awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
h : ¬s ≤[i] 0
hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := 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)
h₁ : ∀ l < k, 0 ≤ (↑s : Set α → ℝ) (s.restrictNonposSeq i l)
this : LE.le (α := ℝ) 0 (∑' (l : ℕ), (↑s : Set α → ℝ) (⋃ (_ : 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
have rw : (↑s : Set α → ℝ) i < ∑' (i_1 : ℕ), (↑s : Set α → ℝ) (⋃ (_ : i_1 < k), s.restrictNonposSeq i i_1)
Tactic: rw [sub_neg]
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
h : ¬s ≤[i] 0
hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := 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)
h₁ : ∀ l < k, 0 ≤ (↑s : Set α → ℝ) (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
have neg : LE.le (α := ℝ) 0 (∑' (l : ℕ), (↑s : Set α → ℝ) (⋃ (_ : l < k), s.restrictNonposSeq i l))
Tactic: suffices 0 ≤ ∑' l : ℕ, s (⋃ _ : l < k, restrictNonposSeq s i l) by
      rw [sub_neg]
      exact lt_of_lt_of_le hi₂ this
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
h : ¬s ≤[i] 0
hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := 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)
h₁ : ∀ l < k, 0 ≤ (↑s : Set α → ℝ) (s.restrictNonposSeq i l)
⊢ LE.le (α := ℝ) 0 (∑' (l : ℕ), (↑s : Set α → ℝ) (⋃ (_ : l < k), s.restrictNonposSeq i l))
have neg : ∀ (i_1 : ℕ), 0 ≤ (↑s : Set α → ℝ) (⋃ (_ : i_1 < k), s.restrictNonposSeq i i_1)
Tactic: refine tsum_nonneg ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
h : ¬s ≤[i] 0
hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := 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)
h₁ : ∀ l < k, 0 ≤ (↑s : Set α → ℝ) (s.restrictNonposSeq i l)
l : ℕ
⊢ 0 ≤ (↑s : Set α → ℝ) (⋃ (_ : l < k), s.restrictNonposSeq i l)
have pos : ∀ (h : l < k), 0 ≤ (↑s : Set α → ℝ) (⋃ (_ : l < k), s.restrictNonposSeq i l)
Tactic: by_cases h : l < k
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
h : ¬s ≤[i] 0
hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := 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)
h₁ : ∀ l < k, 0 ≤ (↑s : Set α → ℝ) (s.restrictNonposSeq i l)
l : ℕ
pos : ∀ (h : l < k), 0 ≤ (↑s : Set α → ℝ) (⋃ (_ : l < k), s.restrictNonposSeq i l)
⊢ 0 ≤ (↑s : Set α → ℝ) (⋃ (_ : l < k), s.restrictNonposSeq i l)
have neg : ∀ (h : ¬l < k), 0 ≤ (↑s : Set α → ℝ) (⋃ (_ : l < k), s.restrictNonposSeq i l)
Tactic: by_cases h : l < k
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
h_1 : ¬s ≤[i] 0
hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := 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)
h₁ : ∀ l < k, 0 ≤ (↑s : Set α → ℝ) (s.restrictNonposSeq i l)
l : ℕ
h : l < k
⊢ 0 ≤ (↑s : Set α → ℝ) (⋃ (_ : l < k), s.restrictNonposSeq i l)
have h₂ : ⋃ (_ : l < k), s.restrictNonposSeq i l = s.restrictNonposSeq i l
Tactic: convert! h₁ _ h
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
h_1 : ¬s ≤[i] 0
hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := 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)
h₁ : ∀ l < k, 0 ≤ (↑s : Set α → ℝ) (s.restrictNonposSeq i l)
l : ℕ
h : l < k
⊢ ⋃ (_ : l < k), s.restrictNonposSeq i l = s.restrictNonposSeq i l
have h₂ : ∀ (x : α), Membership.mem (γ := Set α) (⋃ (_ : l < k), s.restrictNonposSeq i l) x ↔ x ∈ s.restrictNonposSeq i l
Tactic: ext x
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
h_1 : ¬s ≤[i] 0
hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := 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)
h₁ : ∀ l < k, 0 ≤ (↑s : Set α → ℝ) (s.restrictNonposSeq i l)
l : ℕ
h : l < k
x : α
⊢ Membership.mem (γ := Set α) (⋃ (_ : l < k), s.restrictNonposSeq i l) x ↔ x ∈ s.restrictNonposSeq i l
have h₂ : x ∈ s.restrictNonposSeq i l → l < k
Tactic: rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
h_1 : ¬s ≤[i] 0
hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := 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)
h₁ : ∀ l < k, 0 ≤ (↑s : Set α → ℝ) (s.restrictNonposSeq i l)
l : ℕ
h : ¬l < k
⊢ 0 ≤ (↑s : Set α → ℝ) (⋃ (_ : l < k), s.restrictNonposSeq i l)
have h₂ : Eq.{u_1 + 1} (α := Set α) (⋃ (_ : l < k), s.restrictNonposSeq i l) ∅
Tactic: convert! le_of_eq s.empty.symm
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
h_1 : ¬s ≤[i] 0
hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := 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)
h₁ : ∀ l < k, 0 ≤ (↑s : Set α → ℝ) (s.restrictNonposSeq i l)
l : ℕ
h : ¬l < k
⊢ Eq.{u_1 + 1} (α := Set α) (⋃ (_ : l < k), s.restrictNonposSeq i l) ∅
have h₂ : ∀ (x : α), Membership.mem (γ := Set α) (⋃ (_ : l < k), s.restrictNonposSeq i l) x ↔ Membership.mem (γ := Set α) ∅ x
Tactic: ext
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
h_1 : ¬s ≤[i] 0
hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := 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)
h₁ : ∀ l < k, 0 ≤ (↑s : Set α → ℝ) (s.restrictNonposSeq i l)
l : ℕ
h : ¬l < k
x : α
⊢ Membership.mem (γ := Set α) (⋃ (_ : l < k), s.restrictNonposSeq i l) x ↔ Membership.mem (γ := Set α) ∅ x
have h₂ : l < k → x ∉ s.restrictNonposSeq i l
Tactic: simp only [exists_prop, Set.mem_empty_iff_false, Set.mem_iUnion, not_and, iff_false]
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
h : ¬s ≤[i] 0
hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := 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)
a b : ℕ
hab : a ≠ b
⊢ Function.onFun.{1, u_1 + 1, 1} (β := Set α) Disjoint (fun (l : ℕ) ↦ ⋃ (_ : l < k), s.restrictNonposSeq i l) a b
have neg : a < k → Disjoint (s.restrictNonposSeq i a) ((fun (l : ℕ) ↦ (⋃ (_ : l < k), s.restrictNonposSeq i l : Set α)) b)
Tactic: refine Set.disjoint_iUnion_left.mpr fun _ => ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
h : ¬s ≤[i] 0
hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := 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)
a b : ℕ
hab : a ≠ b
h_1 : a < k
⊢ Disjoint (s.restrictNonposSeq i a) ((fun (l : ℕ) ↦ (⋃ (_ : l < k), s.restrictNonposSeq i l : Set α)) b)
have neg : b < k → Disjoint (s.restrictNonposSeq i a) (s.restrictNonposSeq i b)
Tactic: refine Set.disjoint_iUnion_right.mpr fun _ => ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
h : ¬s ≤[i] 0
hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := 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
have neg : ∀ (i_1 : ℕ), ⋃ (_ : i_1 < k), s.restrictNonposSeq i i_1 ⊆ i
Tactic: apply Set.iUnion_subset
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : (↑s : Set α → ℝ) i < 0
h : ¬s ≤[i] 0
hn : ∃ (n : ℕ), s ≤[i \ ⋃ (l : ℕ), ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := 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)
a : ℕ
x : α
⊢ Membership.mem (γ := Set α) (⋃ (_ : a < k), s.restrictNonposSeq i a) x → x ∈ i
have neg : a < k → x ∈ s.restrictNonposSeq i a → x ∈ i
Tactic: simp only [and_imp, exists_prop, Set.mem_iUnion]
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
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
⊢ ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0
have pos : ∀ (h : s ≤[i] 0), ∃ (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
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
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
⊢ ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0
have neg : ∀ (h : ¬s ≤[i] 0), ∃ (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
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
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
⊢ ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0
have neg₂ : ∀ (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
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
Cleared hypotheses: hn
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
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
⊢ ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0
have 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
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
Cleared hypotheses: hn
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : (↑s : Set α → ℝ) i < 0
⊢ ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0
have hi₁ : MeasurableSet i
Tactic: have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : (↑s : Set α → ℝ) i < 0
hi₁ : MeasurableSet i
⊢ ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0
have pos : ∀ (h : s ≤[i] 0), ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0
Tactic: by_cases h : s ≤[i] 0
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : (↑s : Set α → ℝ) i < 0
hi₁ : MeasurableSet i
pos : ∀ (h : s ≤[i] 0), ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0
⊢ ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0
have neg : ∀ (h : ¬s ≤[i] 0), ∃ (j : Set α), MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ (↑s : Set α → ℝ) j < 0
Tactic: by_cases h : s ≤[i] 0