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An automated confluence proof for an infinite rewrite system parametrized over an integrodifferential algebra
 2010. Proceedings of ICMS 2010, LNCS
"... In this paper we present an automated proof for the confluence of a rewrite system for integrodifferential operators (given in Table 1). We also outline a generic prototype implementation of the integrodifferential polynomials—the key tool for this proof—realized using the Theorema system. With it ..."
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In this paper we present an automated proof for the confluence of a rewrite system for integrodifferential operators (given in Table 1). We also outline a generic prototype implementation of the integrodifferential 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
Knowledge Archives in Theorema: A LogicInternal Approach
"... Abstract. Archives are implemented as an extension of Theorema for representing mathematical repositories in a natural way. An archive can be conceived as one large formula in a language consisting of higherorder predicate logic together with a few constructs for structuring knowledge: attaching la ..."
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Abstract. Archives are implemented as an extension of Theorema for representing mathematical repositories in a natural way. An archive can be conceived as one large formula in a language consisting of higherorder predicate logic together with a few constructs for structuring knowledge: attaching labels to subhierarchies, disambiguating symbols by the use of namespaces, importing symbols from other namespaces and specifying the domains of categories and functors as namespaces with variable operations. All these constructs are logicinternal in the sense that they have a natural translation to higherorder logic so that certain aspects of Mathematical Knowledge Management can be realized in the object logic itself. There are a variety of operations on archives, though in this paper we can only sketch a few of them: knowledge retrieval and theory exploration, merging and splitting, insertion and translation to predicate logic.
Computing and Proving with IntegroDifferential Polynomials in Theorema
"... Integrodifferential polynomials are a novel generalization of the wellknown differential polynomials extensively used in differential algebra [17]. They were introduced in [29] as a kind of universal extensions of integrodifferential algebras and have recently been applied in a confluence proof [ ..."
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Integrodifferential polynomials are a novel generalization of the wellknown differential polynomials extensively used in differential algebra [17]. They were introduced in [29] as a kind of universal extensions of integrodifferential algebras and have recently been applied in a confluence proof [34] for the rewrite system
General Polynomial Reduction with TH∃OREM ∀ Functors: Applications to IntegroDifferential Operators and Polynomials
 ISSAC 2008 POSTER ABSTRACTS
, 2008
"... General Polynomial Reduction. We outline a prototype implementation of the algorithms for integrodifferential operators/polynomials in [12]. Our approach based on a generic implementation of noncommutative monoid rings with reduction, programmed in the functors language of the TH∃OREM ∀ system. The ..."
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General Polynomial Reduction. We outline a prototype implementation of the algorithms for integrodifferential operators/polynomials in [12]. Our approach based on a generic implementation of noncommutative monoid rings with reduction, programmed in the functors language of the TH∃OREM ∀ system. The integrodifferential operators—realized by a suitable quotient of noncommutative polynomials over a given integrodifferential algebra—can be used for solving and manipulating boundary problems for linear ordinary differential equations. For describing extensions of integrodifferential algebras algorithmically, we use integrodifferential polynomials. We use a fixed Gröbner basis for normalizing integrodifferential operators. Gröbner bases were invented by Buchberger [2, 3] for commutative polynomials and reinvented in [1] for noncommutative ones. While [9] analyzes the computational aspects of the latter, it does not support two features that are important for our present setting: the usage of infinitely many indeterminates and reduction modulo an (algorithmic) infinite system of polynomials. Among the systems implementing noncommutative Gröbner bases, most address certain special classes (e.g. algebras of solvable type or homogeneous polynomials) which do not include our present case. To our best knowledge, none of these allow polynomials with infinitely many indeterminates and reduction modulo an infinite system of polynomials. For details, see the website