theorem extracted_formal_statement_14.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] (t : ℝ) (f : ℝ → ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ) (ω : Ω)
  (w : (∂μ.tilted fun x => t * X x/∂μ) ω = ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ))
  (h_original_goal :
    (ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ)).toReal • f (X ω) =
      (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ) * f (X ω)) :
  0 ≤ rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ := sorry

State before:
case h
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
t : ℝ
f : ℝ → ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
ω : Ω
w : (∂μ.tilted fun x => t * X x/∂μ) ω = ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ)
⊢ (ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ)).toReal • f (X ω) =
    (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ) * f (X ω)

Tactic: simp only [smul_eq_mul, mul_eq_mul_right_iff, ENNReal.toReal_ofReal_eq_iff]
    left
    have q2 : 0 ≤ μ[fun ω ↦ rexp (t * X ω)] :=
  by
  to_theorem apply integral_nonneg apply Pi.le_def.mpr _ intro i
  simp only [Pi.zero_apply]
  exact exp_nonneg (t * X i)

State after:
[case h.h
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 t : ℝ
 f : ℝ → ℝ
 X : Ω → ℝ
 hX : AEMeasurable X μ
 ω : Ω
 w : (∂μ.tilted fun x => t * X x/∂μ) ω = ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ)
 q2 : 0 ≤ ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ
 ⊢ 0 ≤ rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ]

Relevant lemmas:
error