Automatic Generation of Simple Lemmas from Recursive Definitions Using Decision Procedures – Preliminary Report –

Abstract

Using recent results on integrating induction schemes into decidable theories, a method for generating lemmas useful for reasoning about $\mathcal{T}$-based function definitions is proposed. The method relies on terms in a decidable theory admitting a (finite set of) canonical form scheme(s) and ability to solve parametric equations relating two canonical form schemes with parameters. Using nontrivial examples, it is shown how the method can be used to automatically generate many simple lemmas; these lemmas are likely to be found useful in automatically proving other nontrivial properties of $\mathcal{T}$-based functions, thus unburdening the user of having to provide many simple intermediate lemmas. During the formalization of a problem, after a user inputs $\mathcal{T}$-based definitions, the method can be employed in the background to explore a search space of possible conjectures which can be attempted, thus building a library of lemmas as well as false conjectures. This investigation was motivated by our attempts to automatically generate lemmas arising in proofs of generic, arbitrary data-width parameterized arithmetic circuits. The scope of applicability of the proposed method is broader, however, including generating proofs for proof-carrying codes, certification of proof-carrying code as well as in reasoning about distributed computation algorithms.

This research was partially supported by an NSF ITR award CCR-0113611.