The Mizar Mathematical Library in OMDoc: Translation and Applications

Iancu, Mihnea; Kohlhase, Michael; Rabe, Florian; Urban, Josef

doi:10.1007/s10817-012-9271-4

Cited by 42 publications

(31 citation statements)

References 15 publications

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“…This would also enable reusing purely topological constructions in another areas of mathematicsfor example, fourteen Kuratowski sets can be expressed in the language of group theory and abstract maps with accompanying properties. This also opened the way for explaining rough sets in topological terms and will not be restricted for the Mizar library only, as the translation from the Mizar formalism 7 into other formal languages are available [21]. Additionally we hope to unify the existing approach with newly developed theory of uniform spaces.…”

Section: Discussionmentioning

confidence: 99%

Topological structures as a tool for formal modelling of rough sets

Grabowski

Coghetto²

2017

Annals of Computer Science and Information Systems

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Abstract-In the paper, we present the topological counterpart supporting rich formal apparatus describing properties of rough sets within one of the largest repositories of computerized mathematical knowledge, the Mizar Mathematical Library. The paper develops third and final (after lattice theory and the theory of general binary relations) planned path designed to be linked (via mechanisms of theory merging) with the theory of structures described by Pawlak in the early seventies of the previous century. We propose the revision of the existing topological apparatus offered by Mizar, and give the outline of the formalization of uniform spaces, important objects allowing for further representation of approximation spaces.

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Section: Discussionmentioning

confidence: 99%

Topological structures as a tool for formal modelling of rough sets

Grabowski

Coghetto²

2017

Annals of Computer Science and Information Systems

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“…The statements of the theorems in the whole MML have been exported to the MMT logical framework [34]. This allows the use of various MMT services for MML, such as searching the library or providing proof advice, however does not include an independent verification of the proofs or proof automation.…”

Section: Mmode For Coq or Declarative Proof Language (Dpl) For Coqmentioning

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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“…The need for such cooperation forced the developers of Mizar to implement new intermediate dialects of the language carrying more information (and making it more explicit)-weakly strict and more strict Mizar [3]. Also hyperlinked form of an article was reused [11,36].…”

Section: Current Mizarmentioning

confidence: 99%