. Mechanised reasoning systems and computer algebra systems have apparently different objectives. Their integration is, however, highly desirable, since in many formal proofs both of the two different tasks, proving and calculating, have to be performed. In the context of producing reliable proofs, the question how to ensure correctness when integrating a computer algebra system into a mechanised reasoning system is crucial. In this contribution, we discuss the correctness problems that arise from such an integration and advocate an approach in which the calculations of the computer algebra system are checked at the calculus level of the mechanised reasoning system. We present an implementation which achieves this by adding a verbose mode to the computer algebra system which produces high-level protocol information that can be processed by an interface to derive proof plans. Such a proof plan in turn can be expanded to proofs at different levels of abstraction, so the approach is well-...

. Running a competition for Automated Theorem Proving (ATP) systems is a difficult and arguable venture. However, the potential benefits of such an event by far outweigh the controversial aspects. The motivations for running the CADE-13 ATP System Competition were to contribute to the evaluation of ATP systems, to stimulate ATP research and system development, and to expose ATP systems to researchers both within and outside the ATP community. This paper identifies and discusses the issues that determine the nature of such a competition. Choices and motivated decisions for the CADE-13 competition, with respect to the issues, are given. Key words: Automated theorem proving, competition, design 1. Introduction Running a competition for Automated Theorem Proving (ATP) systems 1 is a difficult and arguable venture. The reasons for this are that existing ATP systems require different amounts of user interaction, are designed for different types of reasoning, are based on different logics...

Advances in the underlying theory of a subdiscipline of AI can result in an apparently impressive improvement in the performance of a system that incorporates the advance. This impression typically comes from observing

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ly logical meta-level characterizations of all admitted logical systems are necessary. These characterization would make use of meta-formulae of the kind formula j ("' j ") which stands for the fact that ' j is a formula in the formal system S j . Correspondingly predicates \Pi j ("\Delta ` j \Gamma"; ß j ), standing for ß j is a proof for \Delta ` j \Gamma in the formal system S j , can be employed. Of course, the meta-language has to be rich enough that the notions of formula and proof can be defined in it. For instance, if we have some derivability relations like \Delta 1 ` j \Gamma 1 ; : : : ; \Delta n ` j \Gamma n \Delta ` j \Gamma RULE j k and ; \Delta ` j \Gamma AXIOM j k we have the following formulae in the meta-language in order to axiomatize the notion of proof \Pi j ("h\Delta 1 i ` j h\Gamma 1 i"; ß 1 ) : : : \Pi j ("h\Delta n i ` j h\Gamma n i"; ß n ) ! \Pi j ("h\Deltai ` j h\Gammai"; RULE j k (ß 1 ; : : : ; ß n )) \Pi...

In this contribution I advocate an open system for formalised mathematical reasoning that is able to capture different mathematical formalisms as well as a wide variety of proof formats. This is assumed to be much more adequate since it more closely reflects the situation in mathematics as a whole. In particular I try to classify different dimension according to which proofs can be classified. 1 Introduction One of the ultimate goals of the mechanisation of proofs is to achieve an increased rigour in proof. Mathematics generally enjoys the prestige of being the correct scientific discipline par excellence. This reputation stems from the requirement that every claim must be justified by a rigorous proof. The ultimate goal of many mechanised reasoning systems is to support mathematicians in the task of constructing such a proof. This is not trivial, since in traditional mathematical practice, proofs are not given in terms of single calculus rules but at a level of abstraction that conv...

THINKER is an automated natural deduction first-order theorem proving program. This paper reports on how it was adapted so as to prove theorems in modal logic. The method employed is an "indirect semantic method", obtained by considering the semantic conditions involved in being a valid argument in these modal logics. The method is extended from propositional modal logic to predicate modal logic, and issues concerning the domain of quantification and "existence in a world's domain " are discussed. Finally, we took at the very interesting issues involved with adding identity to the theorem prover in the realm of modal predicate logic. Various alternatives are discussed.

In this paper we discuss the importance of good representations in mathematics and relate them to general design issues. Good design makes life easy, bad design difficult. For this reason experienced mathematicians spend a significant amount of their time on the design of their concepts. While many formal systems try to support this by providing a high-level language, we argue that more should be learned from the mathematical practice in order to improve the applicability of formal systems.