-- Mathlib/Probability/Moments/MGFAnalytic.lean
lemma iteratedDeriv_two_cgf_eq_integral (h : v ∈ interior (integrableExpSet X μ)) :
    iteratedDeriv 2 (cgf X μ) v
      = μ[fun ω ↦ (X ω - deriv (cgf X μ) v)^2 * exp (v * X ω)] / mgf X μ v

Proof state:
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
⊢ iteratedDeriv 2 (cgf X μ) v = (∫ (x : Ω), (fun ω => (X ω - deriv (cgf X μ) v) ^ 2 * rexp (v * X ω)) x ∂μ) / mgf X μ v

Tactic:
by_cases hμ : μ = 0

Result:
case pos
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : μ = 0
⊢ iteratedDeriv 2 (cgf X μ) v = (∫ (x : Ω), (fun ω => (X ω - deriv (cgf X μ) v) ^ 2 * rexp (v * X ω)) x ∂μ) / mgf X μ v

case neg
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
⊢ iteratedDeriv 2 (cgf X μ) v = (∫ (x : Ω), (fun ω => (X ω - deriv (cgf X μ) v) ^ 2 * rexp (v * X ω)) x ∂μ) / mgf X μ v

Proof state:
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
⊢ (∫ (ω : Ω), X ω ^ 2 * rexp (v * X ω) ∂μ) / mgf X μ v - deriv (cgf X μ) v ^ 2 =
    (∫ (ω : Ω), X ω ^ 2 * rexp (v * X ω) ∂μ - (2 * ∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) * deriv (cgf X μ) v +
        deriv (cgf X μ) v ^ 2 * mgf X μ v) /
      mgf X μ v

Tactic:
rw [add_div, sub_div, sub_add]

Result:
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
⊢ (∫ (ω : Ω), X ω ^ 2 * rexp (v * X ω) ∂μ) / mgf X μ v - deriv (cgf X μ) v ^ 2 =
    (∫ (ω : Ω), X ω ^ 2 * rexp (v * X ω) ∂μ) / mgf X μ v -
      ((2 * ∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) * deriv (cgf X μ) v / mgf X μ v -
        deriv (cgf X μ) v ^ 2 * mgf X μ v / mgf X μ v)

Proof state:
case pos
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : μ = 0
⊢ iteratedDeriv 2 (cgf X μ) v = (∫ (x : Ω), (fun ω => (X ω - deriv (cgf X μ) v) ^ 2 * rexp (v * X ω)) x ∂μ) / mgf X μ v

Tactic:
have : deriv (0 : ℝ → ℝ) = 0 := by ext; exact deriv_const _ 0

Result:
case pos
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : μ = 0
this : deriv 0 = 0
⊢ iteratedDeriv 2 (cgf X μ) v = (∫ (x : Ω), (fun ω => (X ω - deriv (cgf X μ) v) ^ 2 * rexp (v * X ω)) x ∂μ) / mgf X μ v

Proof state:
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
⊢ (∫ (ω : Ω), X ω ^ 2 * rexp (v * X ω) ∂μ) / mgf X μ v - deriv (cgf X μ) v ^ 2 =
    (∫ (ω : Ω), X ω ^ 2 * rexp (v * X ω) ∂μ) / mgf X μ v -
      ((2 * ∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) * deriv (cgf X μ) v / mgf X μ v -
        deriv (cgf X μ) v ^ 2 * mgf X μ v / mgf X μ v)

Tactic:
congr 1

Result:
case e_a
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
⊢ deriv (cgf X μ) v ^ 2 =
    (2 * ∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) * deriv (cgf X μ) v / mgf X μ v -
      deriv (cgf X μ) v ^ 2 * mgf X μ v / mgf X μ v

Proof state:
case e_a
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
⊢ ((∫ (x : Ω), (fun ω => X ω * rexp (v * X ω)) x ∂μ) / mgf X μ v) ^ 2 =
    (2 * ∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) * ((∫ (x : Ω), (fun ω => X ω * rexp (v * X ω)) x ∂μ) / mgf X μ v) /
        mgf X μ v -
      ((∫ (x : Ω), (fun ω => X ω * rexp (v * X ω)) x ∂μ) / mgf X μ v) ^ 2

Tactic:
ring

Result:
no goals

Proof state:
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
⊢ Integrable (fun ω => 2 * X ω * deriv (cgf X μ) v * rexp (v * X ω)) μ

Tactic:
simp_rw [mul_assoc, mul_comm (deriv (cgf X μ) v)]

Result:
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
⊢ Integrable (fun ω => 2 * (X ω * (rexp (v * X ω) * deriv (cgf X μ) v))) μ

Proof state:
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
⊢ Integrable (fun ω => X ω * rexp (v * X ω) * deriv (cgf X μ) v) μ

Tactic:
refine Integrable.mul_const ?_ _

Result:
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
⊢ Integrable (fun ω => X ω * rexp (v * X ω)) μ

Proof state:
case pos
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : μ = 0
this : deriv 0 = 0
⊢ iteratedDeriv 2 (cgf X μ) v = (∫ (x : Ω), (fun ω => (X ω - deriv (cgf X μ) v) ^ 2 * rexp (v * X ω)) x ∂μ) / mgf X μ v

Tactic:
simp [hμ, this, iteratedDeriv_succ]

Result:
no goals

Proof state:
case e_a
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
h_int : Integrable (fun ω => 2 * X ω * deriv (cgf X μ) v * rexp (v * X ω)) μ
⊢ ∫ (ω : Ω), X ω ^ 2 * rexp (v * X ω) ∂μ - (2 * ∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) * deriv (cgf X μ) v +
      deriv (cgf X μ) v ^ 2 * mgf X μ v =
    ∫ (a : Ω), X a ^ 2 * rexp (v * X a) ∂μ - ∫ (a : Ω), 2 * X a * deriv (cgf X μ) v * rexp (v * X a) ∂μ +
      ∫ (a : Ω), deriv (cgf X μ) v ^ 2 * rexp (v * X a) ∂μ

Tactic:
congr

Result:
case e_a.e_a.e_a
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
h_int : Integrable (fun ω => 2 * X ω * deriv (cgf X μ) v * rexp (v * X ω)) μ
⊢ (2 * ∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) * deriv (cgf X μ) v =
    ∫ (a : Ω), 2 * X a * deriv (cgf X μ) v * rexp (v * X a) ∂μ

case e_a.e_a
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
h_int : Integrable (fun ω => 2 * X ω * deriv (cgf X μ) v * rexp (v * X ω)) μ
⊢ deriv (cgf X μ) v ^ 2 * mgf X μ v = ∫ (a : Ω), deriv (cgf X μ) v ^ 2 * rexp (v * X a) ∂μ

Proof state:
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : μ = 0
⊢ deriv 0 = 0

Tactic:
ext

Result:
case h
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : μ = 0
x✝ : ℝ
⊢ deriv 0 x✝ = 0 x✝