Equality in Computer Algebra and Beyond

Abstract

Equality is such a fundamental concept in mathematics that, in fact, we seldom explore it in detail, and tend to regard it as trivial. When it is shown to be non-trivial, we are often surprised. As is often the case, the computerization of mathematical computation in computer algebra systems on the one hand, and mathematical reasoning in theorem provers on the other hand, forces us to explore the issue of equality in greater detail.

Equality in computer proof-assistants

“…Our paper is a result of some thoughts presented in [7] and [3]. However, we are focusing not on computer algebra systems as in [7], and also not on the role of equality in mathematical education (although mathematical proof-assistants are definitely useful in this area).…”

“…However, we are focusing not on computer algebra systems as in [7], and also not on the role of equality in mathematical education (although mathematical proof-assistants are definitely useful in this area). We started from the place where [7] posed some important questions: the area where automated reasoning and computer algebra can interact, and we went ahead of discussions, focusing on real-life implementations of the theoretical ideas: how the automatic proof-checker can cope with the predicate of mathematical equality and corresponding predicate symbol of equality. For our work in this paper, we have chosen MIZAR proof-checker described in [1].…”

The Challenges of Multivalued “Functions”

“…=B to denote what is normally given in the literature as an equality with an = sign, but where one of the purposes of this paper is to question the meaning of that, very overloaded [10], symbol.…”

Effective Set Membership in Computer Algebra and Beyond

“…In [7] we discussed the various ideas of equality that can be found in computer algebra, and showed the importance of distinguishing "I know that X = Y " from "I don't know that X = Y ". In this paper (a fuller version of which is in [8]) we look at effective set membership.…”