Definition ap (n : nat) := N_of_nat n.Definition ap' (n : nat) := N_of_nat (N + n).Fixpoint lx_1 (n : nat) : list ad :=
  match n with
  | O => nil
  | S m => ap m :: lx_1 m
  end.Fixpoint lx'_1 (n : nat) : list ad :=
  match n with
  | O => nil (A:=ad)
  | S m => ap' m :: lx'_1 m
  end.Definition lx := lx_1 N.Definition lx' := lx'_1 N.

Lemma lt_O_n_lx'_1 : forall n : nat, 0 < n -> In (ap' 0) (lx'_1 n).