theorem extracted_formal_statement_40.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {X : Ω → Z} [inst_2 : Nonempty ι]
  [inst_3 : Countable ι] (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (ω' : Fin n → Ω) (i : ι) (i_1 : Fin n → Ω) (q : ∑ x, f i (X (ω' x)) ≤ ∑ x, b)
  (q' : ∑ x, -b ≤ ∑ x_1, f i (X (i_1 x_1))) : ∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (i_1 x_1)) ≤ ↑n * (b + b) := sorry

State before:
case h
Ω : Type u
inst✝¹ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → Z
inst_2 : Nonempty ι
inst_3 : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω' : Fin n → Ω
i : ι
i_1 : Fin n → Ω
q : ∑ x, f i (X (ω' x)) ≤ ∑ x, b
q' : ∑ x, -b ≤ ∑ x_1, f i (X (i_1 x_1))
⊢ ∑ x, f i (X (ω' x)) - ∑ x_1, f i (X (i_1 x_1)) ≤ ↑n * (b + b)

Tactic: simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] at q
    simp only [Finset.sum_neg_distrib, Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] at q'
    linarith

State after:

Relevant lemmas:
Finset.card_univ : ∀ {α : Type u_1} [inst : Fintype α], Finset.univ.card = Fintype.card α
Fintype.card_fin : ∀ (n : ℕ), Fintype.card (Fin n) = n
Finset.sum_neg_distrib : ∀ {α : Type u_3} {β : Type u_4} {s : Finset α} {f : α → β} [inst : SubtractionCommMonoid β],
  ∑ x ∈ s, -f x = -∑ x ∈ s, f x
nsmul_eq_mul : ∀ {α : Type u_1} [inst : NonAssocSemiring α] (n : ℕ) (a : α), n • a = ↑n * a
Finset.sum_const : ∀ {α : Type u_3} {β : Type u_4} {s : Finset α} [inst : AddCommMonoid β] (b : β),
  ∑ _x ∈ s, b = s.card • b