concl.tex

In this paper we have shown how to do type checking for a dependently typed
logic extended with meta-variables. To maintain the important invariant that
terms being evaluated are type correct we work with well-typed approximations
of terms, where potentially ill-typed subterms have been replaced by constants.
We showed that type checking is decidable, that the algorithm is sound and that
the approximated terms are optimal.

We present the type checking algorithm for a simple dependently typed logical
framework {\Core}, but it can be extended to more advanced logics.  This is
evidenced by the fact that we have implemented the algorithm for the
Agda language, supporting for instance, definitions by pattern matching, a
hierarchy of universes and constants with variable arity. The algorithm has
proven to work well with examples of several hundred meta-variables.

There are two main directions of future work. First extending the correctness
proof to a more feature-rich logic. Much of this work has already been done
in the implementation but some work remains in working out the details of the
proofs. The other direction of future work is to build on top of this
algorithm. For instance, a system for implicit arguments or Alf-style
interaction\cite{magnussonnordstrom:alf}.