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Abstract. Heterogeneous specification becomes more and more important because complex systems are often specified using multiple viewpoints, involving multiple formalisms. Moreover, a formal software development process may lead to a change of formalism during the development. However, current research in integrated formal methods only deals with ad-hoc integrations of different formalisms. The heterogeneous tool set (Hets) is a parsing, static analysis and proof management tool combining various such tools for individual specification languages, thus providing a tool for heterogeneous multi-logic specification. Hets is based on a graph of logics and languages (formalized as so-called institutions), their tools, and their translations. This provides a clean semantics of heterogeneous specification, as well as a corresponding proof calculus. For proof management, the calculus of development graphs (known from other large-scale proof management systems) has been adapted to heterogeneous specification. Development graphs provide an overview of the (heterogeneous) specification module hierarchy and the current proof state, and thus may be used for monitoring the overall correctness of a heterogeneous development. 1

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In the domain of qualitative constraint reasoning, a subfield of AI which has evolved in the past 25 years, a large number of calculi for efficient reasoning about spatial and temporal entities has been developed. Reasoning techniques developed for these constraint calculi typically rely on so-called composition tables of the calculus at hand, which allow for replacing semantic reasoning by symbolic operations. Often these composition tables are developed in a quite informal, pictorial manner and hence composition tables are prone to errors. In view of possible safety critical applications of qualitative calculi, however, it is desirable to formally verify these composition tables. In general, the verification of composition tables is a tedious task, in particular in cases where the semantics of the calculus depends on higher-order constructs such as sets. In this paper we address this problem by presenting a heterogeneous proof method that allows for combining a higherorder proof assistance system (such as Isabelle) with an automatic (first order) reasoner (such as SPASS or VAMPIRE). The benefit of this method is that the number of proof obligations that is to be proven interactively with a semi-automatic reasoner can be minimized to an acceptable level.

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 graph-based approaches, such as HITS, or Hyperlink-Induced 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.

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The year 2004 marks 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 proof-checked by a computer. Classical theorem proving procedures of today are based on ingenious search techniques to find a proof for a given theorem in very large search spaces – often in the range of several billion clauses. But in spite of many successful attempts to prove even open mathematical problems automatically, their use in everyday mathematical practice is still limited. The shift

Over the last decade several environments and formalisms for the combination and integration of mathematical software systems have been proposed. Many of these systems aim at a traditional automated theorem proving approach, in which a given conjecture is to be proved or refuted by the cooperation of different reasoning engines. However, they offer little support for experimental mathematics in which new conjectures are constructed by an interleaved process of model computation, model inspection, property conjecture and verification. In particular, despite some previous research in that direction, there are currently no systems available that provide, in an easy to use environment, the flexible combination of diverse reasoning system in a plug-and-play fashion via a high level specification of experiments. [2, 3] presents an integration of more than a dozen different reasoning systems — first order theorem provers, SAT solvers, SMT solvers, model generators, computer algebra, and machine learning systems — in a general bootstrapping algorithm to generate novel theorems in the specialised algebraic domain of

This position paper discusses various issues concerning requirements and design of proof assistant user interfaces (UIs). After a review of some of the difficulties faced by UI projects in academia, it presents a high-level description of proof assistant interaction. This is followed by an exposition of use cases and object identification. Several examples demonstrate the usefulness of these requirement elicitation techniques in the theorem proving domain. The second half of the paper begins with a consideration of the “principle of least effort ” for the design of theorem prover user interfaces. This is followed by a brief review of the “GUI versus text mode ” debate, proposals for better use of GUI facilities and a plea for better support of customisation. The paper ends with a discussion of architecture and system design issues. In particular, it argues for a platform architecture with an extensible set of components and the use of XML protocols for communication between UIs and proof assistant backends.

FoC is a computer algebra library with a strong emphasis on formal certification of its algorithms. We present in this article our work on the link between the FoC language and OMDoc, an emerging XML standard to represent and share mathematical contents. On the one hand, we focus on the elaboration of the documentation system FoCDoc. After an analysis of an OMDoc approach of the documentation we present our own XML implementation (FoCDoc) and how we generate, from a FoC program, documentation files in HTML (MathML), LaTEX and OMDoc. On the other