Extending Formal Fuzzy Sets with Triangular Norms and Conorms

Fuzzy sets is a well-known approach to incomplete or imprecise data. Contrary to the rough sets however, the notion of fuzziness allows for quite natural description in terms of ordinary set theory used by mathematicians and computer scientists. As contemporary mathematics uses more and more methods of computer verification of theorems and discovering their proofs, it is not very strange that also in this area we could observe growing usage of automated proof-assistants. We report on the progress of the development of already well-established framework of fuzzy set theory within one of popular repositories of computerized mathematical knowledge – the Mizar Mathematical Library. Even if the original formal background was created some ten years ago, and during that time it was thoroughly redesigned in order to increase its expressive power and to follow the evolution of underlying proof language, we see the need for further modifications. In this paper, we describe the process of the parametrization of classical operations on fuzzy sets via triangular norms and conorms because as of now, classical union and intersection of corresponding membership functions were defined only based on operations of maximum, and minimum, respectively. We illustrate our development by examples taken from correct and fully verified Mizar code.