Mathematical Theory Exploration in Theorema: Reduction Rings

Abstract

In this paper we present the first-ever computer formalization of the theory of Gr\"obner bases in reduction rings, which is an important theory in computational commutative algebra, in Theorema. Not only the formalization, but also the formal verification of all results has already been fully completed by now; this, in particular, includes the generic implementation and correctness proof of Buchberger's algorithm in reduction rings. Thanks to the seamless integration of proving and computing in Theorema, this implementation can now be used to compute Gr\"obner bases in various different domains directly within the system. Moreover, a substantial part of our formalization is made up solely by "elementary theories" such as sets, numbers and tuples that are themselves independent of reduction rings and may therefore be used as the foundations of future theory explorations in Theorema. In addition, we also report on two general-purpose Theorema tools we developed for an efficient and convenient exploration of mathematical theories: an interactive proving strategy and a "theory analyzer" that already proved extremely useful when creating large structured knowledge bases.