Probability · hard

Statement, exactly as the model and Lean see it

theorem lg_target {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {t : ℝ} (ht : t ∈ interior (ProbabilityTheory.integrableExpSet X μ)) : (∫ (x : Ω), X x ∂μ.tilted fun x => t * X x) = deriv (ProbabilityTheory.cgf X μ) t := by

Mathlib declaration: ProbabilityTheory.integral_tilted_mul_self
Held-out module: Mathlib.Probability.Moments.Tilted
Banned modules (itself + downstream): 2
Reference proof premises (reachable / held-out): 23 / 1

Traces

Each configuration's full record: what was retrieved, what the model wrote, what Lean said back.

No configuration has attempted this theorem yet.

Reference proof

Mathlib's own source for this declaration, shown for comparison. It may use lemmas the prover is not allowed to use.

/-- The integral of `X` against the tilted measure `μ.tilted (t * X ·)` is the first derivative of
the cumulant-generating function of `X` at `t`. -/
lemma integral_tilted_mul_self (ht : t ∈ interior (integrableExpSet X μ)) :
    (μ.tilted (t * X ·))[X] = deriv (cgf X μ) t := by
  simp_rw [integral_tilted_mul_eq_mgf, deriv_cgf ht, ← integral_div, smul_eq_mul]
  congr with ω
  ring