theorem exists_code {n} {f : Vector ℕ n →. ℕ} (hf : Nat.Partrec' f) :
    ∃ c : Code, ∀ v : Vector ℕ n, c.eval v.1 = pure <$> f v := by
  induction' hf with n f hf
  induction hf
  case prim.zero => exact ⟨zero', fun ⟨[], _⟩ => rfl⟩
  case prim.succ => exact ⟨succ, fun ⟨[v], _⟩ => rfl⟩
  case prim.get n i =>
    refine' Fin.succRec (fun n => _) (fun n i IH => _) i
    · exact ⟨head, fun ⟨List.cons a as, _⟩ => by simp [Bind.bind]; rfl⟩
    · obtain ⟨c, h⟩ := IH
      exact ⟨c.comp tail, fun v => by simpa [← Vector.get_tail, Bind.bind] using h v.tail⟩
  case prim.comp m n f g hf hg IHf IHg =>
    simpa [Part.bind_eq_bind] using exists_code.comp IHf IHg
  

Proof state:
case prim.prec
n✝¹ : ℕ
f✝¹ : Vector ℕ n✝¹ →. ℕ
n : ℕ
f : Vector ℕ n → ℕ
n✝ : ℕ
f✝ : Vector ℕ n✝ → ℕ
g✝ : Vector ℕ (n✝ + 2) → ℕ
a✝¹ : Nat.Primrec' f✝
a✝ : Nat.Primrec' g✝
a_ih✝¹ : ∃ c, ∀ (v : Vector ℕ n✝), eval c ↑v = pure <$> ↑f✝ v
a_ih✝ : ∃ c, ∀ (v : Vector ℕ (n✝ + 2)), eval c ↑v = pure <$> ↑g✝ v
⊢ ∃ c,
    ∀ (v : Vector ℕ (n✝ + 1)),
      eval c ↑v =
        pure <$>
          (↑fun v => Nat.rec (f✝ (Vector.tail v)) (fun y IH => g✝ (y ::ᵥ IH ::ᵥ Vector.tail v)) (Vector.head v)) v
case comp
n : ℕ
f : Vector ℕ n →. ℕ
m✝ n✝ : ℕ
f✝ : Vector ℕ n✝ →. ℕ
g✝ : Fin n✝ → Vector ℕ m✝ →. ℕ
a✝¹ : Nat.Partrec' f✝
a✝ : ∀ (i : Fin n✝), Nat.Partrec' (g✝ i)
a_ih✝¹ : ∃ c, ∀ (v : Vector ℕ n✝), eval c ↑v = pure <$> f✝ v
a_ih✝ : ∀ (i : Fin n✝), ∃ c, ∀ (v : Vector ℕ m✝), eval c ↑v = pure <$> g✝ i v
⊢ ∃ c, ∀ (v : Vector ℕ m✝), eval c ↑v = pure <$> (fun v => (Vector.mOfFn fun i => g✝ i v) >>= f✝) v
case rfind
n : ℕ
f : Vector ℕ n →. ℕ
n✝ : ℕ
f✝ : Vector ℕ (n✝ + 1) → ℕ
a✝ : Nat.Partrec' ↑f✝
a_ih✝ : ∃ c, ∀ (v : Vector ℕ (n✝ + 1)), eval c ↑v = pure <$> ↑f✝ v
⊢ ∃ c, ∀ (v : Vector ℕ n✝), eval c ↑v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f✝ (n ::ᵥ v) = 0))) v

Next tactic:
case prim.prec n f g _ _ IHf IHg =>
    obtain ⟨cf, hf⟩ := IHf
    obtain ⟨cg, hg⟩ := IHg
    simp only [Part.map_eq_map, Part.map_some, PFun.coe_val] at hf hg
    refine' ⟨prec cf cg, fun v => _⟩
    rw [← v.cons_head_tail]
    specialize hf v.tail
    replace hg := fun a b => hg (a ::ᵥ b ::ᵥ v.tail)
    simp only [Vector.cons_val, Vector.tail_val] at hf hg
    simp only [Part.map_eq_map, Part.map_some, Vector.cons_val, Vector.tail_cons, Vector.head_cons,
      PFun.coe_val, Vector.tail_val]
    simp only [← Part.pure_eq_some] at hf hg ⊢
    induction' v.head with n _ <;>
      simp [prec, hf, Part.bind_assoc, ← Part.bind_some_eq_map, Part.bind_some,
        show ∀ x, pure x = [x] from fun _ => rfl, Bind.bind, Functor.map]
    suffices ∀ a b, a + b = n →
      (n.succ :: 0 ::
        g (n ::ᵥ Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) n ::ᵥ v.tail) ::
            v.val.tail : List ℕ) ∈
        PFun.fix
          (fun v : List ℕ => Part.bind (cg.eval (v.headI :: v.tail.tail))
            (fun x => Part.some (if v.tail.headI = 0
              then Sum.inl
                (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail : List ℕ)
              else Sum.inr
                (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail))))
          (a :: b :: Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) a :: v.val.tail) by
      erw [Part.eq_some_iff.2 (this 0 n (zero_add n))]
      simp only [List.headI, Part.bind_some, List.tail_cons]
    intro a b e
    induction' b with b IH generalizing a
    · refine' PFun.mem_fix_iff.2 (Or.inl <| Part.eq_some_iff.1 _)
      simp only [hg, ← e, Part.bind_some, List.tail_cons, pure]
      rfl
    · refine' PFun.mem_fix_iff.2 (Or.inr ⟨_, _, IH (a + 1) (by rwa [add_right_comm])⟩)
      simp only [hg, eval, Part.bind_some, Nat.rec_add_one, List.tail_nil, List.tail_cons, pure]
      exact Part.mem_some_iff.2 rfl