Section Nsec.
Variable N : nat.
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 ap_neq_ap' : forall n : nat, n < N -> Neqb (ap n) (ap' n) = false.