Reconstruction of the Mizar type system in the HOL Light system

Abstract. The Mizar system is a system for formalization of mathematics. It contains a relatively sophisticated and rich type system, which makes formalization of mathematics in Mizar more intuitive than in other systems. On the other hand, the Mizar type system is very complex and together with obscure implementation of the Mizar verifier there is concern about correctness of the Mizar system. One of the possible solutions is to translate the Mizar Math Library (MML) to other systems for formalization of mathematics and reverify MML in them. The HOL Light system has been chosen and the necessary first step of proposed translation is to reconstruct the Mizar type system in the HOL Light system, which is the aim of the presented work. The reconstruction is not easy because of complexity of the Mizar type system and different types of logic used in both systems. The basic idea is to represent Mizar types as predicates and express the dynamic part of the Mizar type system as proved theorems in HOL Light. The reconstruction was partialy implemented. The paper gives necessary introduction and summarizes main principles and obstacles encountered during reconstruction