An Automated Confluence Proof for an Infinite Rewrite System via a Gröbner Basis Computation

In this paper we present an automated proof for the confluence of a rewrite system for integro-differential operators (given in Table 1). We also outline a generic prototype implementation of the integro-differential polynomials—the key tool for this proof—realized using the Theorema system. With its generic functor mechanism—detailed in Section 2—we are able to provide a formalization of the theory of integrodifferential algebras. This formalization via functors also allows us to compute in such domains, and in this way to explore the new theory by computational experiments. Integro-differential operators (introduced in [21]) are useful for solving linear boundary problems [25] given by a differential equation with a symbolic right-hand side along with boundary conditions. Boundary problems are of major importance in applications but they are usually treated numerically. A symbolic method for solving such problems was introduced in [20] for the constant coefficients case and extended to a general differential algebra setting in [21]. A very simple example of a boundary problem is the following: Given f ∈C∞[0,1], find g ∈C∞[0,1] such that u′′ = f , u(0) = u(1) = 0.