/-- The `Nat.Partrec.Code.evaln` function is primitive recursive. -/
theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
  have :
    Primrec₂ fun (_ : Unit) (n : ℕ) =>
      let a := ofNat (ℕ × Code) n
      (List.range a.1).map (evaln a.1 a.2) :=
    Primrec.nat_strong_rec _ (hG.comp Primrec.snd).to₂ fun _ p => by
      simp only [G, prod_ofNat_val, ofNat_nat, List.length_map, List.length_range,
        Nat.pair_unpair, Option.some_inj]
      

Proof state:
x✝ : Unit
p : ℕ
⊢ List.map
      (fun n =>
        Nat.rec Option.none
          (fun n_1 n_ih =>
            rec (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2)
              (fun cf cg x x => do
                let x ←
                  Nat.Partrec.Code.lup
                      (List.map
                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                        (List.range p))
                      ((unpair p).1, cf) n
                let y ←
                  Nat.Partrec.Code.lup
                      (List.map
                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                        (List.range p))
                      ((unpair p).1, cg) n
                some (Nat.pair x y))
              (fun cf cg x x => do
                let x ←
                  Nat.Partrec.Code.lup
                      (List.map
                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                        (List.range p))
                      ((unpair p).1, cg) n
                Nat.Partrec.Code.lup
                    (List.map
                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                      (List.range p))
                    ((unpair p).1, cf) x)
              (fun cf cg x x =>
                Nat.rec
                  (Nat.Partrec.Code.lup
                    (List.map
                      (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                      (List.range p))
                    ((unpair p).1, cf) (unpair n).1)
                  (fun n_2 n_ih => do
                    let i ←
                      Nat.Partrec.Code.lup
                          (List.map
                            (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                            (List.range p))
                          (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)
                    Nat.Partrec.Code.lup
                        (List.map
                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                          (List.range p))
                        ((unpair p).1, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))
                  (unpair n).2)
              (fun cf x => do
                let x ←
                  Nat.Partrec.Code.lup
                      (List.map
                        (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                        (List.range p))
                      ((unpair p).1, cf) n
                Nat.rec (some (unpair n).2)
                    (fun n_2 n_ih =>
                      Nat.Partrec.Code.lup
                        (List.map
                          (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
                          (List.range p))
                        (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
                    x)
              (ofNat Code (unpair p).2))
          (unpair p).1)
      (List.range (unpair p).1) =
    List.map (evaln (unpair p).1 (ofNat Code (unpair p).2)) (List.range (unpair p).1)

Next tactic:
refine List.map_congr fun n => ?_