induction 1; intuition; simpl in *.
destruct H.
subst.
eapply NoDup_app in H1.
intuition.
eapply app_NoDup; intuition.
eapply firstn_NoDup.
trivial.
eapply H3.
eapply firstn_In.
eauto.
eapply pred_firstn_In.
eauto.
eauto.
intuition.
eapply H3.
eapply firstn_In.
eauto.
eapply pred_firstn_In.
eauto.
eauto.
intuition.
Qed.

(* Specifications: *)
Theorem allNatsLt_nil_inv :
forall n,
allNatsLt n = nil ->
n = O.