Adam Naumowicz

Mizar in a Nutshell

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.

A Brief Overview of Mizar

Mizar is the name of a formal language derived from informal mathematics and computer software that enables proof-checking of texts written in that language. The system has been actively developed since 1970s, growing into a popular proof assistant accompanied with a huge repository of formalized mathematical knowledge. In this short overview, we give an outline of the key features of the Mizar language, the ideas and theory behind the system, its main applications, and current development.

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.

Improving Mizar Texts with Properties and Requirements

In this paper we report on the current state of implementation of two features of the Mizar system – properties and requirements. These two mechanisms provide elements of basic computer algebra to strengthen the capabilities of the Mizar checker by automation of some frequently used reasoning steps. This, in turn, allows for a notable reduction of the size of users’ input in some reasonings. As a result, the size of the Mizar library can be reduced, and, at the same time, Mizar articles can become more like informal mathematical papers.

Licensing the Mizar mathematical library

We present several steps towards large formal mathematical wikis. The Coq proof assistant together with the CoRN repository are added to the pool of systems handled by the general wiki system described in \cite{DBLP:conf/aisc/UrbanARG10}. A smart re-verification scheme for the large formal libraries in the wiki is suggested for Mizar/MML and Coq/CoRN, based on recently developed precise tracking of mathematical dependencies. We propose to use features of state-of-the-art filesystems to allow real-time cloning and sandboxing of the entire libraries, allowing also to extend the wiki to a true multi-user collaborative area. A number of related issues are discussed.

Formal Mathematics for Mathematicians

The collection of works for this special issue was inspired by the presentations given at the 2011 AMS Special Session on Formal Mathematics for Mathematicians: Developing Large Repositories of Advanced Mathematics. The issue features a collection of articles by practitioners of formalizing proofs who share a deep interest in making computerized mathematics widely available.

Interfacing external CA systems for Grobner bases computation in Mizar proof checking

In this paper, we report on the results of a case study aimed at selecting a prospective CA system to be used by the Mizar proof-checking system for performing computations of Gröbner bases in Mizar’s module responsible for equality calculus. A rudimentary interface has been implemented for each of considered CA systems, and tested in order to assess its feasibility in connection with the Mizar proof checker.

Automating Boolean Set Operations in Mizar Proof Checking with the Aid of an External SAT Solver

In this paper we present the results of an experiment with employing an external SAT solver to strengthen the notion of obviousness of the Mizar proof checker. The presented extension of the Mizar system is based on a version of MiniSAT, called Logic2CNF. The SAT-enhanced Mizar checker is programmed to automatically spawn a new Logic2CNF process whenever it needs to justify any goal that can be solved by reducing it into a corresponding propositional satisfiability problem (equalities based on Boolean operations or set inclusion). The external tool is interfaced within the implementation of Mizar’s requirements directives.

The geometry of generalized Veronese spaces

We introduce and study the notion of a generalized (k-th) Veronese space associated with a partial linear space. Standard geometrical concepts (triangles, strong subspaces etc.) are interpreted in the defined structures (cf. 2.4, 2.11, 3.1). Then some basic features of veronesians are proved, in particular we establish which common geometrical axioms are preserved (cf. 2.6, 3.2, 3.5, 3.4, 3.6, and 4.11). Finally, we determine the automorphism groups of generalized Veronese spaces (cf. 5.10, 5.9, 6.4, and 6.5).

SAT-Enhanced Mizar Proof Checking

In this paper we present an experimental extension of the Mizar system employing an external SAT solver to strengthen the notion of obviousness of the Mizar proof checker. The presented extension is based on a version of MiniSAT, called Logic2CNF. The SAT-enhanced Mizar checker is programmed to automatically spawn a new Logc2CNF process whenever it needs to justify any goal that involves equalities based on Boolean operations.