intuition.
apply polynomial_nz_equiv in H.
apply polynomial_nz_equiv in H0.
unfold polynomial, polynomial_nz in *.
Ltac des :=
match goal with
| [H : exists _, _ |- _] => destruct H
end.
repeat des.
intuition.
exists (max x2 x).
exists (x3 + x0).
exists (x4 + x1).
intuition.
eapply le_trans.
eapply plus_le_compat.
eapply H6; trivial.
eapply H7; trivial.
rewrite mult_plus_distr_r.
repeat rewrite plus_assoc.
eapply plus_le_compat; trivial.
rewrite plus_comm.
rewrite plus_assoc.
eapply plus_le_compat; trivial.
rewrite plus_comm.
eapply plus_le_compat;
eapply mult_le_compat; intuition.
eapply expnat_exp_le; intuition.
eapply expnat_exp_le; intuition.
Qed.

Theorem polynomial_const :
forall c,
polynomial (fun n => c).