Combining Algebra and Universal Algebra in First-Order Theorem Proving: The Case of Commutative Rings

We present a general approach for integrating certain mathematical structures in first-order equational theorem provers. More specifically, we consider theorem proving problems specified by sets of first-order clauses that, contain the axioms of a commutative ring with a unit element. Associative-commutative superposition forms the deductive core of our method, while a convergent rewrite system for commutative rings provides a starting point for more specialized inferences tailored to the given class of formulas. We adopt ideas from the Grobner basis method to show that many inferences of the superposition calculus are redundant. This result is obtained by the judicious application of the simplification techniques afforded by convergent rewriting and by a process called symmetrization that embeds inferences between single clauses and ring axioms.

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