Towards the Construction of a Model of Mizar Concepts

The aim of this paper is to develop a formal theory of Mizar linguistic concepts following the ideas from [14] and [13]. The theory here presented is an abstract of the existing implementation of the Mizar system and is devoted to the formalization of Mizar expressions. The base idea behind the formalization is dependence on variables which is determined by variable-dependence (variables may depend on other variables). The dependence constitutes a Galois connection between opposite poset of dependence-closed set of variables and the sup-semilattice of widening of Mizar types (smooth type widening). In the paper the concepts strictly connected with Mizar expressions are formalized. Among them are quasi-loci, quasi-terms, quasi-adjectives, and quasitypes. The structural induction and operation of substitution are also introduced. The prefix quasi is used to indicate that some rules of construction of Mizar expressions may not be fulfilled. For example, variables, quasi-loci, and quasiterms have no assigned types and, in result, there is no possibility to conduct type-checking of arguments. The other gaps concern inconsistent and out-ofcontext clusters of adjectives in types. Those rules are required in the Mizar identification process. However, the expression appearing in later processes of Mizar checker may not satisfy the rules. So, introduced apparatus is enough and adequate to describe data structures and algorithms from the Mizar checker (like equational classes).