import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_8.{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) :
  let g' := fun t => ∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω;
  let g'' := 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),
    ((fun t => ∫ (x : Ω), (fun ω => id (X ω)) x ∂μ.tilted fun ω => t * X ω) = fun t =>
        (∫ (x : Ω), (fun ω => rexp (t * X ω) * id (X ω)) x ∂μ) / ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) →
      (∫ (x : Ω), (fun ω => id (X ω)) x ∂μ.tilted fun ω => t * X ω) =
          (∫ (x : Ω), (fun ω => rexp (t * X ω) * id (X ω)) x ∂μ) / ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ →
        (∫ (x : Ω), (fun ω => (fun s => s ^ 2) (X ω)) x ∂μ.tilted fun ω => t * X ω) =
            (∫ (x : Ω), (fun ω => rexp (t * X ω) * (fun s => s ^ 2) (X ω)) x ∂μ) /
              ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ →
          HasDerivAt g' (g'' t) t →
            HasDerivAt (fun t => (∫ (x : Ω), rexp (t * X x) * X x ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ)
              ((∫ (x : Ω), rexp (t * X x) * X x ^ 2 ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ -
                (∫ (x : Ω), rexp (t * X x) * X x ∂μ) ^ 2 / (∫ (x : Ω), rexp (t * X x) ∂μ) ^ 2)
              t := 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), (fun ω => id (X ω)) x ∂μ.tilted fun ω => t * X ω) = fun t =>
    (∫ (x : Ω), (fun ω => rexp (t * X ω) * id (X ω)) x ∂μ) / ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ
r01 :
  (∫ (x : Ω), (fun ω => id (X ω)) x ∂μ.tilted fun ω => t * X ω) =
    (∫ (x : Ω), (fun ω => rexp (t * X ω) * id (X ω)) x ∂μ) / ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ
r0' :
  (∫ (x : Ω), (fun ω => (fun s => s ^ 2) (X ω)) x ∂μ.tilted fun ω => t * X ω) =
    (∫ (x : Ω), (fun ω => rexp (t * X ω) * (fun s => s ^ 2) (X ω)) x ∂μ) / ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ
⊢ HasDerivAt g' (g'' t) t

Tactic:
simp only [id_eq] at r0 r0' r01
    dsimp [g', g'']
    rw [r0, r0', r01]
    field_simp
    have p :
  ((μ[fun ω ↦ rexp (t * X ω) * X ω ^ 2]) / μ[fun ω ↦ rexp (t * X ω)]) =
    ((μ[fun ω ↦ rexp (t * X ω) * X ω ^ 2]) * (μ[fun ω ↦ rexp (t * X ω)])) /
      ((μ[fun ω ↦ rexp (t * X ω)]) * (μ[fun ω ↦ rexp (t * X ω)])) :=
  by
  apply Eq.symm (mul_div_mul_right (∫ ω, rexp (t * X ω) * X ω ^ 2 ∂μ) (μ[fun ω ↦ rexp (t * X ω)]) _)
  exact (mgf_pos' (NeZero.ne μ) (integrable_expt_bound hX h)).ne'

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)
 ⊢ HasDerivAt (fun t => (∫ (x : Ω), rexp (t * X x) * X x ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ)
     ((∫ (x : Ω), rexp (t * X x) * X x ^ 2 ∂μ) / ∫ (x : Ω), rexp (t * X x) ∂μ -
       (∫ (x : Ω), rexp (t * X x) * X x ∂μ) ^ 2 / (∫ (x : Ω), rexp (t * X x) ∂μ) ^ 2)
     t]