ) ↦ s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0) hn;
    ∀ (hk₂ : s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l] 0)
      (hmeas : MeasurableSet.{u_1} (α := α) (⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l)),
      (↑s : Set α → ℝ) i - ∑' (i_1 : ℕ), (↑s : Set α → ℝ) (⋃ (_ : i_1 < k), s.restrictNonposSeq i i_1) < 0
neg₈ :
  ∀ (h : ¬s ≤[i] 0) (hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0),
    let k := Nat.find (p := fun (n : ℕ) ↦ s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0) hn;
    ∀ (hk₂ : s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l] 0)
      (hmeas : MeasurableSet.{u_1} (α := α) (⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l)) (i_1 : ℕ),
      MeasurableSet.{u_1} (α := α) (⋃ (_ : i_1 < k), s.restrictNonposSeq i i_1)
neg₉ :
  ∀ (h : ¬s ≤[i] 0) (hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0),
    let k := Nat.find (p := fun (n : ℕ) ↦ s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0) hn;
    ∀ (hk₂ : s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l] 0)
      (hmeas : MeasurableSet.{u_1} (α := α) (⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l)),
      Pairwise (α := ℕ)
        (Function.onFun.{1, u_1 + 1, 1} (β := Set α) Disjoint fun (l : ℕ) ↦ ⋃ (_ : l < k), s.restrictNonposSeq i l)
neg₁₀ :
  ∀ (h : ¬s ≤[i] 0) (hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0),
    let k := Nat.find (p := fun (n : ℕ) ↦ s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0) hn;
    ∀ (hk₂ : s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l] 0)
      (hmeas : MeasurableSet.{u_1} (α := α) (⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l)),
      ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l ⊆ i
neg₁₁ :
  ∀ (h : ¬s ≤[i] 0) (hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0),
    let k := Nat.find (p := fun (n : ℕ) ↦ s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0) hn;
    ∀ (hk₂ : s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l] 0)
      (hmeas : MeasurableSet.{u_1} (α := α) (⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l)) (i_1 : ℕ),
      MeasurableSet.{u_1} (α := α) (⋃ (_ : i_1 < k), s.restrictNonposSeq i i_1)
neg₁₂ :
  ∀ (h : ¬s ≤[i] 0) (hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0),
    let k := Nat.find (p := fun (n : ℕ) ↦ s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0) hn;
    ∀ (hk₂ : s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l] 0)
      (hmeas : MeasurableSet.{u_1} (α := α) (⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l)),
      Pairwise (α := ℕ)
        (Function.onFun.{1, u_1 + 1, 1} (β := Set α) Disjoint fun (l : ℕ) ↦ ⋃ (_ : l < k), s.restrictNonposSeq i l)
neg₁₃ :
  ∀ (h : ¬s ≤[i] 0) (hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0),
    let k := Nat.find (p := fun (n : ℕ) ↦ s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0) hn;
    ∀ (hk₂ : s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l] 0)
      (hmeas : MeasurableSet.{u_1} (α := α) (⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l)),
      ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l ⊆ i
neg₁₄ :
  ∀ (h : ¬s ≤[i] 0) (hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0),
    let k := Nat.find (p := fun (n : ℕ) ↦ s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0) hn;
    ∀ (hk₂ : s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l] 0)
      (hmeas : MeasurableSet.{u_1} (α := α) (⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l)),
      Pairwise (α := ℕ)
        (Function.onFun.{1, u_1 + 1, 1} (β := Set α) Disjoint fun (l : ℕ) ↦ ⋃ (_ : l < k), s.restrictNonposSeq i l)
neg₁₅ neg₁₆ :
  ∀ (h : ¬s ≤[i] 0) (hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0),
    let k := Nat.find (p := fun (n : ℕ) ↦ s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0) hn;
    ∀ (hk₂ : s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l] 0)
      (hmeas : MeasurableSet.{u_1} (α := α) (⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l)),
      ⋃ (l : ℕ), ⋃ (_ : l < k), s.restrictNonposSeq i l ⊆ i
⊢ ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0

Tactic:
classical
  by_cases h : s ≤[i] 0
  · exact ⟨i, hi₁, Set.Subset.refl _, h, hi₂⟩
  push Not at hn
  set k := Nat.find hn
  have hk₂ : s ≤[i \ ⋃ l < k, restrictNonposSeq s i l] 0 := Nat.find_spec hn
  have hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), restrictNonposSeq s i l) :=
    MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  refine ⟨i \ ⋃ l < k, restrictNonposSeq s i l, hi₁.diff hmeas, Set.diff_subset, hk₂, ?_⟩
  rw [of_diff hmeas hi₁, s.of_disjoint_iUnion]
  · have h₁ : ∀ l < k, 0 ≤ s (restrictNonposSeq s i l) := by
      intro l hl
      refine le_of_lt (measure_of_restrictNonposSeq h _ ?_)
      refine mt (restrict_le_zero_subset _ (hi₁.diff ?_) (Set.Subset.refl _)) (Nat.find_min hn hl)
      exact
        MeasurableSet.iUnion fun _ =>
          MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
    suffices 0 ≤ ∑' l : ℕ, s (⋃ _ : l < k, restrictNonposSeq s i l) by
      rw [sub_neg]
      exact lt_of_lt_of_le hi₂ this
    refine tsum_nonneg ?_
    intro l; by_cases h : l < k
    · convert! h₁ _ h
      ext x
      rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
      exact fun _ => h
    · convert! le_of_eq s.empty.symm
      ext; simp only [exists_prop, Set.mem_empty_iff_false, Set.mem_iUnion, not_and, iff_false]
      exact fun h' => False.elim (h h')
  · intro; exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
  · intro a b hab
    refine Set.disjoint_iUnion_left.mpr fun _ => ?_
    refine Set.disjoint_iUnion_right.mpr fun _ => ?_
    exact restrictNonposSeq_disjoint hab
  · apply Set.iUnion_subset
    intro a x
    simp only [and_imp, exists_prop, Set.mem_iUnion]
    intro _ hx
    exact restrictNonposSeq_subset _ hx

Premises used: Iff.mpr, MeasureTheory.VectorMeasure.empty, Set.iUnion_subset, iff_false, and_imp, _private.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet, Set.disjoint_iUnion_left, Set.mem_iUnion, Set.disjoint_iUnion_right, exists_prop, MeasureTheory.VectorMeasure.of_disjoint_iUnion, and_iff_right_iff_imp, Set.mem_empty_iff_false, MeasurableSet.iUnion, le_of_eq, tsum_nonneg, Eq.symm, not_and, Not