Extraction2Extracting ML from Coq, Part 2

(* $Date: 2013-01-16 22:29:57 -0500 (Wed, 16 Jan 2013) $ *)

Let's take a look at a slightly fancier way of using Coq's extraction facilities. Instead of translating a functional program in Coq into a functional program in OCaml, we'll extract the computational content of an interesting proof — the proof that the STLC is normalizing. The result will be an evaluator that takes well-typed STLC terms to their final normal forms.
Note that this is a slightly unusual way of using Coq. In the words of Adam Chlipala...
Many fans of the Curry-Howard correspondence support the idea of extracting programs from proofs. In reality, few users of Coq and related tools do any such thing. Instead, extraction is better thought of as an optimization that reduces the runtime costs of expressive typing. (Chlipala, CPDT)
However, it's interesting to see that it's possible at all! (And to think about what the extracted program does.)

Extracting a Normalizer

Require Import Extraction.
Require Import NormInType.

The NormInType module is a variant of the Norm module with a few significant differences. The essential point is that, during extraction, everything to do with Prop is erased. So, to extract a normalizer from a proof of normalization, we need to carry out the essential bits of the normalization proof in Type rather than Prop.
NormInType also incorporates a copy of the STLC typechecking function from the Typechecking module and its proof of correctness. (The function itself is no different from before, and the correctness proof has just a few small differences because of the changes we made to the basic definitions of STLC.) We need these things because the proof of normalization proceeds by induction on a typing derivation, so the extracted normalization function must be passed a data structure representing a typing derivation. In normdriver.ml, we obtain this derivation from the proof that the typechecking algorithm is sound.

Extraction "norm.ml" normalization type_check type_checking_sound.

Take a look at normdriver.ml to see how this plumbing works in detail.
Finally, we can compile and execute our normalizer in the same way as we did with our evaluator for Imp.
	ocamlc -w -20 -o normdriver norm.mli norm.ml normdriver.ml