induction n; intuition; simpl in *.
omega.
eapply in_or_app.
destruct (lt_dec i n).
left.
eauto.
right.
simpl.
intuition.
Qed.

(* Specifications: *)
Lemma nth_allNatsLt_lt :
forall k n,
n < k ->
nth n (allNatsLt k) n = n.