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Computing and Proving with Integro-Differential Polynomials in Theorema

Integro-differential polynomials are a novel generalization of the well-known differential polynomials extensively used in differential algebra [17]. They were introduced in [29] as a kind of universal extensions of integro-differential algebras and have recently been applied in a confluence proof [34] for the rewrite system underlying the so-called “integro-differential operators”. In this paper we want to elaborate on the computing and proving aspects of integro-differential polynomials, seen from the standpoint of universal algebra and canonical simplifiers (Section 3). The Theorema system (Section 2) turns out to be convenient for such an integration of computing and proving, given that it contains a natural programming language based on a version of higher-order predicate logic. Thus we provide a Theorema implementation for the algebra of integro-differential polynomials, along with sample computations (Section 5). The notion of integro-differential operators (Section 4) introduced in [30] is useful for solving linear boundary problems given by a differential equation with symbolic right-hand side along with boundary conditions [33]. The integrodifferential operators are important since they can express both the problem statement (differential equation and boundary conditions) and its solution operator (which is an integral operator). For a symbolic method for solving boundary problems we refer the reader to [27]. See also [24] for a summary of the algebraic setting for boundary problems. A very simple example of a boundary problem is the following: Given f ∈ C[0, 1], find u ∈ C[0, 1] such that