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 al- lows 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.

A new revised and extended version of our ICFP paper is here. But if you are still looking for for the conference submission, it is here (with its appendix).

The files are now part of the user's contrib under the name of LemmaOverloading.

You can still download the code here for Coq 8.4 and Ssreflect 1.4. In case you have Coq 8.3pl1 and Ssreflect 1.3pl1, you can download the code here. In case you have Coq 8.3 and Ssreflect 1.3, you can download the old code from here. For any problem compiling the files, please do not hesitate in contacting Beta.