Automating the Proofs of Strengthening Lemmas in the Abella Proof Assistant

Abstract

In logical reasoning, it is often the case that only some of a collection of assumptions are needed to reach a conclusion. A strengthening lemma is an assertion that a given conclusion is independent in this sense of a particular assumption. Strengthening lemmas underlie many useful techniques for simplifying proofs in automated and interactive theorem-provers. For example, they underlie a mechanism called subordination that is useful in determining that expressions of a particular type cannot contain objects of another type and in thereby reducing the number of cases to be considered in proving universally quanti ed statements. This thesis concerns the automation of the proofs of strengthening lemmas in a speci cation logic called the logic of hereditary Harrop formulas (HOHH). The Abella Proof Assistant embeds this logic in a way that allows it to prove properties of both the logic itself and of speci cations written in it. Previous research has articulated a (conservative) algorithm for checking if a claimed strengthening lemma is, in fact, true. We provide here an implementation of this algorithm within the setting of Abella. Moreover, we show how to generate an actual proof of the strengthening lemma in Abella from the information computed by the algorithm; such a proof serves as a more trustworthy certifi cate of the correctness of the lemma than the algorithm itself. The results of this work have been incorporated into the Abella system in the form of a \tactic command" that can be invoked within the interactive theorem-prover and that will result in an elaboration of a proof of the lemma and its incorporation into the collection of proven facts about a given speci cation.