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CCoRN, the Constructive Coq Repository at Nijmegan
"... We present CCoRN, the Constructive Coq Repository at Nijmegen. It consists of a library of constructive algebra and analysis, formalized in the theorem prover Coq. In this paper we explain the structure, the contents and the use of the library. Moreover we discuss the motivation and the (possible) ..."
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We present CCoRN, the Constructive Coq Repository at Nijmegen. It consists of a library of constructive algebra and analysis, formalized in the theorem prover Coq. In this paper we explain the structure, the contents and the use of the library. Moreover we discuss the motivation and the (possible) applications of such a library.
A DocumentOriented Coq Plugin for TEXmacs
, 2006
"... This article discusses the integration of the authoring of a mathematical document with the formalisation of the mathematics contained in that document. To achieve this we have started the development of a Coq plugin for the TEXmacs scientific editor, called tmEgg. TEXmacs allows the wysiwyg editing ..."
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This article discusses the integration of the authoring of a mathematical document with the formalisation of the mathematics contained in that document. To achieve this we have started the development of a Coq plugin for the TEXmacs scientific editor, called tmEgg. TEXmacs allows the wysiwyg editing of mathematical documents, much in the style of LATEX. Our plugin allows to integrate into a TEXmacs document mathematics formalised in the Coq proof assistant: formal definitions, lemmas and proofs. The plugin is still under development. Its main current hallmark is a documentconsistent interaction model, instead of the calculatorlike approach usual for TEXmacs plugins. This means that the Coq code in the TEXmacs document is interpreted as one (consistent) Coq file: executing a Coq command in the document means to execute it in the context (state) of all the Coq commands before it. 1
Trustable Communication Between Mathematics Systems
 IN PROC. OF CALCULEMUS 2003
, 2003
"... This paper presents a rigorous, unified framework for facilitating communication between mathematics systems. A mathematics system is given one or more interfaces which oer deductive and computational services to other mathematics systems. To achieve communication between systems, a client inter ..."
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This paper presents a rigorous, unified framework for facilitating communication between mathematics systems. A mathematics system is given one or more interfaces which oer deductive and computational services to other mathematics systems. To achieve communication between systems, a client interface is linked to a server interface by an asymmetric connection consisting of a pair of translations. Answers to requests are trustable in the sense that they are correct provided a small set of prescribed conditions are satis ed. The framework is robust with respect to interface extension and can process requests for abstract services, where the server interface is not fully specified.
A SYNTHESIS OF THE PROCEDURAL AND DECLARATIVE STYLES OF INTERACTIVE THEOREM PROVING
"... Abstract. We propose a synthesis of the two proof styles of interactive theorem proving: the procedural style (where proofs are scripts of commands, like in Coq) and the declarative style (where proofs are texts in a controlled natural language, like in Isabelle/Isar). Our approach combines the adva ..."
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Abstract. We propose a synthesis of the two proof styles of interactive theorem proving: the procedural style (where proofs are scripts of commands, like in Coq) and the declarative style (where proofs are texts in a controlled natural language, like in Isabelle/Isar). Our approach combines the advantages of the declarative style – the possibility to write formal proofs like normal mathematical text – and the procedural style – strong automation and help with shaping the proofs, including determining the statements of intermediate steps. Our approach is new, and differs significantly from the ways in which the procedural and declarative proof styles have been combined before in the Isabelle, Ssreflect and Matita systems. Our approach is generic and can be implemented on top of any procedural interactive theorem prover, regardless of its architecture and logical foundations. To show the viability of our proposed approach, we fully implemented it as a proof interface called miz3, on top of the HOL Light interactive theorem prover. The declarative language that this interface uses is a slight variant of the language of the Mizar system, and can be used for any interactive theorem prover regardless of its logical foundations. The miz3 interface allows easy access to the full set of tactics and formal libraries of HOL Light, and as such has ‘industrial strength’. Our approach gives a way to automatically convert any procedural proof to a declarative counterpart, where the converted proof is similar in size to the original. As all declarative systems have essentially the same proof language, this gives a straightforward way to port proofs between interactive theorem provers. 1.
A graphbased approach towards discerning inherent structures in a digital library of formal mathematics
 In Lecture Notes in Computer Science
, 2004
"... Abstract. As the amount of online formal mathematical content grows, for example through active efforts such as the Mathweb [21], MOWGLI [4], Formal Digital Library, or FDL [1], and others, it becomes increasingly valuable to find automated means to manage this data and capture semantics such as rel ..."
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Abstract. As the amount of online formal mathematical content grows, for example through active efforts such as the Mathweb [21], MOWGLI [4], Formal Digital Library, or FDL [1], and others, it becomes increasingly valuable to find automated means to manage this data and capture semantics such as relatedness and significance. We apply graphbased approaches, such as HITS, or HyperlinkInduced Topic Search, [11] used for World Wide Web document search and analysis, to formal mathematical data collections. The nodes of the graphs we analyze are theorems and definitions, and the links are logical dependencies. By exploiting this link structure, we show how one may extract organizational and relatedness information from a collection of digital formal math. We discuss the value of the information we can extract, yielding potential applications in math search tools, theorem proving, and education.
An Expert System for the Flexible Processing of XMLBased Mathematical Knowledge in a PrologEnvironment
, 2003
"... In this paper, we describe techniques for querying and transforming Xml{based mathematical knowledge. The Xml{documents are transformed into an equivalent Prolog{structure called eld notation, which serves as our Document Object Model (DOM). ..."
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In this paper, we describe techniques for querying and transforming Xml{based mathematical knowledge. The Xml{documents are transformed into an equivalent Prolog{structure called eld notation, which serves as our Document Object Model (DOM).
EMANI, ERAM AND OTHER EUROPEAN ACTIVITIES CONTRIBUTING TO A GLOBAL DIGITAL LIBRARY IN MATHEMATICS
"... With the rapidly growing activities in electronic publishing ideas came up to install global repositories which deal with three mainstreams in this enterprise: storing the electronic material currently available, pursuing projects to solve the archiving problem for this material with the ambition to ..."
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With the rapidly growing activities in electronic publishing ideas came up to install global repositories which deal with three mainstreams in this enterprise: storing the electronic material currently available, pursuing projects to solve the archiving problem for this material with the ambition to preserve the content in readable form for future generations, and to capture the printed literature in digital versions providing good access and search facilities for the readers. Longterm availability of published research articles in mathematics and easy access to them is a strong need for researchers working with mathematics. Hence in this domain some pioneering projects have been established addressing the above mentioned problems. The talk will describe some of these activities and the plan to develop a global Digital Library in Mathematics (DLM). For example, in the archiving area as a special project for mathematics the Electronic Mathematics Archives Network Initiative (EMANI) had been designed. Having in mind that a distributed architecture would be more suitable and reduce the load on the partners for such a project, a network is proposed, which also might be a more open approach for extending the project from a initially restricted solution to a more comprehensive enterprise. For the core of the network, a cooperational system of reference libraries and content providers like publishers and editors has been be set up. On the side of the libraries the following partners have agreed to set up a prototype for the archive: the Tsinghua University Library in Beijing, the
EMANI – Leader and Follower for the WDML
"... Abstract. The rapidly growing activities in electronic publishing lead to the request to develop global repositories, which care about three fundamental activities: to collect and store the electronic material currently available, to pursue projects for solving the longterm archiving problem for th ..."
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Abstract. The rapidly growing activities in electronic publishing lead to the request to develop global repositories, which care about three fundamental activities: to collect and store the electronic material currently available, to pursue projects for solving the longterm archiving problem for this material with the ambition to preserve the content in readable form for future generations, and to capture the printed literature in digital versions providing good access and search facilities for the readers. Longterm availability of published research articles in mathematics and easy access to them is a strong need for researchers working with mathematics. The article will describe some new developments for two main projects in this subject: the plan to develop a global World Digital Mathematical Library (WDML or DML) and the main project dealing with a coordinated archiving of digital documents in mathematics, the Electronic Mathematics Archiving Network Initiative (EMANI). For the core of the EMANI network, a cooperational system of reference libraries and content providers like publishers and editors has been be set up. Both systems share a lot of common work packages. Hence discussions in one
Merging procedural and declarative proof
"... Abstract. There are two different styles for writing natural deduction proofs: the ‘Gentzen ’ style in which a proof is a tree with the conclusion at the root and the assumptions at the leaves, and the ‘Fitch ’ style (also called ‘flag ’ style) in which a proof consists of lines that are grouped tog ..."
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Abstract. There are two different styles for writing natural deduction proofs: the ‘Gentzen ’ style in which a proof is a tree with the conclusion at the root and the assumptions at the leaves, and the ‘Fitch ’ style (also called ‘flag ’ style) in which a proof consists of lines that are grouped together in nested boxes. In the world of proof assistants these two kinds of natural deduction correspond to procedural proofs (tactic scripts that work on one or more subgoals, like those of the Coq, HOL and PVS systems), and declarative proofs (like those of the Mizar and Isabelle/Isar languages). In this paper we give an algorithm for converting tree style proofs to flag style proofs. We then present a rewrite system that simplifies the results. This algorithm can be used to convert arbitrary procedural proofs to declarative proofs. It does not work on the level of the proof terms (the basic inferences of the system), but on the level of the statements that the user sees in the goals when constructing the proof. The algorithm from this paper has been implemented in the ProofWeb interface to Coq. In ProofWeb a proof that is given as a Coq proof script (even with arbitrary Coq tactics) can be displayed both as a tree style and as a flag style proof. 1