-- File: FoML/FoML/Bennett.lean
Theorem statement:
theorem extracted_formal_statement_34.{u_1} {Ω : Type u_1} [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) (i : Fin n)
  (hbdd : ∀ᵐ (ω : Ω) ∂μ, rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)) (ω : Ω)
  (hω : rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)) (h_original_goal : ‖rexp (τ * X i ω)‖ ≤ rexp τ) :
  rexp (τ * X i ω) ≤ rexp τ := sorry

Tactic state before:
case h
Ω : Type u_1
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
i : Fin n
hbdd : ∀ᵐ (ω : Ω) ∂μ, rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)
ω : Ω
hω : rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)
⊢ ‖rexp (τ * X i ω)‖ ≤ rexp τ

Tactic: rw [norm_eq_abs, abs_of_nonneg (exp_pos _).le]

Tactic state after:
[case h
 Ω : Type u_1
 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
 i : Fin n
 hbdd : ∀ᵐ (ω : Ω) ∂μ, rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)
 ω : Ω
 hω : rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)
 ⊢ rexp (τ * X i ω) ≤ rexp τ]

Premises used:
Real.exp_pos : ∀ (x : ℝ), 0 < rexp x
Real.norm_eq_abs : ∀ (r : ℝ), ‖r‖ = |r|
abs_of_nonneg : ∀ {α : Type u_1} [inst : Lattice α] [inst_2 : AddGroup α] {a : α} [inst_3 : AddLeftMono α],
  0 ≤ a → |a| = a