Mizar: An Impression

This note presents an introduction to the Mizar system, followed by a brief comparison between Mizar and Coq. Appended are a Mizar grammar and an annotated example of a complete Mizar article. 1 What It Is Mizar is a system for representing mathematical proof in a computer such that a program can check the correctness. It has been developed by Andrzej Trybulec and his team in Bialystok, Poland since about 1973. The Mizar language is quite close to the language of informal mathematics (‘the mathematical vernacular’). Mizar is based on a ZF-like set theory with classical first order logic. Part of the Mizar project is the development of a large database of mathematics which currently numbers 587 ‘articles’ taking 41 megabytes (even without the proofs it’s 6 megabytes). A major project currently underway is the translation to Mizar (under direction of Grzegorz Bancerek) of a real mathematics book: ‘A Compendium of Continuous Lattices’ [1]. Mizar has had a number of dialects. ‘Mizar MSE’ (also called ‘baby Mizar’) is a toy language not meant for real applications. The current implementation of the full Mizar language is called ‘PC Mizar’. It’s written in Turbo/Borland Pascal and runs only under DOS/Windows. Functionally it’s basically one program, ‘mizf’, that non-interactively checks a Mizar file for its correctness. The Mizar system doesn’t have much documentation. The text that most resembles a manual is ‘An Outline of PC Mizar’ by Michaà l Muzalewski [3]. A way to explore the Mizar language that works surprisingly well is to study the Mizar syntax (see appendix B on page 16 for a context free grammar) and for specific constructions to search the Mizar library for examples. 2 Definitions A Mizar article consists of an ‘environ’ header which indicates what other articles from the Mizar library it refers to, followed by a sequence of definitions and theorems. A definition has the general shape: definition let arguments; assume preconditions; func pattern -> type means :label: statement; An electronic version of this manual is available on the World Wide Web at the address