-- Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
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), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
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)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ A
hE₃ : ↑0 E < ↑s E
k : ℕ
hk₁ : 1 ≤ bdd k
hk₂ : 1 / ↑(bdd k) < ↑s E
⊢ A ⊆ i \ ⋃ l, ⋃ (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l

Tactic:
apply Set.diff_subset_diff_right

Result:
case h
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
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), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
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)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ A
hE₃ : ↑0 E < ↑s E
k : ℕ
hk₁ : 1 ≤ bdd k
hk₂ : 1 / ↑(bdd k) < ↑s E
⊢ ⋃ l, ⋃ (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l ⊆
    ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l

Proof state:
case pos.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
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), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
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)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ A
hE₃ : ↑0 E < ↑s E
k : ℕ
hk₁ : 1 ≤ bdd k
hk₂ : 1 / ↑(bdd k) < ↑s E
hA' : A ⊆ i \ ⋃ l, ⋃ (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
⊢ 1 /
      (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s
              (i \ ⋃ l, ⋃ (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) -
            1) +
        1) <
    ↑s E

Tactic:
convert hk₂

Result:
case h.e'_3.h.e'_6
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
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), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
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)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ A
hE₃ : ↑0 E < ↑s E
k : ℕ
hk₁ : 1 ≤ bdd k
hk₂ : 1 / ↑(bdd k) < ↑s E
hA' : A ⊆ i \ ⋃ l, ⋃ (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l
⊢ ↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s
            (i \ ⋃ l, ⋃ (_ : l ≤ k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) -
          1) +
      1 =
    ↑(bdd k)

Proof state:
case intro.refine_1
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
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), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
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)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : ∀ (b : ℝ), ∃ i, ∀ (a : ℕ), i ≤ a → b ≤ ↑(bdd a)
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ A
hE₃ : ↑0 E < ↑s E
k : ℕ
hk : ∀ (a : ℕ), k ≤ a → (1 / ↑s E + 1) ⊔ 1 ≤ ↑(bdd a)
hle : 1 ≤ ↑(bdd k)
⊢ 1 ≤ bdd k

Tactic:
norm_cast at hle

Result:
no goals