-- Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
theorem restrictNonposSeq_disjoint' {n m : ℕ} (h : n < m) :
    restrictNonposSeq s i n ∩ restrictNonposSeq s i m = ∅

Proof state:
case intro.succ
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n : ℕ
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
n✝ : ℕ
h : n < n✝ + 1
hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i (n✝ + 1)
⊢ False

Tactic:
rw [restrictNonposSeq] at hx₂

Result:
case intro.succ
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n : ℕ
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
n✝ : ℕ
h : n < n✝ + 1
hx₂ :
  x ∈
    MeasureTheory.SignedMeasure.someExistsOneDivLT s
      (i \
        ⋃ k,
          ⋃ (H : k ≤ n✝),
            let_fun this := ⋯;
            MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ False

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n m : ℕ
h : n < m
⊢ ∀ (x : α),
    x ∉ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m

Tactic:
rintro x ⟨hx₁, hx₂⟩

Result:
case intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n m : ℕ
h : n < m
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i m
⊢ False

Proof state:
case intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n m : ℕ
h : n < m
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i m
⊢ False

Tactic:
cases m

Result:
case intro.zero
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n : ℕ
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
h : n < 0
hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i 0
⊢ False

case intro.succ
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n : ℕ
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
n✝ : ℕ
h : n < n✝ + 1
hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i (n✝ + 1)
⊢ False

Proof state:
case intro.succ
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n : ℕ
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
n✝ : ℕ
h : n < n✝ + 1
hx₂ :
  x ∈
    MeasureTheory.SignedMeasure.someExistsOneDivLT s
      (i \
        ⋃ k,
          ⋃ (H : k ≤ n✝),
            let_fun this := ⋯;
            MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ False

Tactic:
exact
  (someExistsOneDivLT_subset hx₂).2
    (Set.mem_iUnion.2 ⟨n, Set.mem_iUnion.2 ⟨Nat.lt_succ_iff.mp h, hx₁⟩⟩)

Result:
no goals

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n m : ℕ
h : n < m
⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅

Tactic:
rw [Set.eq_empty_iff_forall_not_mem]

Result:
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n m : ℕ
h : n < m
⊢ ∀ (x : α),
    x ∉ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m

Proof state:
case intro.zero
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n : ℕ
x : α
hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n
h : n < 0
hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i 0
⊢ False

Tactic:
omega

Result:
no goals