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Four Decades of Mizar

Adam Grabowski, Artur Korniłowicz, Adam Naumowicz
2015 Journal of automated reasoning  
This special issue is dedicated to works related to Mizar, the theorem proving project started by Andrzej Trybulec in the 1970s, and other automated proof checking systems used for formalizing mathematics  ...  In the period of over 40 years, a number of different versions of the Mizar system have been investigated.  ...  Karol Pąk in his submission "Improving Legibility of Formal Proofs Based on the Close Reference Principle is NP-Hard" investigates proof graphs available in the MML measuring information flow.  ... 
doi:10.1007/s10817-015-9345-1 fatcat:wclwjz6eqzbvbis7w5tdolu5n4

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

Adam Naumowicz
2015 Journal of automated reasoning  
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.  ...  In particular, the automation in processing of inclusions and equalities as compared to the initial implementation of Boolean operations only [15] resulted in a significant improvement.  ... 
doi:10.1007/s10817-015-9332-6 fatcat:oq5uq4kls5ellhi6ryoh6una3q

Improving Legibility of Formal Proofs Based on the Close Reference Principle is NP-Hard

Karol Pąk
2015 Journal of automated reasoning  
Proof development in proof assistants such as HOL, Coq, Mizar, etc. is an activity where authors usually produce proofs by typing out proof scripts or system tactics.  ...  To analyze the legibility of natural deduction proofs, we investigate proof graphs that represent the flow of information in given reasoning.  ...  This interpretation has a lot in common with the construction then implemented in Mizar, Isabelle/Isar and other systems where the proof style inspired by Mizar is implemented: Declare [25] , Mizar Mode  ... 
doi:10.1007/s10817-015-9337-1 fatcat:re4mixffqjed3hsoocwcghc7dm

Presentation and Manipulation of Mizar Properties in an Isabelle Object Logic [chapter]

Cezary Kaliszyk, Karol Pąk
2017 Lecture Notes in Computer Science  
In this paper, we discuss various elements of the Mizar language and the possible ways how these can be represented in the Isabelle framework in order to allow a suitable way of working in typed set theory  ...  We also compare the language elements corresponding to the actual proofs, with special care for implicit proof expansions not present in Isabelle.  ...  Acknowledgements We thank Chad Brown for the discussions on the various set-theoretic foundations.  ... 
doi:10.1007/978-3-319-62075-6_14 fatcat:irrggzjxlrfedbirdcwe32i7oa

Progress in the Independent Certification of Mizar Mathematical Library in Isabelle

Cezary Kaliszyk, Karol Pąk
2017 Proceedings of the 2017 Federated Conference on Computer Science and Information Systems  
In this paper, we present the progress in the development of an independent certification mechanism of Mizar proofs based on the Isabelle logical framework.  ...  We improve the mechanism for defining Mizar structures and show that it permits simpler validation of proof developments involving such objects.  ...  The Mizar system [5] is one of the oldest computer systems used to certify proofs.  ... 
doi:10.15439/2017f289 dblp:conf/fedcsis/KaliszykP17 fatcat:muqh2uxfi5az5mgffdqfnahf6y

Methods of Lemma Extraction in Natural Deduction Proofs

Karol Pąk
2012 Journal of automated reasoning  
Additionally, we present methods for improving the legibility of the reasoning that remains after the extraction of the lemmas.  ...  The existing examples of natural deduction proofs, either declarative or procedural, indicate that often the legibility of proof scripts is of secondary importance to the authors.  ...  Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s)  ... 
doi:10.1007/s10817-012-9267-0 fatcat:umkypcbtgrc47h5wjyorsobnr4

Isabelle Import Infrastructure for the Mizar Mathematical Library [chapter]

Cezary Kaliszyk, Karol Pąk
2018 Lecture Notes in Computer Science  
To show that the imported 100 articles give rise to a usable Isabelle environment, we use the environment to formalize proofs in the Isabelle/Mizar environment using the imported types and their properties  ...  We present an infrastructure that allows importing an initial part of the Mizar Mathematical Library into the Isabelle/Mizar object logic.  ...  unfolded in the XML format, can be identified and folded, which is useful to improve the legibility of proof texts.  ... 
doi:10.1007/978-3-319-96812-4_13 fatcat:puwm4uo5xjfxvniy5qn6t2ltou

Improving legibility of natural deduction proofs is not trivial

Karol Pąk, Henk Barendregt
2014 Logical Methods in Computer Science  
In formal proof checking environments such as Mizar it is not merely the validity of mathematical formulas that is evaluated in the process of adoption to the body of accepted formalizations, but also  ...  To better understand the notion of readability of formal proofs, and to assess and improve their readability, we propose in this paper a method of improving proof readability based on Behaghel's First  ...  Another direction of improving the legibility of Mizar proof scripts is the automation of the most frequently performed operations in reasoning, which was explored by Naumowicz and Byliński [27] .  ... 
doi:10.2168/lmcs-10(3:23)2014 fatcat:7iw5folfnne7pkte4n2p75czrm

Towards Intuitive Reasoning in Axiomatic Geometry

Maximilian Doré, Krysia Broda
2019 Electronic Proceedings in Theoretical Computer Science  
Improvements in automated theorem provers (ATP) in recent years now mean they can prove many of these intermediate lemmas.  ...  Geometrical texts can thereby easily be formalized in Elfe, leaving only the cornerstones of a proof to be derived by the user.  ...  Figure 21 : The proof in MIZAR [10] and ISAR/ISABELLE [11] The proof texts in ELFE, MIZAR and ISABELLE for the simple lemma are all quite legible and provide insight into why the proof succeeds.  ... 
doi:10.4204/eptcs.290.4 fatcat:uhjx3tsmyvgn7jagtn2kyrslau

Point-and-write --- Documenting Formal Mathematics by Reference [article]

Carst Tankink and Christoph Lange and Josef Urban
2012 arXiv   pre-print
This paper describes the design and implementation of mechanisms for light-weight inclusion of formal mathematics in informal mathematical writings, particularly in a Web-based setting.  ...  This is conceptually done in three stages: (i) by choosing a suitable representation layer (based on RDF) for encoding the information about available resources of formal mathematics, (ii) by exporting  ...  While the OMDoc ontology defines vocabulary that seems suitable also for many Mizar internal proof steps, the current Mizar implementation only annotates the main top-level Mizar items, together with the  ... 
arXiv:1204.5094v2 fatcat:bufnydfvqng5liipnandmnh4ba

Point-and-Write – Documenting Formal Mathematics by Reference [chapter]

Carst Tankink, Christoph Lange, Josef Urban
2012 Lecture Notes in Computer Science  
This paper describes the design and implementation of mechanisms for light-weight inclusion of formal mathematics in informal mathematical writings, particularly in a Web-based setting.  ...  This is conceptually done in three stages: (i) by choosing a suitable representation layer (based on RDF) for encoding the information about available resources of formal mathematics, (ii) by exporting  ...  While the OMDoc ontology defines vocabulary that seems suitable also for many Mizar internal proof steps, the current Mizar implementation only annotates the main top-level Mizar items, together with the  ... 
doi:10.1007/978-3-642-31374-5_12 fatcat:eove6tyutjevnki3lislofwhei

The Isabelle/Naproche Natural Language Proof Assistant [chapter]

Adrian De Lon, Peter Koepke, Anton Lorenzen, Adrian Marti, Marcel Schütz, Makarius Wenzel
2021 Lecture Notes in Computer Science  
"Image missing" is included in the current version of the Isabelle/PIDE which allows comfortable editing and asynchronous proof-checking of ForTheL texts.  ...  Abstract"Image missing" is an emerging natural proof assistant that accepts input in the controlled natural language ForTheL.  ...  The development of the Mizar system [11] was accompanied or even driven by the stepwise adaptation of its language to standard mathematical proof methods and logical foundations.  ... 
doi:10.1007/978-3-030-79876-5_36 fatcat:2ox6tmqltvelpppwkrpketx46m

Towards Formal Proof Metrics [chapter]

David Aspinall, Cezary Kaliszyk
2016 Lecture Notes in Computer Science  
In this paper we investigate the idea of defining proof metrics by analogy with software metrics.  ...  Some proofs have involved many authors, years of effort, and resulted in large, complex interdependent sets of proof "source code" files.  ...  We're grateful to colleagues Iain Whiteside, Ajitha Rajan and the DReaM group at Edinburgh for discussions. The referees provided useful remarks.  ... 
doi:10.1007/978-3-662-49665-7_19 fatcat:iv4g4azu4vgutho6oq32n3f2c4

A User-friendly Interface for a Lightweight Verification System

Andrei Lapets, Assaf Kfoury
2012 Electronical Notes in Theoretical Computer Science  
(3) immediate validation feedback in the form of reformatted raw input.  ...  The system provides validation capabilities by utilizing a database of propositions governing common mathematical concepts.  ...  The ability to implicitly invoke propositions dealing with the algebra of sets was essential in making this process manageable and in allowing the resulting proofs to be legible.  ... 
doi:10.1016/j.entcs.2012.06.004 fatcat:culh3mn67fci5edwavu6tkvv3u

Univalent Foundations and the UniMath Library [article]

Anthony Bordg
2018 arXiv   pre-print
the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2).  ...  We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of  ...  Acknowledgments The author would like to thank Benedikt Ahrens, Thierry Coquand, and an anonymous referee for their useful comments and suggestions.  ... 
arXiv:1710.02723v6 fatcat:lnrwbdf4erdjtgxphptn2lql5e
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