private theorem hG : Primrec G := by
  have a := (Primrec.ofNat (ℕ × Code)).comp (Primrec.list_length (α := List (Option ℕ)))
  have k := Primrec.fst.comp a
  refine' Primrec.option_some.comp (Primrec.list_map (Primrec.list_range.comp k) (_ : Primrec _))
  replace k := k.comp (Primrec.fst (β := ℕ))
  have n := Primrec.snd (α := List (List (Option ℕ))) (β := ℕ)
  refine' Primrec.nat_casesOn k (_root_.Primrec.const Option.none) (_ : Primrec _)
  have k := k.comp (Primrec.fst (β := ℕ))
  have n := n.comp (Primrec.fst (β := ℕ))
  have k' := Primrec.snd (α := List (List (Option ℕ)) × ℕ) (β := ℕ)
  have c := Primrec.snd.comp (a.comp <| (Primrec.fst (β := ℕ)).comp (Primrec.fst (β := ℕ)))
  apply
    Nat.Partrec.Code.rec_prim c
      (_root_.Primrec.const (some 0))
      (Primrec.option_some.comp (_root_.Primrec.succ.comp n))
      (Primrec.option_some.comp (Primrec.fst.comp <| Primrec.unpair.comp n))
      (Primrec.option_some.comp (Primrec.snd.comp <| Primrec.unpair.comp n))
  · have L := (Primrec.fst.comp Primrec.fst).comp
      (Primrec.fst (α := (List (List (Option ℕ)) × ℕ) × ℕ)
        (β := Code × Code × Option ℕ × Option ℕ))
    have k := k.comp (Primrec.fst (β := Code × Code × Option ℕ × Option ℕ))
    have n := n.comp (Primrec.fst (β := Code × Code × Option ℕ × Option ℕ))
    have cf := Primrec.fst.comp (Primrec.snd (α := (List (List (Option ℕ)) × ℕ) × ℕ)
        (β := Code × Code × Option ℕ × Option ℕ))
    have cg := (Primrec.fst.comp Primrec.snd).comp
      (Primrec.snd (α := (List (List (Option ℕ)) × ℕ) × ℕ)
        (β := Code × Code × Option ℕ × Option ℕ))
    refine Primrec.option_bind (hlup.comp <| L.pair <| (k.pair cf).pair n) ?_
    unfold Primrec₂
    conv =>
      congr
      · ext p
        dsimp only []
        erw [Option.bind_eq_bind, ← Option.map_eq_bind]
    refine Primrec.option_map ((hlup.comp <| L.pair <| (k.pair cg).pair n).comp Primrec.fst) ?_
    unfold Primrec₂
    exact Primrec₂.natPair.comp (Primrec.snd.comp Primrec.fst) Primrec.snd
  · have L := (Primrec.fst.comp Primrec.fst).comp
      (Primrec.fst (α := (List (List (Option ℕ)) × ℕ) × ℕ)
        (β := Code × Code × Option ℕ × Option ℕ))
    have k := k.comp (Primrec.fst (β := Code × Code × Option ℕ × Option ℕ))
    have n := n.comp (Primrec.fst (β := Code × Code × Option ℕ × Option ℕ))
    have cf := Primrec.fst.comp (Primrec.snd (α := (List (List (Option ℕ)) × ℕ) × ℕ)
        (β := Code × Code × Option ℕ × Option ℕ))
    have cg := (Primrec.fst.comp Primrec.snd).comp
      (Primrec.snd (α := (List (List (Option ℕ)) × ℕ) × ℕ)
        (β := Code × Code × Option ℕ × Option ℕ))
    refine Primrec.option_bind (hlup.comp <| L.pair <| (k.pair cg).pair n) ?_
    unfold Primrec₂
    have h :=
      hlup.comp ((L.comp Primrec.fst).pair <| ((k.pair cf).comp Primrec.fst).pair Primrec.snd)
    exact h
  · have L := (Primrec.fst.comp Primrec.fst).comp
      (Primrec.fst (α := (List (List (Option ℕ)) × ℕ) × ℕ)
        (β := Code × Code × Option ℕ × Option ℕ))
    have k := k.comp (Primrec.fst (β := Code × Code × Option ℕ × Option ℕ))
    have n := n.comp (Primrec.fst (β := Code × Code × Option ℕ × Option ℕ))
    have cf := Primrec.fst.comp (Primrec.snd (α := (List (List (Option ℕ)) × ℕ) × ℕ)
        (β := Code × Code × Option ℕ × Option ℕ))
    have cg := (Primrec.fst.comp Primrec.snd).comp
      (Primrec.snd (α := (List (List (Option ℕ)) × ℕ) × ℕ)
        (β := Code × Code × Option ℕ × Option ℕ))
    have z := Primrec.fst.comp (Primrec.unpair.comp n)
    

Proof state:
case hpc
a : Primrec fun a => ofNat (ℕ × Code) (List.length a)
k✝¹ : Primrec fun a => (ofNat (ℕ × Code) (List.length a.1)).1
n✝¹ : Primrec Prod.snd
k✝ : Primrec fun a => (ofNat (ℕ × Code) (List.length a.1.1)).1
n✝ : Primrec fun a => a.1.2
k' : Primrec Prod.snd
c : Primrec fun a => (ofNat (ℕ × Code) (List.length a.1.1)).2
L : Primrec fun a => a.1.1.1
k : Primrec fun a => (ofNat (ℕ × Code) (List.length a.1.1.1)).1
n : Primrec fun a => a.1.1.2
cf : Primrec fun a => a.2.1
cg : Primrec fun a => a.2.2.1
z : Primrec fun a => (unpair a.1.1.2).1
⊢ Primrec fun a =>
    let z := (unpair a.1.1.2).1;
    Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (ℕ × Code) (List.length a.1.1.1)).1, a.2.1) z)
      fun y => do
      let i ← Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (ℕ × Code) (List.length a.1.1.1)).2) (Nat.pair z y)
      Nat.Partrec.Code.lup a.1.1.1 ((ofNat (ℕ × Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i))

Next tactic:
refine'
      Primrec.nat_casesOn (Primrec.snd.comp (Primrec.unpair.comp n))
        (hlup.comp <| L.pair <| (k.pair cf).pair z)
        (_ : Primrec _)