We present a new framework for formalizing mathematics in untyped set theory using auto2. Using this framework, we formalize in Isabelle FOL the entire chain of development from the axioms of set theory to the definition of the fundamental group for an arbitrary topological space. The auto2 prover is used as the sole automation tool, and enables succinct proof scripts throughout the project.
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 dierent 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.  In this paper we explore the possibility of emulating the Mizar environment as close as possible inside the Isabelle logical framework. We introduce adaptations to the Isabelle FOL object logic that correspond to the logic of Mizar, as well as Isar inner syntax notations that correspond to these of the Mizar language. We show how Isabelle types can be used to differentiate between the syntactic categories of the Mizar language, such as sets and Mizar types including modes and attributes, and show how they interact with the basic constructs of the Tarski-Grothendieck set theory. We discuss Mizar definitions and provide simple abbreviations that allow the introduction of Mizar predicates, functions, attributes and modes using the Isabelle Pure language elements for introducing definitions and theorems. We finally consider the definite and indefinite description operators in Mizar and their use to introduce definitions by “means” and “equals”. We demonstrate the usability of the environment on a sample Mizar-style formalization, with cluster inferences and “by” steps performed manually. This paper is intended to be a practical reference manual for basic Mizar terminology which may be helpful to get started using the system. The paper describes most important aspects of the Mizar language as well as some features of the verification software.