Scheme-based Theorem Discovery and Concept Invention

This book describes an approach to automatically invent/explore new mathematical theories, with the goal of producing results comparable to those produced by humans, as represented, for example, in the libraries of proof assistants. The approach described is based on schemes, which are formulae in higher-order logic. It shows that it is possible to automate the instantiation process of schemes to generate conjectures and definitions. It also shows how the new definitions and the lemmata discovered during the exploration of a theory can be used, not only to help with the proof obligations during the exploration, but also to reduce redundancies inherent in most theory-formation systems. It describes how to exploit associative-commutative (AC) operators using ordered rewriting to avoid AC variations of the same instantiation. All ideas contained in this book are implemented in an automated tool, called IsaScheme, which employs Knuth-Bendix completion and recent automatic inductive proof methods. This systematic and comprehensive introduction to the scheme-based theory exploration will be welcome by researchers and graduate students alike.