Scheme-Based Synthesis of Inductive Theories
We describe 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 the Isabelle proof assistant. Our approach is
based on ‘schemes’, which are terms in higher-order logic. We show that it is possible to automate the instantiation process
of schemes to generate conjectures and definitions. We also show how the new definitions and the lemmata discovered during
the exploration of the 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. We implemented our ideas in an automated tool, called IsaScheme,
which employs Knuth-Bendix completion and recent automatic inductive proof tools. We have evaluated our system in a theory
of natural numbers and a theory of lists.