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We examine a problem for machine supported metatheory. There are statements true about a theory that are true of some (but only some) extensions; however standard theory-structuring facilities do not support selective inheritance. We use the example of the deduction theorem for modal logic and show how a statement about a theory can explicitly formalize the closure conditions extensions should satisfy for it to remain true. We show how metatheories based on inductive denitions allow theories and general metatheorems to be organized this way, and report on a case study using the theory FS0 . 1 Introduction Hierarchical theory structuring plays an important role in the application of theorem provers to nontrivial problems, and many systems provide support for it. For example HOL [6], Isabelle [13] and their predecessor LCF [7] support simple theory hierarchies. In these systems a theory is a specication of a language, using types and typed constants, and a collection of rules...

. This paper introduces the imps proof script mechanism and some practical methods for exploiting it. 1 Introduction imps, an Interactive Mathematical Proof System [4, 2], is intended to serve three ultimate purposes: { To provide mathematics education with a mathematics laboratory for students to develop axiomatic theories, proofs, and rigorous methods of symbolic computation. { To provide mathematical research with mechanized support covering a range of concrete and abstract mathematics, eventually with the help of a large theory library of formal mathematics. { To allow applied formal methods to use exible approaches to formalizing problem domains and proof techniques, in showing software or hardware correctness. Thus, the goal of imps is to provide mechanical support for traditional methods and activities of mathematics, and for traditional styles of mathematical proof. Other automated theorem provers may be intended for quite dierent sorts of problems, and they can theref...

In this paper an outline is given of an approach to formally reasoning about importation, parameterisation and instantiation of specifications written in a modular extension of the Z language (called Sum). Interpretation and instantiation of theories in first order logic are well understood. We illustrate how to use these results directly to provide a framework within which we can soundly and efficiently reason about modular specifications. A reasoning environment within the Ergo 4:1 theorem prover has been constructed that provides the theory management, construction and extension facilities needed to support such a reasoning process. Sum specifications are mapped to the appropriate Ergo structures by a straightforward translation process. A simple example in Sum is presented to demonstrate the use of these theory extension mechanisms. As far as the authors are aware, no other system offers interpreted automated support for reasoning about parameterisation and instantiation of modular...

A first step in systematically engineering better interfaces for theorem proving assistants (TPAs) is to assess what has already been achieved in the domain. We examine three TPAs employing quite different styles of interaction. We consider the support provided by the interfaces for each of four mechanisms for efficient interactive proof: planning, reuse, reflection and articulation. Common themes are observed, as are strengths and weaknesses of the interfaces and we discuss the general issues, attempting to abstract away from the particular artifacts studied. 1 Introduction As formal methods unburden themselves of notions of uniquitous proof, recognition is growing for the value of formal proof in certain areas, for example verifying algorithms, see [7] and exploring properties of complex interactions, see [17]. Furthermore the tools necessary to check and to manage large and complicated proofs are emerging allowing significant proofs to be performed interactively. Such theorem provi...

Abstract. In this contribution we address two related questions. Firstly, we want to shed light on the question how to use a representation formalism to represent a given problem. Secondly, we want to find out how different formalizations are related and in particular how it is possible to check that one formalization entails another. The latter question is a tough nut for mathematical knowledge management systems, since it amounts to the question, how a system can recognize that a solution to a problem is already available, although possibly in disguise. As our starting point we take McCarthy’s 1964 mutilated checkerboard challenge problem for proof procedures and compare some of its different formalizations. 1

Locales are the theory development modules of the Isabelle proof assistant. Interpretation is a powerful technique of theorem reuse which facilitates their automatic transport to other contexts. This paper is concerned with the interpretation of locales in the context of other locales. Our main concern is to make interpretation an effective tool in an interactive proof environment. Interpretation dependencies between locales are maintained explicitly, by means of a development graph, so that theorems proved in one locale can be propagated to other locales that interpret it. Proof tools in Isabelle are controlled by sets of default theorems they use. These sets are required to be finite, but can become infinite in the presence of arbitrary interpretations. We show that finiteness can be maintained.

Introduction An important problem for automated reasoning is to develop mechanisms for integration of special purpose reasoning systems into general purpose reasoning systems, and into other systems. In this abstract we first point out some of the problems to be solved in order to have reasoning modules that can be effectively integrated, and suggest the notion of a logical service as a way of viewing integrable reasoning modules. We then present some preliminary results from ongoing work aimed at gaining a better understanding of this notion. Finally, we illustrate some of the ideas by describing a series of reasoners specializing in linear arithmetic using the concepts developed. What sort of reasoning components can usefully plug and play? A yes/no type black box decider is not adequate in general. In order for reasoning modules to be usefully integrated they must be able to interact with other components in non-trivial ways. Such modules may need to accept information incr

In this paper we develop a formal mathematical model for processes, along the lines of Communicating Sequential Processes (CSP). Our development is completely general and is capable of dealing with time. Rather than building semantic models on a traditional trace-based failures model, we replace the set of traces by the set of elements of a monoid. In the case of untimed CSP, this monoid is taken to be the set of all event sequences under concatenation; mathematically this is the free monoid generated by the event alphabet. For timed CSP, one natural monoid that may be considered is the free monoid generated by the set of atomic events together with the wait operators W (t) for t a real number with the relations W (t + s) = W (t)W (s): Supported by the United States Army cecom under contract DAAB07-94-C-H601, through MITRE's Technology Program. Author's address: The MITRE Corporation, 202 Burlington Rd, Bedford MA 01730-1420 USA; Telephone: 617-271-2749; Fax: 617-2713816; E-mail: jt...

IPR is an automatic theorem-proving system intended particularly for use in higher-level mathematics. It discovers the proofs of theorems in mathematics applying known theorems and definitions. Theorems and definitions are stored in the knowledge base in the form of sequents rather than formulas or rewrite rules. Because there is more easily-accessible information in a sequent than there is in the formula it represents, a simple algorithm can be used to search the knowledge base for the most useful theorem or definition to be used in the theorem-proving process. This paper describes how the sequents in the knowledge base are formed from theorems stated by the user and how the knowledge base is used in the theorem-proving process. An example of a theorem proved and the English proof output are also given. 1 Introduction The motivating goal behind this work is to develop a theorem-proving system which will be useful to both an expert and a non-expert in the attempt to prove theorems in ...