While it took, as they say, a bit of *doing*, I have a completed draft of a paper on my website that I believe provides a really elegant solution to what has been a very, very annoying problem for some time: writing down a proof called the *completeness of focusing*. Don't worry, the paper explains what that means: it has to, because one of the things the paper argues is that most existing statements of the completeness of focusing aren't general enough! In the process, this writeup gave me a good (enough) excuse to talk in a very general way about a lot of fundamental phenomena around the idea of polarization in logic that we've been noticing in recent years.

Anyway, the draft is here - Structural Focalization. The accompanying Twelf development is, of course, on the Twelf wiki: Focusing. I've thrown the paper and the Twelf code up on arXiv while I figure out while I figure out what to do with it; comments would be greatly appreciated!

I don't want to have a post that's just "hey here's a paper," so here's an addition, a follow-up to the rather popular "Totally Nameless Representation" post from awhile back. I still prove weakening in Agda by exactly the technique presented there, which commenter thecod called the "presheaf method." But for natural deduction systems, I almost never use the term metric approach that was the core of what I was presenting in "totally nameless representation," since it works just fine to use the approach where term `M` of type `A` (the one you're substituting for `x`) is in context `Γ` and the term `N` (the one with `x` free) is in context `Γ' ++ A :: Γ` - the free variable `x` is allowed to be in the middle of the context, in other words; you don't have to use exchange on the second term, so you're always calling induction on something Agda recognizes as structurally smaller.

This worked for natural deduction systems, but I didn't think it would work for sequent calculi, since you needed to toy around with both contexts and induct on both terms in presentation of cut for a sequent calculus. However, for a focused sequent calculus like what I present in the paper, you still can do without the totally nameless metric! If you set things up right, the *rightist* substitutions (where you work on decomposing the right term, the one with the variable free) allow you to extend the context only of the first term, and the *leftist* substitutions (where you work on decomposing the left term, the one you're substituting for the variable) allow you to work on the second term, and the two are only connected by the *principal* substitutions (which reduce the type, which more or less lets you get away with anything as far as the induction principle is concerned).

A code example of this, which could use some more commentary, can be found on GitHub: Polar.agda.