Over the last month or so I've started to formalise the proofs of the DDC core language in Coq. The existing proofs were just getting too big to manage by hand, and typing them all up in Latex was seeming like a less and less fun thing to do.

Being a Coq-newbie I started with Benjamin Pierce and co's Software Foundations course notes. I had done a couple of pages of Coq tutorial before, but never tried to use it for anything real. I've found the course notes to provide a fairly smooth ride, though I got antsy during the IMP formalisation and skipped straight to the lambda calculus one. After about two weeks of evenings I was able to redo the simply typed lambda calculus (STLC) one by myself, but hit a wall trying to scale up to System-F. To cut a longer story short, I've come to appreciate (along with many others) what a massive hack capture avoiding substitution is.

I'm fast moving towards the viewpoint that the deBruijn and/or locally nameless representation is actually the "real" lambda calculus, whereas named binders are just a layer of syntactic sugar that confuses everything (and gives you cavities). I've ended up just settling on vanilla deBruijn indices for further work, mainly because the examples I've seen so far using the locally nameless representation rely on a heavy amount of Coq Ltac magic that I don't understand yet. I'm also more than willing to trade time for cleverness. By that I mean I'm willing to commit more time for a less clever solution, provided I know that the approach I take will get the job done and I won't have to start all over again using a different formalisation of binders.

Underneath all of this is an itch in my mind saying that I was already writing a perfectly good paper about the DDC core language, and I don't want to start another one right now about "How to formalise DDC Core in Coq". Anyway, I've made it through Progress and Preservation for STLC using the deBruijn representation, including all the list lemmas, and am feeling fairly good about it so far. The proofs are in the main DDC tree at http://code.ouroborus.net/ddc/ddc-head/proof/.

Easy tickets to get started with: on the trac.

## Tuesday, April 19, 2011

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Twelf will completely change your mind about variable binding. HOAS makes dealing with names much easier. For proofs with low proof-theoretic complexity (like progress and preservation, but not like strong normalization), I think Twelf is a much more intuitive route.

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