Abstract-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.

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?

No abstract

Building a repository of proof-checked mathematical knowledge is without any doubt a lot of work, and besides the actual formalization process there also is the task of maintaining the repository. Thus it seems obvious to keep a repsoitory 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 and provide a number of examples supporting our thesis.

Abstract-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.