induction k; intuition; simpl in *.
destruct (lt_dec n k).
rewrite app_nth1.
eauto.
rewrite allNatsLt_length.
trivial.
rewrite app_nth2.
rewrite allNatsLt_length.
assert (k = n).
omega.
rewrite H0.
rewrite minus_diag.
simpl.
trivial.
rewrite allNatsLt_length.
omega.
Qed.

(* Specifications: *)
Lemma nth_allNatsLt :
forall k n,
nth n (allNatsLt k) n = n.