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Integrated Semantic Browsing of the Mizar Mathematical Library for Authoring Mizar Articles
 Proceeding of the Third International Conference on Mathematical Knowledge Management
, 2004
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Gradual computerisation/formalisation of mathematical texts into Mizar
 From Insight to Proof: Festschrift in Honour of Andrzej Trybulec
"... Abstract. We explain in this paper the gradual computerisation process of an ordinary mathematical text into more formal versions ending with a fully formalised Mizar text. The process is part of the MathLang–Mizar project and is divided into a number of steps (called aspects). The first three aspec ..."
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Abstract. We explain in this paper the gradual computerisation process of an ordinary mathematical text into more formal versions ending with a fully formalised Mizar text. The process is part of the MathLang–Mizar project and is divided into a number of steps (called aspects). The first three aspects (CGa, TSa and DRa) are the same for any MathLang–TP project where TP is any proof checker (e.g., Mizar, Coq, Isabelle, etc). These first three aspects are theoretically formalised and implemented and provide the mathematician and/or TP user with useful tools/automation. Using TSa, the mathematician edits his mathematical text just as he would use L ATEX, but at the same time he sees the mathematical text as it appears on his paper. TSa also gives the mathematician easy editing facilities to help assign to parts of the text, grammatical and mathematical roles and to relate different parts through a number of mathematical, rethorical and structural relations. MathLang would then automatically produce CGa and DRa versions of the text, checks
Mathematical Knowledge Management in MIZAR
 Proc. of MKM 2001
, 2001
"... We report on how mathematics is done in the Mizar system. Mizar oers a language for writing mathematics and provides software for proofchecking. Mizar is used to build Mizar Mathematical Library (MML). This is a long term project aiming at building a comprehensive library of mathematical knowled ..."
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We report on how mathematics is done in the Mizar system. Mizar oers a language for writing mathematics and provides software for proofchecking. Mizar is used to build Mizar Mathematical Library (MML). This is a long term project aiming at building a comprehensive library of mathematical knowledge. The language and the checking software evolve and the evolution is driven by the growing MML.
Narrative structure of mathematical texts
 In preparation, available at http://www.macs.hw.ac.uk/~mm20
, 2007
"... Abstract. There are many styles for the narrative structure of a mathematical document. Each mathematician has its own conventions and traditions about labeling portions of texts (e.g., chapter, section, theorem or proof) and identifying statements according to their logical importance (e.g., theore ..."
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Abstract. There are many styles for the narrative structure of a mathematical document. Each mathematician has its own conventions and traditions about labeling portions of texts (e.g., chapter, section, theorem or proof) and identifying statements according to their logical importance (e.g., theorem is more important than lemma). Such narrative/structuring labels guide the reader’s navigation of the text and form the key components in the reasoning structure of the theory reflected in the text. We present in this paper a method to computerise the narrative structure of a text which includes the relationships between labeled text entities. These labels and relations are input by the user on top of their natural language text. This narrative structure is then automatically analysed to check its consistency. This automatic analysis consists of two phases: (1) checking the correct usage of labels and relations (i.e., that a “proof” justifies a “theorem ” but cannot justify an “axiom”) and (2) checking that the logical precedences in the document are selfconsistent. The development of this method was driven by the experience of computerising a number of mathematical documents (covering different authoring styles). We illustrate how such computerised narrative structure could be used for further manipulations, i.e. to build a skeleton of a formal document in a formal system like Mizar, Coq or Isabelle. 1
Computer Supported Formal Work: Towards a Digital Mathematical Assistant
 STUDIES IN LOGIC, GRAMMAR AND RHETORIC
, 2007
"... The year 2004 marked the fiftieth birthday of the first computer generated proof of a mathematical theorem: “the sum of two even numbers is again an even number ” (with Martin Davis ’ implementation of Presburger Arithmetic in 1954). While Martin Davis and later the research community of automated ..."
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The year 2004 marked the fiftieth birthday of the first computer generated proof of a mathematical theorem: “the sum of two even numbers is again an even number ” (with Martin Davis ’ implementation of Presburger Arithmetic in 1954). While Martin Davis and later the research community of automated deduction used machine oriented calculi to find the proof for a theorem by automatic means, the Automath project of N.G. de Bruijn – more modest in its aims with respect to automation – showed in the late 1960s and early 70s that a complete mathematical textbook could be coded and proofchecked by a computer. Roughly at the same time in 1973, the Mizar project started as an attempt to reconstruct mathematics based on computers. Since 1989, the most important activity in the Mizar project has been the development of a database for mathematics. International cooperation resulted in creating a database which includes more than 7000 definitions of mathematical concepts and more than 42000 theorems. The work by
Commutative Algebra in the Mizar System
, 2001
"... We report on the development of algebra in the Mizar system. This includes the construction of formal multivariate power series and polynomials as well as the de nition of ideals up to a proof of the Hilbert basis theorem. We present how the algebraic structures are handled and how we inherited the ..."
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We report on the development of algebra in the Mizar system. This includes the construction of formal multivariate power series and polynomials as well as the de nition of ideals up to a proof of the Hilbert basis theorem. We present how the algebraic structures are handled and how we inherited the past developments from the Mizar Mathematical Library (MML). The MML evolves and past contributions are revised and generalized. Our work on formal power series caused a number of such revisions. It seems that revising past developments with an intent to generalize them is a necessity when building a data base of formalized mathematics. This poses a question: how much generalization is best?
Unified Proof Style for Teaching Mathematics
"... Structured derivations were introduced by Back and von Wright as an extension of the calculational proof style originally proposed by E.W. Dijkstra and his colleagues. Structured derivations added nested subderivations and inherited assumptions to the original calculational style. This paper introdu ..."
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Structured derivations were introduced by Back and von Wright as an extension of the calculational proof style originally proposed by E.W. Dijkstra and his colleagues. Structured derivations added nested subderivations and inherited assumptions to the original calculational style. This paper introduces a further extension of the structured derivation format, and gives a precise syntax and semantics for the extended proof style. The extension provides a unification of the tree main proof styles used in mathematics today: Hilbertstyle forward chaining proofs, Gentzenstyle backward chaining proofs and algebraic derivations and calculations (in particular, Dijkstra’s calculational proof style). Each of these proof styles can be directly modelled as an extended structured derivation. Even more importantly, the three proof styles can be freely intermixed in a single structured derivation, allowing different proof styles to be used in different parts of the derivation, each time choosing the proof style that is most suitable for the (sub)problem at hand. We describe here (extended) structured derivations, feature by feature, and
New Auxiliary Software for MML Database Management
"... Abstract – Mizar Mathematical Library (MML) [7] is a database containing more than 900 articles verified automatically by the MIZAR system making it one of the biggest databases of computerchecked mathematical knowledge in the world. The submission process of new articles, the great number of revisi ..."
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Abstract – Mizar Mathematical Library (MML) [7] is a database containing more than 900 articles verified automatically by the MIZAR system making it one of the biggest databases of computerchecked mathematical knowledge in the world. The submission process of new articles, the great number of revisions and many experiments on the MML database depend on the existence of various auxiliary applications which have been created, usually ad hoc, since the beginning of the MIZAR system. With the start of some recent experiments with datamining and research on the robustness of the MIZAR system [3], there is a need to create new software for realizing these tasks. This paper discusses the motivation for creating such programs and presents the design and reciprocal relationships between programs aimed at datamining and research on the robustness of the MIZAR system. 1. Motivation