The Role of the Mizar Mathematical Library for Interactive Proof Development in Mizar

Abstract

The Mizar system is one of the pioneering systems aimed at supporting mathematical proof development on a computer that have laid the groundwork for and eventually have evolved into modern interactive proof assistants. We claim that an important milestone in the development of these systems was the creation of organized libraries accumulating all previously available formalized knowledge in such a way that new works could effectively re-use all previously collected notions. In the case of Mizar, the turning point of its development was the decision to start building the Mizar Mathematical Library as a centrally-managed knowledge base maintained together with the formalization language and the verification system. In this paper we show the process of forming this library, the evolution of its design principles, and also present some data showing its current use with the modern version of the Mizar proof checker, but also as a rich corpus of semantically linked mathematical data in various areas including web-based and natural language proof presentation, maths education, and machine learning based automated theorem proving.

Exploration of neural machine translation in autoformalization of mathematics in Mizar

“…By comparison, none of the existing proof assistants can provide a corpus of this size. Among all proof assistants, Mizar has more than 55 thousand theorem statements and more than 3 million lines of code in its Mizar Mathematical Library (MML) [6]. This is the largest formal library that the ITP community can provide that has a comparable size to a natural language corpus.…”

A Tale of Two Set Theories

“…Mizar types must be inhabited and this obligation must be proven by a user directly in the definition of a given type or before the first use if a type has the form of intersection of types. Parallel to the system development, the Mizar community puts a significant effort into building the Mizar Mathematical Library (MML) [4]. The MML is the comprehensive repository of currently formalized mathematics in the Mizar system.…”

“…for X being set st X ∈ A() holds F(X) ∈ U() The proof uses a function that maps each x in A() to {F(x)}. 4…”

A formal proof of Hensel's lemma over the p-adic integers

“…This snapshot, and a map between this paper and the formalization, can be found on the author's website. 3 The code blocks presented in this paper should be read as schematic, not literal: we sometimes change names, omit universe levels, and swap implicit and explicit arguments for the sake of presentation.…”