Mizar Attributes: A Technique to Encode Mathematical Knowledge into Type Systems

At first glance Mizar attributes look like unary predicates over mathematical objects enabling a more natural writing and reading. Attributes in Mizar, however, serve additional, more important purposes concerning typing of mathematical objects: Using attributes not only new (sub)types can be introduced, but also the user can characterize further relations between types and in this way make available existing notations for new objects. Thereby it should be stressed that these type relations can stand for elaborated mathematical theorems. This paper describes the properties and benefits of Mizar attributes from a user’s perspective. We comprehend the development of Mizar attributes, and give examples highlighting their use — essentially in the area of algebra. Concluding we discuss their impact on building mathematical repositories.