Lambda the Ultimate

How to Make Ad Hoc Proof Automation Less Ad Hoc
Georges Gonthier, Beta Ziliani, Aleksandar Nanevski, and Derek Dreyer, to appear in ICFP 2011

Most interactive theorem provers provide support for some form of user-customizable proof automation. In a number of popular systems, such as Coq and Isabelle, this automation is achieved primarily through tactics, which are programmed in a separate language from that of the prover’s base logic. While tactics are clearly useful in practice, they can be difficult to maintain and compose because, unlike lemmas, their behavior cannot be specified within the expressive type system of the prover itself.

We propose a novel approach to proof automation in Coq that allows the user to specify the behavior of custom automated routines in terms of Coq’s own type system. Our approach involves a sophisticated application of Coq’s canonical structures, which generalize Haskell type classes and facilitate a flexible style of dependently-typed logic programming. Specifically, just as Haskell type classes are used to infer the canonical implementation of an overloaded term at a given type, canonical structures can be used to infer the canonical proof of an overloaded lemma for a given instantiation of its parameters. We present a series of design patterns for canonical structure programming that enable one to carefully and predictably coax Coq’s type inference engine into triggering the execution of user-supplied algorithms during unification, and we illustrate these patterns through several realistic examples drawn from Hoare Type Theory. We assume no prior knowledge of Coq and describe the relevant aspects of Coq type inference from first principles.

If you've ever toyed with Coq but run into the difficulties that many encounter in trying to construct robust, comprehensible proof scripts using tactics, which manipulate the proof state and can leave you with the "ground" of the proof rather than the "figure," if you will, in addition to being fragile in the face of change, you may wish to give this a read. It frankly never would have occurred to me to try to turn Ltac scripts into lemmas at all. This is much more appealing than most other approaches to the subject I've seen.