Christoph Schwarzweller

Commutative Algebra in the Mizar System

We report on the development of algebra in the Mizar system. This includes the construction of formal multivariate power series and polynomials as well as the definition of ideals up to a proof of the Hilbert basis theorem. We present how the algebraic structures are handled and how we inherited the past developments from the Mizar Mathematical Library (MML). The MML evolves and past contributions are revised and generalized. Our work on formal power series caused a number of such revisions. It seems that revising past developments with an intent to generalize them is a necessity when building a database of formalized mathematics. This poses a question: how much generalization is best?

Translating mathematical vernacular into knowledge repositories

Defining functions is a major topic when building mathematical repositories. Though relatively easy in mathematical vernacular, function definitions rise a number of questions and problems in fully formal languages (see [4]). This becomes even more important for repositories in which properties of the defined functions are not only stated, but also proved correct. In this paper we investigate function definitions in the Mizar system. Though most of them are straightforward and follow the intuition, we also found a number of examples differing from mathematical vernacular or where different solutions seem equally reasonable. Sometimes there even do not seem to exist solutions not somehow “ignoring mathematical vernacular”. So the question is: Should we seek for some kind of standard, that is a “formal mathematical vernacular”, or should we accept that different authors prefer different styles?

Rough Concept Analysis – Theory Development in the Mizar System

Theories play an important role in building mathematical knowledge repositories. Organizing knowledge in theories is an obvious approach to cope with the growing number of definitions, theorems, and proofs. However, they are also a matter of subject on their own: developing a new piece of mathematics often relies on extending or combining already developed theories in this way reusing definitions as well as theorems. We believe that this aspect of theory development is crucial for mathematical knowledge management.

On duplication in mathematical repositories

Building a repository of proof-checked mathematical knowledge is without any doubt a lot of work, and besides the actual formalization process there is also the task of maintaining the repository. Thus it seems obvious to keep a repository as small as possible, in particular each piece of mathematical knowledge should be formalized only once. In this paper, however, we claim that it might be reasonable or even necessary to duplicate knowledge in a mathematical repository. We analyze different situations and reasons for doing so, provide a number of examples supporting our thesis and discuss some implications for building mathematical repositories.

Equality in computer proof-assistants

Equality is fundamental notion of logic and mathematics as a whole. If computer-supported formalization of knowledge is taken into account, sooner or later one should precisely declare the intended meaning/interpretation of the primitive predicate symbol of equality. In the paper we draw some issues how computerized proof-assistants can deal with this notion, and at the same time, we propose solutions, which are not contradictory with mathematical tradition and readability of source code. Our discussion is illustrated with examples taken from the implementation of the MIZAR system.

On algebraic hierarchies in mathematical repository of Mizar

Mathematics, especially algebra, uses plenty of structures: groups, rings, integral domains, fields, vector spaces to name a few of the most basic ones. Classes of structures are closely connected - usually by inclusion - naturally leading to hierarchies that has been reproduced in different forms in different mathematical repositories. In this paper we give a brief overview of some existing algebraic hierarchies and report on the latest developments in the Mizar computerized proof assistant system. In particular we present a detailed algebraic hierarchy that has been defined in Mizar and discuss extensions of the hierarchy towards more involved domains. Taking fully formal approach into account we meet new difficulties comparing with its informal mathematical framework.

Modular Integer Arithmetic

Modular Integer Arithmetic In this article we show the correctness of integer arithmetic based on Chinese Remainder theorem as described e.g. in [11]: Integers are transformed to finite sequences of modular integers, on which the arithmetic operations are performed. Retransformation of the results to the integers is then accomplished by means of the Chinese Remainder theorem. The method presented is a typical example for computing in homomorphic images.

Schur's Theorem on the Stability of Networks

Schurs Theorem on the Stability of Networks A complex polynomial is called a Hurwitz polynomial if all its roots have a real part smaller than zero. This kind of polynomial plays an all-dominant role in stability checks of electrical networks. In this article we prove Schurs criterion [17] that allows to decide whether a polynomial p(x) is Hurwitz without explicitly computing its roots: Schurs recursive algorithm successively constructs polynomials pi(x) of lesser degree by division with x - c, ℜ {c} < 0, such that pi(x) is Hurwitz if and only if p(x) is.

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.

The First Isomorphism Theorem and Other Properties of Rings

Summary Different properties of rings and fields are discussed [12], [41] and [17]. We introduce ring homomorphisms, their kernels and images, and prove the First Isomorphism Theorem, namely that for a homomorphism f : R → S we have R/ker(f) ≅ Im(f). Then we define prime and irreducible elements and show that every principal ideal domain is factorial. Finally we show that polynomial rings over fields are Euclidean and hence also factorial