theorem extracted_formal_statement_103.{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] [inst_2 : Countable ι] {X : Ω → Z}
  (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (ω : (Fin n → Ω) × (Fin n → Ω))
  (h_original_goal : ‖(↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|‖ ≤ ↑n * b) :
  (2 ^ n)⁻¹ * |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|| ≤ ↑n * b := sorry

State before:
case h
Ω : Type u
inst✝² : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
⊢ ‖(↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|‖ ≤ ↑n * b

Tactic: simp only [Signs.card, Nat.cast_pow, Nat.cast_ofNat, Int.reduceNeg, norm_mul, norm_inv, norm_pow,
  norm_ofNat, norm_eq_abs]

State after:
[case h
 Ω : Type u
 inst✝² : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝¹ : IsProbabilityMeasure μ
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 ⊢ (2 ^ n)⁻¹ * |∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|| ≤ ↑n * b]

Relevant lemmas:
Real.norm_ofNat : ∀ (n : ℕ) [inst : n.AtLeastTwo], ‖OfNat.ofNat n‖ = OfNat.ofNat n
norm_mul : ∀ {α : Type u_2} [inst : Norm α] [inst_3 : Mul α] [inst_4 : NormMulClass α] (a b : α), ‖a * b‖ = ‖a‖ * ‖b‖
Real.norm_eq_abs : ∀ (r : ℝ), ‖r‖ = |r|
norm_pow : ∀ {α : Type u_2} [inst : SeminormedRing α] [inst_3 : NormOneClass α] [inst_4 : NormMulClass α] (a : α)
  (n : ℕ), ‖a ^ n‖ = ‖a‖ ^ n
norm_inv : ∀ {α : Type u_2} [inst : NormedDivisionRing α] (a : α), ‖a⁻¹‖ = ‖a‖⁻¹
Nat.cast_pow : ∀ {α : Type u_1} [inst : Semiring α] (m n : ℕ), ↑(m ^ n) = ↑m ^ n
Signs.card : ∀ (n : ℕ), Fintype.card (Signs n) = 2 ^ n
Nat.cast_ofNat : ∀ {R : Type u_1} {n : ℕ} [inst : NatCast R] [inst_3 : n.AtLeastTwo], ↑(OfNat.ofNat n) = OfNat.ofNat n