Groebner Bases in Theorema

In this talk we show how the theory of Groebner bases can be represented in the computer system

Theorema

, a system initiated by Bruno Buchberger in the mid-nineties. The main purpose of

Theorema

is to serve mathematical theory exploration and, in particular, automated reasoning. However, it is also an essential aspect of the

Theorema

philosophy that the system also provides good facilities for carrying out computations. The main difference between

Theorema

and ordinary computer algebra systems is that in

Theorema

one can both program (and, hence, compute) and prove (generate and verify proofs of theorems and algorithms). In fact, algorithms / programs in

Theorema

are just equational (recursive) statements in predicate logic and their application to data is just a special case of simplification w.r.t. equational logic as part of predicate logic.

We present one representation of Groebner bases theory among many possible “views” on the theory. In this representation, we use functors to construct hierarchies of domains (e. g. for power products, monomials, polynomials, etc.) in a nicely structured way, which is meant to be a model for gradually more efficient implementations based on more refined and powerful theorems or at least programming tricks, data structures, etc.