Wednesday, May 25, 2011

Proofs and mutual recursion

Quick update. I'm still working on mechanising the proofs for the DDC core language in Coq. So far I've made it through Progress, Preservation and the correspondence between Big and Small step semantics for Simply Typed Lambda Calculus (STLC), System-F, System-F2, and PCF. Extending the STLC proof with polymorphism and fixpoints etc was surprisingly straightforward. It only took about 4 hours each to create the other versions of the proof.

The next qualitative hurdle appears to be handling the mutual recursion that arises when we extend the language in other ways. To add case expressions, I need to represent case alternatives. Case alternatives contain expressions, and expressions contain (lists of) alternatives.

Unfortunately, with mutually recursive definitions, it appears that Coq can't automatically generate all the induction schemes I need. I could be wrong on this though, as I'm still a Coq newbie. I've figured out that the Coq command "Combined Schemes", will make some of the schemes I might want, but it doesn't cover all of the use-cases. The problem (I think) is that although expressions are defined in terms of lists of alternatives, the lists themselves are polymorphic and aren't directly defined in terms of alternatives. For this reason, I can't convince "Combined Scheme" to make me a strong enough induction scheme, so I must write down and prove this myself.

At least, that's what I'm inferring from reading the proof of soundness for the STG machine described in a paper by Maciej Pirog and Dariusz Biernacki. I'm still working through it.

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