theorem extracted_formal_statement_1.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) →
        (∀ᵐ (ω : Ω) ∂μ, ∀ (i : Fin n), |X i ω| ≤ 1) →
          ∀ (ω : Ω), (∀ (i : Fin n), |X i ω| ≤ 1) → ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ) := sorry

State before:
case h
Ω : Type u
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
S : Ω → ℝ := fun ω => ∑ i, X i ω
hS_meas : Measurable S
f : ℝ → ℝ := fun x => rexp (τ * x)
f_meas : Measurable f
f_measω : Measurable fun ω => rexp (τ * S ω)
h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
h_all : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : Fin n), |X i ω| ≤ 1
ω : Ω
hω : ∀ (i : Fin n), |X i ω| ≤ 1
⊢ ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)

Tactic: rw [norm_eq_abs, abs_of_nonneg (exp_pos _).le]
    rw [exp_le_exp]
    suffices S ω ≤ n by nlinarith
    unfold S
    calc
  ∑ i, X i ω ≤ ∑ i, |X i ω| := by gcongr; exact le_abs_self (X _ ω)
  _ ≤ ∑ i, 1 := by gcongr; exact hω _
  _ = n := by simp [Finset.sum_const, nsmul_eq_mul, mul_one]

State after:

Relevant lemmas:
Real.exp_pos : ∀ (x : ℝ), 0 < rexp x
Real.norm_eq_abs : ∀ (r : ℝ), ‖r‖ = |r|
nsmul_eq_mul : ∀ {α : Type u_1} [inst : NonAssocSemiring α] (n : ℕ) (a : α), n • a = ↑n * a
Real.exp_le_exp : ∀ {x y : ℝ}, rexp x ≤ rexp y ↔ x ≤ y
mul_one : ∀ {M : Type u} [inst : MulOneClass M] (a : M), a * 1 = 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
le_abs_self : ∀ {α : Type u_1} [inst : Lattice α] [inst_2 : AddGroup α] (a : α), a ≤ |a|
abs_of_nonneg : ∀ {α : Type u_1} [inst : Lattice α] [inst_2 : AddGroup α] {a : α} [inst_3 : AddLeftMono α],
  0 ≤ a → |a| = a