Gröbner Bases - Theory Refinement in the Mizar System

We argue that for building mathematical knowledge repositories a broad development of theories is of major importance. Organizing mathematical knowledge in theories is an obvious approach to cope with the immense number of topics, definitions, theorems, and proofs in a general repository that is not restricted to a special field. However, concrete mathematical objects are often reinterpreted as special instances of a general theory, in this way reusing and refining existing developments. We believe that in order to become widely accepted mathematical knowledge management systems have to adopt this flexibility and to provide collections of well-developed theories. As an example we describe the Mizar development of the theory of Grobner bases, a theory which is built upon the theory of polynomials, ring (ideal) theory, and the theory of rewriting systems. Here, polynomials are considered both as ring elements and elements of rewriting systems. Both theories (and polynomials) already have been formalized in Mizar and are therefore refined and reused. Our work also includes a number of theorems that, to our knowledge, have been proved mechanically for the first time.