-- Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
    (hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0

Proof state:
case neg.hd
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
a b : ℕ
hab : a ≠ b
⊢ Function.onFun Disjoint (fun l => ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) a b

Tactic:
refine Set.disjoint_iUnion_left.mpr fun _ => ?_

Result:
case neg.hd
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
a b : ℕ
hab : a ≠ b
x✝ : a < k
⊢ Disjoint (MeasureTheory.SignedMeasure.restrictNonposSeq s i a)
    ((fun l => ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) b)

Proof state:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
hn : ¬∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
h : ¬s ≤[i] 0
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ ↑s j < 0

Tactic:
push_neg at hn

Result:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ ↑s j < 0

Proof state:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ l < k, 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : ¬l < k
⊢ 0 ≤ ↑s (⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)

Tactic:
convert le_of_eq s.empty.symm

Result:
case h.e'_4.h.e'_7
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ l < k, 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : ¬l < k
⊢ ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l = ∅

Proof state:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ↑s i - ∑' (i_1 : ℕ), ↑s (⋃ (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) < 0

Tactic:
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 _

Result:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ l < k, 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
⊢ ↑s i - ∑' (i_1 : ℕ), ↑s (⋃ (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) < 0

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
hl : l < k
⊢ ¬s ≤[i \ ⋃ k, ⋃ (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k] 0

Tactic:
refine mt (restrict_le_zero_subset _ (hi₁.diff ?_) (Set.Subset.refl _)) (Nat.find_min hn hl)

Result:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
hl : l < k
⊢ MeasurableSet (⋃ k, ⋃ (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)

Proof state:
case h.e'_4.h.e'_7.h
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ l < k, 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : ¬l < k
x✝ : α
⊢ x✝ ∈ ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l ↔ x✝ ∈ ∅

Tactic:
simp only [exists_prop, Set.mem_empty_iff_false, Set.mem_iUnion, not_and, iff_false]

Result:
case h.e'_4.h.e'_7.h
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ l < k, 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : ¬l < k
x✝ : α
⊢ l < k → x✝ ∉ MeasureTheory.SignedMeasure.restrictNonposSeq s i l