theorem extracted_formal_statement_50.{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)
  (h_original_goal :
    (∫ (x : Fin n → Ω), ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))|‖ ∂Measure.pi fun x => μ) ≤
      ∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ) :
  (fun a => ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (a k)))|‖) ≤ fun a => ⨆ i, ∑ k, (b + b) := sorry

State before:
case refine_2.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
⊢ (∫ (x : Fin n → Ω), ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))|‖ ∂Measure.pi fun x => μ) ≤
    ∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ

Tactic: apply integral_mono
    · exact integrable_aux5 hf hf' ω
    ·
  simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, smul_add, nsmul_eq_mul, ciSup_const,
    integrable_const]

State after:
[case refine_2.h.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
 ⊢ (fun a => ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (a k)))|‖) ≤ fun a => ⨆ i, ∑ k, (b + b)]

Relevant lemmas:
MeasureTheory.integral_mono : ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {f g : α → ℝ},
  Integrable f μ → Integrable g μ → f ≤ g → ∫ (a : α), f a ∂μ ≤ ∫ (a : α), g a ∂μ
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
nsmul_eq_mul : ∀ {α : Type u_1} [inst : NonAssocSemiring α] (n : ℕ) (a : α), n • a = ↑n * a
MeasureTheory.integrable_const : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : Measure α}
  [inst : NormedAddCommGroup β] [inst_4 : IsFiniteMeasure μ] (c : β), Integrable (fun x => c) μ
smul_add : ∀ {M : Type u_1} {A : Type u_7} [inst : AddZeroClass A] [inst_4 : DistribSMul M A] (a : M) (b₁ b₂ : A),
  a • (b₁ + b₂) = a • b₁ + a • b₂
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
integrable_aux5 : ∀ {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ} {ι : Type u_1} {f : ι → Z → ℝ}
  {μ : Measure Ω} [inst_4 : IsProbabilityMeasure μ] [inst_5 : Nonempty ι] [inst_6 : Countable ι] {X : Ω → Z},
  (∀ (i : ι), Measurable (f i ∘ X)) →
    ∀ {b : ℝ},
      (∀ (i : ι) (z : Z), |f i z| ≤ b) →
        ∀ (ω : Fin n → Ω), Integrable (fun ω' => ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (ω' k)))|‖) (Measure.pi fun x => μ)
ciSup_const : ∀ {α : Type u_1} {ι : Sort u_4} [inst : ConditionallyCompleteLattice α] [hι : Nonempty ι] {a : α},
  ⨆ x, a = a