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])⟩)
      

Proof state:
case intro.intro.succ.succ
n✝² : ℕ
f✝¹ : Vector ℕ n✝² →. ℕ
n✝¹ : ℕ
f✝ : Vector ℕ n✝¹ → ℕ
n✝ : ℕ
f : Vector ℕ n✝ → ℕ
g : Vector ℕ (n✝ + 2) → ℕ
a✝¹ : Nat.Primrec' f
a✝ : Nat.Primrec' g
cf cg : Code
v : Vector ℕ (n✝ + 1)
hf : eval cf (List.tail ↑v) = pure (pure (f (Vector.tail v)))
hg : ∀ (a b : ℕ), eval cg (a :: b :: List.tail ↑v) = pure (pure (g (a ::ᵥ b ::ᵥ Vector.tail v)))
n : ℕ
n_ih✝ :
  eval (prec cf cg) (n :: List.tail ↑v) =
    pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::ᵥ IH ::ᵥ Vector.tail v)) n))
b : ℕ
IH :
  ∀ (a : ℕ),
    a + b = n →
      Nat.succ n ::
          0 ::
            g (n ::ᵥ Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::ᵥ IH ::ᵥ Vector.tail v)) n ::ᵥ Vector.tail v) ::
              List.tail ↑v ∈
        PFun.fix
          (fun v =>
            Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>
              Part.some
                (if List.headI (List.tail v) = 0 then
                  Sum.inl
                    (Nat.succ (List.headI v) ::
                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))
                else
                  Sum.inr
                    (Nat.succ (List.headI v) ::
                      Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))
          (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::ᵥ IH ::ᵥ Vector.tail v)) a :: List.tail ↑v)
a : ℕ
e : a + Nat.succ b = n
⊢ Sum.inr
      ((a + 1) ::
        b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::ᵥ IH ::ᵥ Vector.tail v)) (a + 1) :: List.tail ↑v) ∈
    Part.bind
      (eval cg
        (List.headI
            (a ::
              Nat.succ b ::
                Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::ᵥ IH ::ᵥ Vector.tail v)) a :: List.tail ↑v) ::
          List.tail
            (List.tail
              (a ::
                Nat.succ b ::
                  Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::ᵥ IH ::ᵥ Vector.tail v)) a :: List.tail ↑v))))
      fun x =>
      Part.some
        (if
            List.headI
                (List.tail
                  (a ::
                    Nat.succ b ::
                      Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::ᵥ IH ::ᵥ Vector.tail v)) a :: List.tail ↑v)) =
              0 then
          Sum.inl
            (Nat.succ
                (List.headI
                  (a ::
                    Nat.succ b ::
                      Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::ᵥ IH ::ᵥ Vector.tail v)) a :: List.tail ↑v)) ::
              Nat.pred
                  (List.headI
                    (List.tail
                      (a ::
                        Nat.succ b ::
                          Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::ᵥ IH ::ᵥ Vector.tail v)) a ::
                            List.tail ↑v))) ::
                List.headI x ::
                  List.tail
                    (List.tail
                      (List.tail
                        (a ::
                          Nat.succ b ::
                            Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::ᵥ IH ::ᵥ Vector.tail v)) a ::
                              List.tail ↑v))))
        else
          Sum.inr
            (Nat.succ
                (List.headI
                  (a ::
                    Nat.succ b ::
                      Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::ᵥ IH ::ᵥ Vector.tail v)) a :: List.tail ↑v)) ::
              Nat.pred
                  (List.headI
                    (List.tail
                      (a ::
                        Nat.succ b ::
                          Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::ᵥ IH ::ᵥ Vector.tail v)) a ::
                            List.tail ↑v))) ::
                List.headI x ::
                  List.tail
                    (List.tail
                      (List.tail
                        (a ::
                          Nat.succ b ::
                            Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::ᵥ IH ::ᵥ Vector.tail v)) a ::
                              List.tail ↑v)))))

Next tactic:
simp only [hg, eval, Part.bind_some, Nat.rec_add_one, List.tail_nil, List.tail_cons, pure]