Mizar: State-of-the-art and Beyond

Bancerek, Grzegorz; Byliński, Czesław; Grabowski, Adam; Korniłowicz, Artur; Matuszewski, Roman; Naumowicz, Adam; PaźK, Karol; Urban, Josef

doi:10.1007/978-3-319-20615-8_17

Cited by 186 publications

(112 citation statements)

References 41 publications

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“…From our point of view, the most significant application of Pell's equation was done in the proof of Matiyasevich's theorem [6] that we try to formalize in the Mizar system [1]. That theorem states that every computable enumerable set is diophantine.…”

Section: Motivationmentioning

confidence: 99%

Pell’s Equation

Acewicz

Pąk

2017

Formalized Mathematics

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Section: Motivationmentioning

confidence: 99%

Pell’s Equation

Acewicz

Pąk

2017

Formalized Mathematics

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“…They are used to guarantee the correctness of compilers [1], operating systems [2], hardware [3], as well as to certify mathematical results that involve computation [4]. The Mizar system [5] is one of the oldest computer systems used to certify proofs. Its library, the Mizar Mathematical Library [6] (MML) contains today more than 1200 articles and 60000 proved theorems mainly about mathematics.…”

Section: Introductionmentioning

confidence: 99%

Progress in the Independent Certiﬁcation of Mizar Mathematical Library in Isabelle

Kaliszyk

Pąk

2017

Proceedings of the 2017 Federated Conference on Computer Science and Information Systems

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Abstract-The Mizar Mathematical Library is one of the largest collections of machine understandable formal proofs encompassing many areas of today mathematics including results from algebra, analysis, topology, and lattice theory. The Mizar system has so far been the only tool able to completely process, certify, and make use of these developments. In this paper, we present the progress in the development of an independent certification mechanism of Mizar proofs based on the Isabelle logical framework. The approach allows rechecking the Mizar formal proofs based on a more succinct and more precisely specified formal infrastructure. Additionally, it necessitates a full formal specification of the mechanisms that ensure the correctness of the defined objects, in particular, the proofs that such mechanisms are correct. The development already covers an important part of the Mizar library foundations. We improve the mechanism for defining Mizar structures and show that it permits simpler validation of proof developments involving such objects. To demonstrate this, we perform a complete translation of the Mizar net of basic algebraic structures including their attributes and certify all the corresponding proofs in Isabelle.

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“…T HE Mizar system [1], [2], [3] is a computer proofassistant system used for encoding formal mathematical data (definitions and theorems) and verifying mathematical proofs. The system comprises a dedicated formal computer language -the Mizar language, a collection of command-line tools including the VERIFIER and a repository of formal texts -Mizar Mathematical Library (MML) that have been written in the Mizar language and machine-verified for their logical correctness by the VERIFIER.…”

Section: Introductionmentioning

confidence: 99%

Introducing Euclidean Relations to Mizar

Naumowicz

Korniłowicz

2017

Proceedings of the 2017 Federated Conference on Computer Science and Information Systems

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Abstract-In this paper we present the methodology of implementing a new enhancement of the Mizar proof checker based on enabling special processing of Euclidean predicates, i.e. binary predicates which fulfill a specific variant of transitivity postulated by Euclid. Typically, every proof step in formal mathematical reasoning is associated with a formula to be proved and a list of references used to justify the formula. With the proposed enhancement, the Euclidean property of given relations can be registered during their definition, and so the verification of some proof steps related to these relations can be automated to avoid explicit referencing.

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Mizar: State-of-the-art and Beyond

Cited by 186 publications

References 41 publications

Pell’s Equation

Pell’s Equation

Progress in the Independent Certiﬁcation of Mizar Mathematical Library in Isabelle

Introducing Euclidean Relations to Mizar

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