/-- 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]
      refine List.map_congr fun n => ?_
      have : List.range p = List.range (Nat.pair p.unpair.1 (encode (ofNat Code p.unpair.2))) := by
        simp
      rw [this]
      generalize p.unpair.1 = k
      generalize ofNat Code p.unpair.2 = c
      intro nk
      cases' k with k'
      · simp [evaln]
      let k := k' + 1
      

Proof state:
case succ
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
c : Code
k' : ℕ
nk : n ∈ List.range (Nat.succ k')
k : ℕ := k' + 1
⊢ 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 (Nat.pair (Nat.succ k') (encode c))))
                  (Nat.succ k', 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 (Nat.pair (Nat.succ k') (encode c))))
                  (Nat.succ k', 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 (Nat.pair (Nat.succ k') (encode c))))
                  (Nat.succ k', 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 (Nat.pair (Nat.succ k') (encode c))))
                (Nat.succ k', 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 (Nat.pair (Nat.succ k') (encode c))))
                (Nat.succ k', 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 (Nat.pair (Nat.succ k') (encode c))))
                      (n_1, c) (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 (Nat.pair (Nat.succ k') (encode c))))
                    (Nat.succ k', 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 (Nat.pair (Nat.succ k') (encode c))))
                  (Nat.succ k', 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 (Nat.pair (Nat.succ k') (encode c))))
                    (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
                x)
          c)
      (Nat.succ k') =
    evaln (Nat.succ k') c n

Next tactic:
simp only [show k'.succ = k from rfl]