import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_111.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] [inst_1 : NeZero μ] (a b : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ)
  (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) (t : ℝ)
  (r0 :
    (fun t => ∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) = fun t =>
      (∫ (x : Ω), rexp (t * X x) * X x ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ)
  (r01 :
    (∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) = (∫ (x : Ω), rexp (t * X x) * X x ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ)
  (r0' :
    (∫ (x : Ω), X x ^ 2 ∂μ.tilted fun ω => t * X ω) =
      (∫ (x : Ω), rexp (t * X x) * X x ^ 2 ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ)
  (p :
    (∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) / ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ =
      ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) /
        ((∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ))
  (h_original_goal :
    ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) /
          (∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) ^ 2 -
        (∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω) x ∂μ) ^ 2 / (∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) ^ 2 =
      ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * mgf X μ t -
          (∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω) x ∂μ) /
        mgf X μ t ^ 2) :
  ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) /
        (∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) ^ 2 -
      (∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω) x ∂μ) ^ 2 / (∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) ^ 2 =
    ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * mgf X μ t -
        (∫ (ω : Ω), (fun ω => rexp (t * X ω) * X ω) ω ∂μ) ^ 2) /
      mgf X μ t ^ 2 := sorry

Tactic state before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : NeZero μ
a b : ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
g' : ℝ → ℝ := fun t => ∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω
g'' : ℝ → ℝ :=
  fun t => (∫ (x : Ω), (X ^ 2) x ∂μ.tilted fun ω => t * X ω) - (∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) ^ 2
t : ℝ
r0 :
  (fun t => ∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) = fun t =>
    (∫ (x : Ω), rexp (t * X x) * X x ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ
r01 : (∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) = (∫ (x : Ω), rexp (t * X x) * X x ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ
r0' :
  (∫ (x : Ω), X x ^ 2 ∂μ.tilted fun ω => t * X ω) =
    (∫ (x : Ω), rexp (t * X x) * X x ^ 2 ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ
p :
  (∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) / ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ =
    ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) /
      ((∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ)
⊢ ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) /
        (∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) ^ 2 -
      (∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω) x ∂μ) ^ 2 / (∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) ^ 2 =
    ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * mgf X μ t -
        (∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω) x ∂μ) /
      mgf X μ t ^ 2

Tactic:
rw [Eq.symm (pow_two (∫ ω, (fun ω ↦ rexp (t * X ω) * X ω) ω ∂μ))]

Tactic state after:
[Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝¹ : IsFiniteMeasure μ
 inst✝ : NeZero μ
 a b : ℝ
 X : Ω → ℝ
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 g' : ℝ → ℝ := fun t => ∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω
 g'' : ℝ → ℝ :=
   fun t => (∫ (x : Ω), (X ^ 2) x ∂μ.tilted fun ω => t * X ω) - (∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) ^ 2
 t : ℝ
 r0 :
   (fun t => ∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) = fun t =>
     (∫ (x : Ω), rexp (t * X x) * X x ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ
 r01 : (∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) = (∫ (x : Ω), rexp (t * X x) * X x ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ
 r0' :
   (∫ (x : Ω), X x ^ 2 ∂μ.tilted fun ω => t * X ω) =
     (∫ (x : Ω), rexp (t * X x) * X x ^ 2 ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ
 p :
   (∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) / ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ =
     ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) /
       ((∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ)
 ⊢ ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) /
         (∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) ^ 2 -
       (∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω) x ∂μ) ^ 2 / (∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) ^ 2 =
     ((∫ (x : Ω), (fun ω => rexp (t * X ω) * X ω ^ 2) x ∂μ) * mgf X μ t -
         (∫ (ω : Ω), (fun ω => rexp (t * X ω) * X ω) ω ∂μ) ^ 2) /
       mgf X μ t ^ 2]