/-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
    (hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
    (hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
    (hpr : Primrec fun a : α × Code × Code × σ × σ => pr a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2)
    {co : α → Code → Code → σ → σ → σ}
    (hco : Primrec fun a : α × Code × Code × σ × σ => co a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2)
    {pc : α → Code → Code → σ → σ → σ}
    (hpc : Primrec fun a : α × Code × Code × σ × σ => pc a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2)
    {rf : α → Code → σ → σ} (hrf : Primrec fun a : α × Code × σ => rf a.1 a.2.1 a.2.2) :
    let F (a : α) (c : Code) : σ :=
      Nat.Partrec.Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)
    Primrec fun a => F a (c a) := by
  intros F
  let G₁ : (α × List σ) × ℕ × ℕ → Option σ := fun p =>
    let a := p.1.1
    let IH := p.1.2
    let n := p.2.1
    let m := p.2.2
    (IH.get? m).bind fun s =>
      (IH.get? m.unpair.1).bind fun s₁ =>
        (IH.get? m.unpair.2).map fun s₂ =>
          cond n.bodd
            (cond n.div2.bodd (rf a (ofNat Code m) s)
              (pc a (ofNat Code m.unpair.1) (ofNat Code m.unpair.2) s₁ s₂))
            (cond n.div2.bodd (co a (ofNat Code m.unpair.1) (ofNat Code m.unpair.2) s₁ s₂)
              (pr a (ofNat Code m.unpair.1) (ofNat Code m.unpair.2) s₁ s₂))
  have : Primrec G₁ := by
    refine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _
    unfold Primrec₂
    refine' option_bind ((list_get?.comp (snd.comp fst) (fst.comp <| Primrec.unpair.comp
            (snd.comp snd))).comp fst) _
    unfold Primrec₂
    refine'
      option_map
        ((list_get?.comp (snd.comp fst) (snd.comp <| Primrec.unpair.comp (snd.comp snd))).comp <|
          fst.comp fst)
        _
    have a : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.1.1.1) :=
      fst.comp (fst.comp <| fst.comp <| fst.comp fst)
    have n : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.1.2.1) :=
      fst.comp (snd.comp <| fst.comp <| fst.comp fst)
    have m : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.1.2.2) :=
      snd.comp (snd.comp <| fst.comp <| fst.comp fst)
    have m₁ := fst.comp (Primrec.unpair.comp m)
    have m₂ := snd.comp (Primrec.unpair.comp m)
    have s : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.2) :=
      snd.comp (fst.comp fst)
    have s₁ : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.2) :=
      snd.comp fst
    have s₂ : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.2) :=
      snd
    have h₁ := hrf.comp <| a.pair (((Primrec.ofNat Code).comp m).pair s)
    have h₂ := hpc.comp <| a.pair (((Primrec.ofNat Code).comp m₁).pair <|
      ((Primrec.ofNat Code).comp m₂).pair <| s₁.pair s₂)
    have h₃ := hco.comp <| a.pair
      (((Primrec.ofNat Code).comp m₁).pair <| ((Primrec.ofNat Code).comp m₂).pair <| s₁.pair s₂)
    have h₄ := hpr.comp <| a.pair
      (((Primrec.ofNat Code).comp m₁).pair <| ((Primrec.ofNat Code).comp m₂).pair <| s₁.pair s₂)
    unfold Primrec₂
    exact
      (nat_bodd.comp n).cond
        ((nat_bodd.comp <| nat_div2.comp n).cond h₁ h₂)
        (cond (nat_bodd.comp <| nat_div2.comp n) h₃ h₄)
  let G : α → List σ → Option σ := fun a IH =>
    IH.length.casesOn (some (z a)) fun n =>
      n.casesOn (some (s a)) fun n =>
        n.casesOn (some (l a)) fun n =>
          n.casesOn (some (r a)) fun n => G₁ ((a, IH), n, n.div2.div2)
  have : Primrec₂ G := by
    unfold Primrec₂
    refine nat_casesOn (list_length.comp snd) (option_some_iff.2 (hz.comp fst)) ?_
    unfold Primrec₂
    refine nat_casesOn snd (option_some_iff.2 (hs.comp (fst.comp fst))) ?_
    unfold Primrec₂
    refine nat_casesOn snd (option_some_iff.2 (hl.comp (fst.comp <| fst.comp fst))) ?_
    unfold Primrec₂
    refine nat_casesOn snd (option_some_iff.2 (hr.comp (fst.comp <| fst.comp <| fst.comp fst))) ?_
    unfold Primrec₂
    exact this.comp <|
      ((fst.pair snd).comp <| fst.comp <| fst.comp <| fst.comp <| fst).pair <|
          snd.pair <| nat_div2.comp <| nat_div2.comp snd
  refine'
    ((nat_strong_rec (fun a n => F a (ofNat Code n)) this.to₂ fun a n => _).comp
      _root_.Primrec.id <| encode_iff.2 hc).of_eq
      fun a => by simp
  simp (config := { zeta := false })
  iterate 4 cases' n with n; · simp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]; rfl
  simp only []
  rw [List.length_map, List.length_range]
  let m := n.div2.div2
  show
    G₁ ((a, (List.range (n + 4)).map fun n => F a (ofNat Code n)), n, m) =
      some (F a (ofNat Code (n + 4)))
  have hm : m < n + 4 := by
    simp only [div2_val]
    

Proof state:
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2
co : α → Code → Code → σ → σ → σ
hco : Primrec fun a => co a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2
pc : α → Code → Code → σ → σ → σ
hpc : Primrec fun a => pc a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2
rf : α → Code → σ → σ
hrf : Primrec fun a => rf a.1 a.2.1 a.2.2
F : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)
G₁ : (α × List σ) × ℕ × ℕ → Option σ :=
  fun p =>
    let a := p.1.1;
    let IH := p.1.2;
    let n := p.2.1;
    let m := p.2.2;
    Option.bind (List.get? IH m) fun s =>
      Option.bind (List.get? IH (unpair m).1) fun s₁ =>
        Option.map
          (fun s₂ =>
            bif bodd n then
              bif bodd (div2 n) then rf a (ofNat Code m) s
              else pc a (ofNat Code (unpair m).1) (ofNat Code (unpair m).2) s₁ s₂
            else
              bif bodd (div2 n) then co a (ofNat Code (unpair m).1) (ofNat Code (unpair m).2) s₁ s₂
              else pr a (ofNat Code (unpair m).1) (ofNat Code (unpair m).2) s₁ s₂)
          (List.get? IH (unpair m).2)
this✝ : Primrec G₁
G : α → List σ → Option σ :=
  fun a IH =>
    Nat.casesOn (List.length IH) (some (z a)) fun n =>
      Nat.casesOn n (some (s a)) fun n =>
        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))
this : Primrec₂ G
a : α
n : ℕ
m : ℕ := div2 (div2 n)
⊢ n / 2 / 2 < n + 4

Next tactic:
exact
      lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _))
        (Nat.succ_le_succ (Nat.le_add_right _ _))