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.
Highlights
Around 1970s, the advances in computer technology and its popularization together with the proliferation of more user-friendly programming languages allowed the mathematical community to initiate several seminal projects like de Bruijn’s Automath [20], Milner’s LCF [58] or Glushkov’s Evidence Algorithm [54]
From the very beginning Trybulec postulated a language and a computer system for recording mathematical papers in such a way that [35,56]: (a) the papers could be stored in a computer and later, at least partially, translated into natural languages, (b) the papers would be formal and concise, (c) it would form a basis for construction of an automated information system for mathematics, (d) it would facilitate detection of errors, verification of references, elimination of repeated theorems, etc., (e) it would open a way to machine assisted education of the art of proving theorems, (f) it would enable automated generation of input into typesetting systems
A crucial factor that helped to establish Mizar’s position among leading proof assistants, stand the test of time and be developed in the direction of achieving these goals, was the realization of the fact, that large-scale formalizations require developing methods of efficient accumulating, maintaining and re-use of previously generated mathematical content. This approach is today taken by developers of dedicated formal libraries e.g. the Isabelle based Archive of Formal Proofs [14], as well as all large formalization projects, both in mathematics
Summary
Around 1970s, the advances in computer technology and its popularization together with the proliferation of more user-friendly programming languages allowed the mathematical community to initiate several seminal projects like de Bruijn’s Automath [20], Milner’s LCF [58] or Glushkov’s Evidence Algorithm [54]. The current form of the arithmetic in the MML was shaped around the year 2003 in the course of developing a series of encyclopedic articles XCMPLX* and XREAL* extracted from the library in order to simplify the browsing for selected useful properties of real, complex, and extended real numbers.
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