Protected access using lightweight capabilities

The following is some SML code that implements the natural numbers using peano arithmetic in the SML type system. These numbers can be used to protect access to list functions. I haven't yet figured out if addition is possible, but I'm hoping that it is. It could be really handy!

```signature SLIST =
sig
type 'a nat
type Z
type 'a S
val S : 'a nat -> 'a S nat
val P : 'a S nat -> 'a nat
val Z : unit -> Z nat
val zerop : 'a nat -> bool
val toInt : 'a nat -> int

type ('elt,'n) slist

val snil : unit -> ('elt, Z nat) slist
val scons : 'elt -> ('elt, 'n nat) slist -> ('elt, 'n S nat) slist
val :+: : 'elt * ('elt, 'n nat) slist -> ('elt, 'n S nat) slist
val shd : ('elt, 'n S nat) slist -> 'elt
val stl : ('elt, 'n S nat) slist -> ('elt, 'n nat) slist
val slen : ('elt, 'n S nat) slist -> int

end

structure SList :> SLIST =
struct
(* encode integer types *)
type 'a nat = int
type Z = unit
type 'a S = unit
fun S i = i+1;
fun P i = i-1;
fun Z () = 0
fun zerop 0 = true
| zerop _ = false
fun toInt d = d

type ('elt, 'n) slist = 'elt list * 'n

fun snil () = ([],0)

fun scons elt sl =
let val (l,i) = sl
in ((elt::l),S i)
end

infixr :+:
fun x :+: y = scons x y

fun shd sl =
let val (h::t,i) = sl
in h
end

fun stl sl  =
let val (h::t,i) = sl
in (t,P i)
end

fun slen sl =
let val (_,i) = sl
in i
end

end

open SList
infixr :+:

val mylist = 1 :+: 2 :+: 3 :+: snil();
val the_tail = stl mylist;
(*
This doesn't even compile!
*)
```

Managing state in Prolog monadically, using DCGs.

Prolog is a beautiful language which makes a lot of irritating rudimentary rule application and search easy. I have found it is particularly nice when trying to deal with compilers which involve rule based transformation from a source language L to a target language L'.

However, the management of these rules generally requires keeping track of a context, and this context has to be explicitly threaded through the entire application, which involves a lot of irritating and error prone sequence variables. This often leads to your code looking something a bit like this:

compile(seq(a,b),(ResultA,ResultB),S0,S2) :- compile(a,ResultA,S0,S1), compile(b,ResultB,S1,S2).
While not the worst thing, I've found it irritating and ugly, and I've made a lot of mistakes with incorrectly sequenced variables. It's much easier to see sequence made explicitly textually in the code.

While they were not designed for this task, but rather for parsing, DCGs turn out to be a convenient …

Generating etags automatically when needed

Have you ever wanted M-. (the emacs command which finds the definition of the term under the cursor) to just "do the right thing" and go to the most current definition site, but were in a language that didn't have an inferior process set-up to query about source locations correctly (as is done in lisp, ocaml and some other languages with sophisticated emacs interfaces)?

Well, fret no more. Here is an approach that will let you save the appropriate files and regenerate your TAGS file automatically when things change assuring that M-. takes you to the appropriate place.

You will have to reset the tags-table-list or set it when you first use M-. and you'll want to change the language given to find and etags in the 'create-prolog-tags function (as you're probably not using prolog), but otherwise it shouldn't require much customisation.

And finally, you will need to run etags once manually, or run 'M-x create-prolog-tags' in order to get the initia…

Decidable Equality in Agda

So I've been playing with typing various things in System-F which previously I had left with auxiliary well-formedness conditions. This includes substitutions and contexts, both of which are interesting to have well typed versions of. Since I've been learning Agda, it seemed sensible to carry out this work in that language, as there is nothing like a problem to help you learn a language.

In the course of proving properties, I ran into the age old problem of showing that equivalence is decidable between two objects. In this particular case, I need to be able to show the decidability of equality over types in System F in order to have formation rules for variable contexts. We'd like a context Γ to have (x:A) only if (x:B) does not occur in Γ when (A ≠ B). For us to have statements about whether two types are equal or not, we're going to need to be able to decide if that's true using a terminating procedure.

And so we arrive at our story. In Coq, equality is som…