intuition.
unfold polynomial, polynomial_nz in *.
do 3 (destruct H).
exists (S x).
exists (S x0).
exists (x0 + S x1).
intuition.
rewrite H.
rewrite plus_assoc.
eapply plus_le_compat; intuition.
simpl.
destruct (eq_nat_dec x 0).
subst.
simpl.
repeat rewrite mult_1_r.
rewrite <- plus_0_l at 1.
eapply plus_le_compat; intuition.
rewrite <- plus_0_l at 1.
rewrite <- plus_0_r at 1.
eapply plus_le_compat; intuition.
eapply plus_le_compat; intuition.
eapply mult_le_compat; intuition.
destruct (eq_nat_dec n 0); subst.
rewrite expnat_0;
omega.
rewrite <- mult_1_l at 1.
eapply mult_le_compat; intuition.
Qed.

Theorem polynomial_plus :
forall f1 f2 ,
polynomial f1 ->
polynomial f2 ->
polynomial (fun n => f1 n + f2 n).