induction n; intuition.
simpl in *.
eapply le_trans.
2:{
eapply plus_le_compat.
eapply IHn.
eapply mult_le_compat.
eapply le_refl.
eapply IHn.
}
rewrite mult_plus_distr_l.
repeat rewrite mult_assoc.
repeat rewrite plus_assoc.
eapply plus_le_compat.
rewrite plus_comm.
eapply plus_le_compat.
rewrite <- (plus_0_r (expnat k n)) at 1.
eapply plus_le_compat.
omega.
intuition.
intuition.
rewrite (mult_comm k n).
rewrite <- (mult_assoc n).
destruct n; simpl; intuition.
Qed.

(* Specifications: *)
Lemma expnat_base_S_same : forall n,
n > 0 ->
(2 * (expnat n n) <= expnat (S n) n)%nat.