Grzegorz Bancerek

Mizar: State-of-the-art and Beyond

Mizari¾?is one of the pioneering systems for mathematics formalization, which still has ani¾?active user community. The project has been in constant development since 1973, when Andrzej Trybulec designed the fundamentals of ai¾?language capable of rigorously encoding mathematical knowledge in ai¾?computerized environment which guarantees its full logical correctness. Since then, the system with its feature-rich language devised to approximate mathematics writing has influenced other formalization projects and has given rise to ai¾?number of Mizari¾?modes implemented on top of other systems. However, the information about the system as ai¾?whole is not readily available to developers of other systems. Various papers describing Mizari¾?features have been rather incremental and focused only on particular newly implemented Mizari¾?aspects. The objective of the current paper is to give ai¾?survey of the most important Mizari¾?features that distinguish it from other popular proof checkers. We also go ai¾?step further and describe most important current trends and lines of development that go beyond the state-of-the-art system.

Information Retrieval in MML

Mizar, a proof-checking system, is used to build the Mizar Mathematical Library (MML). This is a long term project aiming at building a comprehensive library of mathematical knowledge. We describe issues concerning information retrieval, i.e., searching, browsing and presentation of MML contents. A web-based tool providing such functionalities is being implemented by G. Bancerek. We hope that our observations are helpful when solving similar problems for other repositories of formalized mathematics.

On the Structure of Mizar Types

Abstract The aim of this paper is to develop a formal theory of Mizar types. The examples are extracted from Mizar Mathematical Library (MML), some of them are simplified or presented in a bit different way. The presented theory is an approach to the structure of Mizar types as a sup-semilattice with widening (subtyping) relation as the order. It is an abstraction from the existing implementation of the Mizar verifier by Andrzej Trybulec and Czeslaw Bylinski. The theory describes the structure of types of the base fragment of Mizar language.

A Compendium of Continuous Lattices in MIZAR

This paper reports on the MIZAR formalization of the theory of continuous lattices as presented in Gierz et al.: A Compendium of Continuous Lattices, 1980. By a MIZAR formalization we mean a formulation of theorems, definitions, and proofs written in the MIZAR language whose correctness is verified by the MIZAR processor. This effort was originally motivated by the question of whether or not the MIZAR system was sufficiently developed for the task of expressing advanced mathematics. The current state of the formalization – 57 MIZAR articles written by 16 authors – indicates that in principle the MIZAR system has successfully met the challenge. To our knowledge it is the most sizable effort aimed at mechanically checking some substantial and relatively recent field of advanced mathematics. However, it does not mean that doing mathematics in MIZAR is as simple as doing mathematics traditionally (if doing mathematics is simple at all). The work of formalizing the material of the Gierz et al. compendium has (i) prompted many improvements of the MIZAR proof checking system, (ii) caused numerous revisions of the the MIZAR data base, and (iii) contributed to the “to do” list of further changes to the MIZAR system.

Information retrieval and rendering with MML query

Mizar, a proof-checking system, is used to build the Mizar Mathematical Library (MML). MML Query is a semantics-based tool for managing the mathematical knowledge in Mizar including searching, browsing and presentation of the evolving MML content. The tool is becoming widely used as an aid for Mizar authors and plays an essential role in the ongoing reorganization of MML. We present new features of MML Query implemented in the third release and describe the possibilities offered by them.

Integrated Semantic Browsing of the Mizar Mathematical Library for Authoring Mizar Articles

The Mizar system is equipped with a very large library containing tens of thousands of theorems and thousands of definitions, which often use overloaded notation. For efficient authoring of new Mizar articles it is necessary to have good tools for searching and browsing this library. It would be ideal if such tools were simple, intuitive and easy to access. Particularly, they should provide interactive and integrated support during authoring Mizar articles.

Presenting and Explaining Mizar

The Mizar proof language has both many human-friendly presentation features, and also firm semantical level allowing rigorous proof checking. Both the presentation features and the semantics are important for users, and an ideal Mizar presentation should be both human-friendly (i.e. very close to textbook presentations), and also allowing fast access to the detailed semantics and detailed proof explanations. This poses several questions, problems and choices when presenting original Mizar texts, presenting results of semantic queries over the Mizar library, and also when presenting texts produced directly on the semantical level, e.g. by automated theorem provers. This paper discusses solutions to these problems, and particularly implements an initial system for presenting detailed explanations of atomic Mizar inferences. This is done by the cooperation of the Mizar XML presentation tools, the MML Query system, and automated theorem provers working on the MPTP semantic translation of Mizar.

Introduction to Matroids

Introduction to Matroids The paper includes elements of the theory of matroids [23]. The formalization is done according to [12]. MML identifier: MATROID0, version: 7.9.03 4.108.1028

Sorting by Exchanging

Sorting by Exchanging We show that exchanging of pairs in an array which are in incorrect order leads to sorted array. It justifies correctness of Bubble Sort, Insertion Sort, and Quicksort.

Program Algebra over an Algebra

Summary We introduce an algebra with free variables, an algebra with undefined values, a program algebra over a term algebra, an algebra with integers, and an algebra with arrays. Program algebra is defined as universal algebra with assignments. Programs depend on the set of generators with supporting variables and supporting terms which determine the value of free variables in the next state. The execution of a program is changing state according to successor function using supporting terms.