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Coq Extraction
What is it ?
The extraction is a mechanism allowing to produce functional code
(Ocaml/Haskell/Scheme) out of any Coq terms (either programs or
Who did it ?
The current implementation (from version 7.0 up to now) has been done
by P. Letouzey during his PhD, helped by J.C. Filliâtre and supervised
by C. Paulin.
An earlier implementation (versions 6.x) was due to B. Werner and
C. Paulin.
Where can we find more information ?
- Coq Reference Manual includes a full chapter about extraction
- P. Letouzey's PhD thesis [3] forms a complete document about
both theory and implementation and test-cases of Coq-extraction
- A more recent article [4] proposes a short overview of extraction
- earlier documents [1] [2] may also be useful.
Why a complete re-implementation ?
Extraction code has been completely rewritten since version V6.3.
1) Principles
The main goal of the new extraction is to handle any Coq term, even
those upon sort Type, and to produce code that always compiles.
Thus it will never answer something like "Not an ML type", but rather
a dummy term like the ML unit.
Translation between Coq and ML is based upon the following principles:
- Terms of sort Prop don't have any computational meaning, so they are
merged into one ML term "__". This part is done according to P. Letouzey's
works [1] and [2].
This dummy constant "__" used to be implemented by the unit (), but
we recently found that this constant might be applied in some cases.
So "__" is now in Ocaml a fixpoint that forgets its arguments:
let __ = let rec f _ = Obj.repr f in Obj.repr f
- Terms that are type schemes (i.e. something of type ( : )( : )...s with
s a sort ) don't have any ML counterpart at the term level, since they
are types transformers. In fact they do not have any computational
meaning either. So we also merge them into that dummy term "__".
- A Coq term gives a ML term or a ML type depending of its type:
type schemes will (try to) give ML types, and all other terms give ML terms.
And the rest of the translation is (almost) straightforward: an inductive
gives an inductive, etc...
This gives ML code that have no special reason to typecheck, due
to the incompatibilities between Coq and ML typing systems. In fact
most of the time everything goes right.
We now verify during extraction that the produced code is typecheckable,
and if it is not we insert unsafe type casting at critical points in the
code, with either "Obj.magic" in Ocaml or "unsafeCoerce" in Haskell.
2) Differences with previous extraction (V6.3 and before)
2.a) The pros
The ability to extract every Coq term, as explain in the previous
The ability to extract from a file an ML module (cf Extraction Library in the
You can have a taste of extraction directly at the toplevel by
using the "Extraction <ident>" or the "Recursive Extraction <ident>".
This toplevel extraction was already there in V6.3, but was printing
Fw terms. It now prints in the language of your choice:
Ocaml, Haskell or Scheme.
The optimization done on extracted code has been ported between
V6.3 and V7 and enhanced, and in particular the mechanism of automatic
2.b) The cons
The presence of some parasite "__" as dummy arguments
in functions. This denotes the rests of a proof part. The previous
extraction was able to remove them totally. The current implementation
removes a good deal of them, but not all.
This problem is due to extraction upon Type.
For example, let's take this pathological term:
(if b then Set else Prop) : Type
The only way to know if this is an Set (to keep) or a Prop (to remove)
is to compute the boolean b, and we do not want to do that during
There is no more "ML import" feature. You can compensate by using
Axioms, and then "Extract Constant ..."
Exécution de termes de preuves: une nouvelle méthode d'extraction
pour le Calcul des Constructions Inductives, Pierre Letouzey,
DEA thesis, 2000,
A New Extraction for Coq, Pierre Letouzey,
Types 2002 Post-Workshop Proceedings.
Programmation fonctionnelle certifiée: l'extraction de programmes
dans l'assistant Coq. Pierre Letouzey, PhD thesis, 2004.
Coq Extraction, An overview. Pierre Letouzey. CiE2008.
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