Mechanically Verified Calculational Abstract Interpretation

Abstract: Calculational abstract interpretation, long advocated by Cousot, is a technique for deriving correct-by-construction abstract interpreters from the formal semantics of programming languages. This paper addresses the problem of deriving correct-by-verified-construction abstract interpreters with the use of a proof assistant. We identify several technical challenges to overcome with the aim of supporting verified calculational abstract interpretation that is faithful to existing pencil-and-paper proofs, supports calculation with Galois connections generally, and enables the extraction of verified static analyzers from these proofs. To meet these challenges, we develop a theory of Galois connections in monadic style that include a specification effect. Effectful calculations may reason classically, while pure calculations have extractable computational content. Moving between the worlds of specification and implementation is enabled by our metatheory. To validate our approach, we give the first mechanically verified proof of correctness for Cousot's "Calculational design of a generic abstract interpreter." Our proof "by calculus" closely follows the original paper-and-pencil proof and supports the extraction of a verified static analyzer.