/-- The hard part of the Lusin separation theorem saying that two disjoint analytic sets are
contained in disjoint Borel sets (see the full statement in `AnalyticSet.measurablySeparable`).
Here, we prove this when our analytic sets are the ranges of functions from `ℕ → ℕ`.
-/
theorem measurablySeparable_range_of_disjoint [T2Space α] [MeasurableSpace α]
    [OpensMeasurableSpace α] {f g : (ℕ → ℕ) → α} (hf : Continuous f) (hg : Continuous g)
    (h : Disjoint (range f) (range g)) : MeasurablySeparable (range f) (range g) := by
  /- We follow [Kechris, *Classical Descriptive Set Theory* (Theorem 14.7)][kechris1995].
    If the ranges are not Borel-separated, then one can find two cylinders of length one whose
    images are not Borel-separated, and then two smaller cylinders of length two whose images are
    not Borel-separated, and so on. One thus gets two sequences of cylinders, that decrease to two
    points `x` and `y`. Their images are different by the disjointness assumption, hence contained
    in two disjoint open sets by the T2 property. By continuity, long enough cylinders around `x`
    and `y` have images which are separated by these two disjoint open sets, a contradiction.
    -/
  by_contra hfg
  have I : ∀ n x y, ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y', x' ∈ cylinder x n ∧ y' ∈ cylinder y n ∧
      ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) := by
    intro n x y
    contrapose!
    intro H
    rw [← iUnion_cylinder_update x n, ← iUnion_cylinder_update y n, image_iUnion, image_iUnion]
    refine' MeasurablySeparable.iUnion fun i j => _
    exact H _ _ (update_mem_cylinder _ _ _) (update_mem_cylinder _ _ _)
  -- consider the set of pairs of cylinders of some length whose images are not Borel-separated
  let A :=
    { p : ℕ × (ℕ → ℕ) × (ℕ → ℕ) //
      ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
  -- for each such pair, one can find longer cylinders whose images are not Borel-separated either
  have : ∀ p : A, ∃ q : A,
      q.1.1 = p.1.1 + 1 ∧ q.1.2.1 ∈ cylinder p.1.2.1 p.1.1 ∧ q.1.2.2 ∈ cylinder p.1.2.2 p.1.1 := by
    rintro ⟨⟨n, x, y⟩, hp⟩
    rcases I n x y hp with ⟨x', y', hx', hy', h'⟩
    exact ⟨⟨⟨n + 1, x', y'⟩, h'⟩, rfl, hx', hy'⟩
  choose F hFn hFx hFy using this
  let p0 : A := ⟨⟨0, fun _ => 0, fun _ => 0⟩, by simp [hfg]⟩
  -- construct inductively decreasing sequences of cylinders whose images are not separated
  let p : ℕ → A := fun n => F^[n] p0
  have prec : ∀ n, p (n + 1) = F (p n) := fun n => by simp only [iterate_succ', Function.comp]
  -- check that at the `n`-th step we deal with cylinders of length `n`
  have pn_fst : ∀ n, (p n).1.1 = n := by
    intro n
    induction' n with n IH
    · rfl
    · 

Proof state:
case succ
α : Type u_1
ι : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous f
hg : Continuous g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A :=
  { val := (0, fun x => 0, fun x => 0),
    property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }
p : ℕ → A := fun n => F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
n : ℕ
IH : (↑(p n)).1 = n
⊢ (↑(p (Nat.succ n))).1 = Nat.succ n

Next tactic:
simp only [prec, hFn, IH]