theorem extracted_formal_statement_15.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} {t a b : ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (ht : t < 0) (h_original_goal : Integrable (fun ω => rexp (-t * -X ω)) μ) : -t > 0 := sorry

State before:
case inl
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t a b : ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
ht : t < 0
⊢ Integrable (fun ω => rexp (-t * -X ω)) μ

Tactic: apply integrable_expt (-t) _ _ hX.neg

State after:
[case inl
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 X : Ω → ℝ
 t a b : ℝ
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 ht : t < 0
 ⊢ ∀ᵐ (ω : Ω) ∂μ, -X ω ≤ ?m.21027,
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 X : Ω → ℝ
 t a b : ℝ
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 ht : t < 0
 ⊢ ℝ,
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 X : Ω → ℝ
 t a b : ℝ
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 ht : t < 0
 ⊢ -t > 0]

Relevant lemmas: