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
  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
  case comp m n f g _ _ IHf IHg => exact exists_code.comp IHf IHg
  case rfind n f _ IHf =>
    obtain ⟨cf, hf⟩ := IHf; refine' ⟨rfind cf, fun v => _⟩
    replace hf := fun a => hf (a ::ᵥ v)
    simp only [Part.map_eq_map, Part.map_some, Vector.cons_val, PFun.coe_val,
      show ∀ x, pure x = [x] from fun _ => rfl] at hf ⊢
    refine' Part.ext fun x => _
    simp only [rfind, Part.bind_eq_bind, Part.pure_eq_some, Part.map_eq_map, Part.bind_some,
      exists_prop, cons_eval, comp_eval, fix_eval, tail_eval, succ_eval, zero'_eval,
      List.headI_nil, List.headI_cons, pred_eval, Part.map_some, false_eq_decide_iff,
      Part.mem_bind_iff, List.length, Part.mem_map_iff, Nat.mem_rfind, List.tail_nil,
      List.tail_cons, true_eq_decide_iff, Part.mem_some_iff, Part.map_bind]
    constructor
    · rintro ⟨v', h1, rfl⟩
      suffices ∀ v₁ : List ℕ, v' ∈ PFun.fix
        (fun v => (cf.eval v).bind fun y => Part.some <|
          if y.headI = 0 then Sum.inl (v.headI.succ :: v.tail)
            else Sum.inr (v.headI.succ :: v.tail)) v₁ →
        ∀ n, (v₁ = n :: v.val) → (∀ m < n, ¬f (m ::ᵥ v) = 0) →
          ∃ a : ℕ,
            (f (a ::ᵥ v) = 0 ∧ ∀ {m : ℕ}, m < a → ¬f (m ::ᵥ v) = 0) ∧ [a] = [v'.headI.pred]
        by exact this _ h1 0 rfl (by rintro _ ⟨⟩)
      clear h1
      intro v₀ h1
      refine' PFun.fixInduction h1 fun v₁ h2 IH => _
      clear h1
      rintro n rfl hm
      

Proof state:
case intro.mp.intro.intro
n✝¹ : ℕ
f✝ : Vector ℕ n✝¹ →. ℕ
n✝ : ℕ
f : Vector ℕ (n✝ + 1) → ℕ
a✝ : Nat.Partrec' ↑f
cf : Code
v : Vector ℕ n✝
hf : ∀ (a : ℕ), eval cf (a :: ↑v) = Part.some [f (a ::ᵥ v)]
v' v₀ : List ℕ
n : ℕ
h2 :
  v' ∈
    PFun.fix
      (fun v =>
        Part.bind (eval cf v) fun y =>
          Part.some
            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)
            else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))
      (n :: ↑v)
IH :
  ∀ (a'' : List ℕ),
    (Sum.inr a'' ∈
        Part.bind (eval cf (n :: ↑v)) fun y =>
          Part.some
            (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: ↑v)) :: List.tail (n :: ↑v))
            else Sum.inr (Nat.succ (List.headI (n :: ↑v)) :: List.tail (n :: ↑v)))) →
      ∀ (n : ℕ),
        a'' = n :: ↑v →
          (∀ m < n, ¬f (m ::ᵥ v) = 0) →
            ∃ a, (f (a ::ᵥ v) = 0 ∧ ∀ {m : ℕ}, m < a → ¬f (m ::ᵥ v) = 0) ∧ [a] = [Nat.pred (List.headI v')]
hm : ∀ m < n, ¬f (m ::ᵥ v) = 0
⊢ ∃ a, (f (a ::ᵥ v) = 0 ∧ ∀ {m : ℕ}, m < a → ¬f (m ::ᵥ v) = 0) ∧ [a] = [Nat.pred (List.headI v')]

Next tactic:
have := PFun.mem_fix_iff.1 h2