  The Apagodu-Zeilberger algorithm can be used for computing annihilating
operators for definite sums over hypergeometric terms, or for definite
integrals over hyperexponential functions. In this paper, we propose a
generalization of this algorithm which is applicable to arbitrary
$\partial$-finite functions. In analogy to the hypergeometric case, we
introduce the notion of proper $\partial$-finite functions. We show that the
algorithm always succeeds for these functions, and we give a tight a priori
bound for the order of the output operator.
