Abstract not found

www.cs.mdx.ac.uk/imp Authors of traditional mathematical texts often have difficulty balancing the amount of contextual information and proof detail. We propose a simple hypermedia framework that can assist in the organisation and presentation of mathematical theorems and definitions. We describe the application of the framework to convert an existing course in general topology to a web-based set of materials. A pilot study of the materials indicated a high level of user satisfaction. We discuss further lines of investigation, in particular, the presentation of larger bodies of work. 1

New successes in dealing with set theories by means of state-of-the-art theoremprovers may ensue from terse and concise axiomatizations, such as can be moulded in the framework of the (fully equational) Tarski-Givant map calculus. In this paper we carry out this task in detail, setting the ground for a number of experiments. Key words: Set theory, relation algebras, first-order theorem-proving, algebraic logic. 1 Introduction Like other mature fields of mathematics, Set Theory deserves sustained efforts that bring to light richer and richer decidable fragments of it [5], general inference rules for reasoning in it [23, 2], effective proof strategies based on its domain-knowledge, and so forth. Advances in this specialized area of automated reasoning tend, in spite of their steadiness, to be slow compared to the overall progress in the field. Many experiments with set theories have hence been carried out with standard theorem-proving systems. Still today such experiments pose consider...

The theorem of Sylow is proved in Isabelle HOL. We follow the proof by Wielandt that is more general than the original and uses a non-trivial combinatorial identity. The mathematical proof is explained in some detail leading on to the mechanization of group theory and the necessary combinatorics in Isabelle. We present the mechanization of the proof in detail giving reference to theorems contained in an appendix. Some weak points of the experiment with respect to a natural treatment of abstract algebraic reasoning give rise to a discussion of the use of module systems to represent abstract algebra in theorem provers. Drawing from that, we present tentative ideas for further research into a section concept for Isabelle.

www.uclic.ucl.ac.uk/imp Abstract. Hierarchical proof presentations are ubiquitous within logic and computer science, but have made little impact on mathematics in general. The reasons for this are not currently known, and need to be understood if mathematical knowledge management systems are to gain acceptance in the mathematical community. We report on some initial experiments with three users of a set of web-based hierarchical proofs, which suggest that usability problems could be a factor. In order to better understand these problems we present a theoretical analysis of hierarchical proofs using Cognitive Dimensions [6]. The analysis allows us to formulate some concrete hypotheses about the usability of hierarchical proof presentations. 1

As a part of ongoing research on automated reasoning in set theory, we focus here on an example of a computer proof that involves a recursive definition.

his format recursively consists of a list of key proof steps, each justied by a subproof. The result is a hierarchy of nested lists of proof steps: the higher level steps give the outline of the proof, the lower level steps the details. The obvious adaptation to hypertext is to hide or reveal the subproofs under the reader's control. This was rst done by Grundy [1] for a calculational proof style. However, a major drawback is that key insights may be contained within a hidden subproof | but a reader has no way of knowing this at a glance. We believe that a more sophisticated approach should involve summarising the hidden parts. In particular, references to key steps, proof methods or imported results can allow the reader to understand the proof step without seeing the details. Incorporating Context Apart from understanding the logical argument of a proof, a reader also needs to understand its signicance. This comes from both the rationale for its su