Definitional Expansions in Mizar

Korniłowicz, Artur

doi:10.1007/s10817-015-9331-7

Cited by 17 publications

(3 citation statements)

References 23 publications

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“…Interestingly, in Mizar Ver. 8.1.01 from 2012, to filter out some futile definitional expansions automatically generated by a newly implemented mechanism in the Mizar checker [ 52 ], the most general root type was restored into the HIDDEN file, but under a new name: object.…”

Section: Axiomaticsmentioning

confidence: 99%

The Role of the Mizar Mathematical Library for Interactive Proof Development in Mizar

Bancerek¹,

et al. 2017

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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.

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Section: Axiomaticsmentioning

confidence: 99%

The Role of the Mizar Mathematical Library for Interactive Proof Development in Mizar

Bancerek¹,

et al. 2017

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“…The proofs in our certification contain more steps than the Mizar ones. This is mainly due to the lack of the Mizar automation in our system, e.g., type inference, equational calculus [38], definitional expansions [39]. Additionally, we still have to directly indicate the background information, such as registrations, that are processed automatically by Mizar.…”

Section: A Field Formalizationmentioning

confidence: 99%

Progress in the Independent Certiﬁcation of Mizar Mathematical Library in Isabelle

Kaliszyk

Pąk

2017

Annals of 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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“…on_line, which shortens the notation and provides future connections with Hilbert's axioms. Of course, at the end we are left with the proofs that the set of formulas (obtained by the so-called definitional expansions [24]), chosen formula can be deducted, but our attribute-steered approach seems to be better, not in terms of the efficiency of proving, but readability and usefulness for human mathematician.…”

Section: The Original Self-contained Approachmentioning

confidence: 99%

Tarski's geometry modelled in Mizar computerized proof assistant

Grabowski¹

2016

Annals of Computer Science and Information Systems

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Abstract-In the paper, we discuss the formal approach to Tarski geometry axioms modelled with the help of the Mizar computerized proof assistant system. Although our basic development was inspired by Julien Narboux's Coq pseudo-code and is dated back to 2014, there are significant steps in the formalization of geometry done in the last decade of the previous century. Taking this into account, we will propose the reuse of existing results within this new framework (including Hilbert's axiomatic approach), with the ultimate future goal to encode the textbook Metamathematische Methoden in der Geometrie by Schwabhäuser, Szmielew and Tarski. We try however to go much further from the use of simple predicates in the direction of the use of structures with their inheritance, attributes as a tool of more human-friendly namespaces for axioms, and registrations of clusters to obtain more automation (with the possible use of external equational theorem provers like Otter/Prover9).

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Definitional Expansions in Mizar

Cited by 17 publications

References 23 publications

The Role of the Mizar Mathematical Library for Interactive Proof Development in Mizar

The Role of the Mizar Mathematical Library for Interactive Proof Development in Mizar

Progress in the Independent Certiﬁcation of Mizar Mathematical Library in Isabelle

Tarski's geometry modelled in Mizar computerized proof assistant

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