| Authors: | Jean-Christophe Filliâtre and Pierre Letouzey |
|---|
We present here the |Coq| extraction commands, used to build certified and relatively efficient functional programs, extracting them from either |Coq| functions or |Coq| proofs of specifications. The functional languages available as output are currently |OCaml|, Haskell and Scheme. In the following, "ML" will be used (abusively) to refer to any of the three.
Before using any of the commands or options described in this chapter,
the extraction framework should first be loaded explicitly
via Require Extraction, or via the more robust
From Coq Require Extraction.
Note that in earlier versions of Coq, these commands and options were
directly available without any preliminary Require.
.. coqtop:: in Require Extraction.
Note
In the following, a qualified identifier :token:`qualid` can be used to refer to any kind of |Coq| global "object" : constant, inductive type, inductive constructor or module name.
The next two commands are meant to be used for rapid preview of extraction. They both display extracted term(s) inside |Coq|.
.. cmd:: Extraction @qualid Extraction of the mentioned object in the |Coq| toplevel.
.. cmd:: Recursive Extraction {+ @qualid }
Recursive extraction of all the mentioned objects and
all their dependencies in the |Coq| toplevel.
All the following commands produce real ML files. User can choose to produce one monolithic file or one file per |Coq| library.
.. cmd:: Extraction @string {+ @qualid }
Recursive extraction of all the mentioned objects and all
their dependencies in one monolithic file :token:`string`.
Global and local identifiers are renamed according to the chosen ML
language to fulfill its syntactic conventions, keeping original
names as much as possible.
.. cmd:: Extraction Library @ident Extraction of the whole |Coq| library :n:`@ident.v` to an ML module :n:`@ident.ml`. In case of name clash, identifiers are here renamed using prefixes ``coq_`` or ``Coq_`` to ensure a session-independent renaming.
.. cmd:: Recursive Extraction Library @ident Extraction of the |Coq| library :n:`@ident.v` and all other modules :n:`@ident.v` depends on.
.. cmd:: Separate Extraction {+ @qualid }
Recursive extraction of all the mentioned objects and all
their dependencies, just as :n:`Extraction @string {+ @qualid }`,
but instead of producing one monolithic file, this command splits
the produced code in separate ML files, one per corresponding Coq
``.v`` file. This command is hence quite similar to
:cmd:`Recursive Extraction Library`, except that only the needed
parts of Coq libraries are extracted instead of the whole.
The naming convention in case of name clash is the same one as
:cmd:`Extraction Library`: identifiers are here renamed using prefixes
``coq_`` or ``Coq_``.
The following command is meant to help automatic testing of
the extraction, see for instance the test-suite directory
in the |Coq| sources.
.. cmd:: Extraction TestCompile {+ @qualid }
All the mentioned objects and all their dependencies are extracted
to a temporary |OCaml| file, just as in ``Extraction "file"``. Then
this temporary file and its signature are compiled with the same
|OCaml| compiler used to built |Coq|. This command succeeds only
if the extraction and the |OCaml| compilation succeed. It fails
if the current target language of the extraction is not |OCaml|.
.. cmd:: Extraction Language {| OCaml | Haskell | Scheme }
:name: Extraction Language
The ability to fix target language is the first and more important
of the extraction options. Default is ``OCaml``.
Since |OCaml| is a strict language, the extracted code has to be optimized in order to be efficient (for instance, when using induction principles we do not want to compute all the recursive calls but only the needed ones). So the extraction mechanism provides an automatic optimization routine that will be called each time the user wants to generate an |OCaml| program. The optimizations can be split in two groups: the type-preserving ones (essentially constant inlining and reductions) and the non type-preserving ones (some function abstractions of dummy types are removed when it is deemed safe in order to have more elegant types). Therefore some constants may not appear in the resulting monolithic |OCaml| program. In the case of modular extraction, even if some inlining is done, the inlined constants are nevertheless printed, to ensure session-independent programs.
Concerning Haskell, type-preserving optimizations are less useful because of laziness. We still make some optimizations, for example in order to produce more readable code.
The type-preserving optimizations are controlled by the following |Coq| flags and commands:
.. flag:: Extraction Optimize Default is on. This controls all type-preserving optimizations made on the ML terms (mostly reduction of dummy beta/iota redexes, but also simplifications on Cases, etc). Turn this flag off if you want a ML term as close as possible to the Coq term.
.. flag:: Extraction Conservative Types Default is off. This controls the non type-preserving optimizations made on ML terms (which try to avoid function abstraction of dummy types). Turn this flag on to make sure that ``e:t`` implies that ``e':t'`` where ``e'`` and ``t'`` are the extracted code of ``e`` and ``t`` respectively.
.. flag:: Extraction KeepSingleton Default is off. Normally, when the extraction of an inductive type produces a singleton type (i.e. a type with only one constructor, and only one argument to this constructor), the inductive structure is removed and this type is seen as an alias to the inner type. The typical example is ``sig``. This flag allows disabling this optimization when one wishes to preserve the inductive structure of types.
.. flag:: Extraction AutoInline Default is on. The extraction mechanism inlines the bodies of some defined constants, according to some heuristics like size of bodies, uselessness of some arguments, etc. Those heuristics are not always perfect; if you want to disable this feature, turn this flag off.
.. cmd:: Extraction Inline {+ @qualid }
In addition to the automatic inline feature, the constants
mentioned by this command will always be inlined during extraction.
.. cmd:: Extraction NoInline {+ @qualid }
Conversely, the constants mentioned by this command will
never be inlined during extraction.
.. cmd:: Print Extraction Inline Prints the current state of the table recording the custom inlinings declared by the two previous commands.
.. cmd:: Reset Extraction Inline Empties the table recording the custom inlinings (see the previous commands).
Inlining and printing of a constant declaration:
The user can explicitly ask for a constant to be extracted by two means:
- by mentioning it on the extraction command line
- by extracting the whole |Coq| module of this constant.
In both cases, the declaration of this constant will be present in the produced file. But this same constant may or may not be inlined in the following terms, depending on the automatic/custom inlining mechanism.
For the constants non-explicitly required but needed for dependency reasons, there are two cases:
- If an inlining decision is taken, whether automatically or not, all occurrences of this constant are replaced by its extracted body, and this constant is not declared in the generated file.
- If no inlining decision is taken, the constant is normally declared in the produced file.
The following command provides some extra manual control on the code elimination performed during extraction, in a way which is independent but complementary to the main elimination principles of extraction (logical parts and types).
.. cmd:: Extraction Implicit @qualid [ {+ @ident } ]
This experimental command allows declaring some arguments of
:token:`qualid` as implicit, i.e. useless in extracted code and hence to
be removed by extraction. Here :token:`qualid` can be any function or
inductive constructor, and the given :token:`ident` are the names of
the concerned arguments. In fact, an argument can also be referred
by a number indicating its position, starting from 1.
When an actual extraction takes place, an error is normally raised if the :cmd:`Extraction Implicit` declarations cannot be honored, that is if any of the implicit arguments still occurs in the final code. This behavior can be relaxed via the following flag:
.. flag:: Extraction SafeImplicits Default is on. When this flag is off, a warning is emitted instead of an error if some implicit arguments still occur in the final code of an extraction. This way, the extracted code may be obtained nonetheless and reviewed manually to locate the source of the issue (in the code, some comments mark the location of these remaining implicit arguments). Note that this extracted code might not compile or run properly, depending of the use of these remaining implicit arguments.
Extraction will fail if it encounters an informative axiom not realized. A warning will be issued if it encounters a logical axiom, to remind the user that inconsistent logical axioms may lead to incorrect or non-terminating extracted terms.
It is possible to assume some axioms while developing a proof. Since these axioms can be any kind of proposition or object or type, they may perfectly well have some computational content. But a program must be a closed term, and of course the system cannot guess the program which realizes an axiom. Therefore, it is possible to tell the system what ML term corresponds to a given axiom.
.. cmd:: Extract Constant @qualid => @string Give an ML extraction for the given constant. The :token:`string` may be an identifier or a quoted string.
.. cmd:: Extract Inlined Constant @qualid => @string
Same as the previous one, except that the given ML terms will
be inlined everywhere instead of being declared via a ``let``.
.. note::
This command is sugar for an :cmd:`Extract Constant` followed
by a :cmd:`Extraction Inline`. Hence a :cmd:`Reset Extraction Inline`
will have an effect on the realized and inlined axiom.
Caution!
It is the responsibility of the user to ensure that the ML terms given to realize the axioms do have the expected types. In fact, the strings containing realizing code are just copied to the extracted files. The extraction recognizes whether the realized axiom should become a ML type constant or a ML object declaration. For example:
.. coqtop:: in Axiom X:Set. Axiom x:X. Extract Constant X => "int". Extract Constant x => "0".
Notice that in the case of type scheme axiom (i.e. whose type is an arity, that is a sequence of product finished by a sort), then some type variables have to be given (as quoted strings). The syntax is then:
.. cmdv:: Extract Constant @qualid {+ @string } => @string
:undocumented:
The number of type variables is checked by the system. For example:
.. coqtop:: in Axiom Y : Set -> Set -> Set. Extract Constant Y "'a" "'b" => " 'a * 'b ".
Realizing an axiom via :cmd:`Extract Constant` is only useful in the
case of an informative axiom (of sort Type or Set). A logical axiom
has no computational content and hence will not appear in extracted
terms. But a warning is nonetheless issued if extraction encounters a
logical axiom. This warning reminds user that inconsistent logical
axioms may lead to incorrect or non-terminating extracted terms.
If an informative axiom has not been realized before an extraction, a
warning is also issued and the definition of the axiom is filled with
an exception labeled AXIOM TO BE REALIZED. The user must then
search these exceptions inside the extracted file and replace them by
real code.
The system also provides a mechanism to specify ML terms for inductive types and constructors. For instance, the user may want to use the ML native boolean type instead of the |Coq| one. The syntax is the following:
.. cmd:: Extract Inductive @qualid => @string__1 [ {+ @string } ]
Give an ML extraction for the given inductive type. You must specify
extractions for the type itself (:n:`@string__1`) and all its
constructors (all the :n:`@string` between square brackets). In this form,
the ML extraction must be an ML inductive datatype, and the native
pattern matching of the language will be used.
When :n:`@string__1` matches the name of the type of characters or strings
(``char`` and ``string`` for OCaml, ``Prelude.Char`` and ``Prelude.String``
for Haskell), extraction of literals is handled in a specialized way, so as
to generate literals in the target language. This feature requires the type
designated by :n:`@qualid` to be registered as the standard char or string type,
using the :cmd:`Register` command.
.. cmdv:: Extract Inductive @qualid => @string [ {+ @string } ] @string
Same as before, with a final extra :token:`string` that indicates how to
perform pattern matching over this inductive type. In this form,
the ML extraction could be an arbitrary type.
For an inductive type with :math:`k` constructors, the function used to
emulate the pattern matching should expect :math:`k+1` arguments, first the :math:`k`
branches in functional form, and then the inductive element to
destruct. For instance, the match branch ``| S n => foo`` gives the
functional form ``(fun n -> foo)``. Note that a constructor with no
arguments is considered to have one unit argument, in order to block
early evaluation of the branch: ``| O => bar`` leads to the functional
form ``(fun () -> bar)``. For instance, when extracting :g:`nat`
into |OCaml| ``int``, the code to be provided has type:
``(unit->'a)->(int->'a)->int->'a``.
Caution!
As for :cmd:`Extract Constant`, this command should be used with care:
- The ML code provided by the user is currently not checked at all by extraction, even for syntax errors.
- Extracting an inductive type to a pre-existing ML inductive type
is quite sound. But extracting to a general type (by providing an
ad-hoc pattern matching) will often not be fully rigorously
correct. For instance, when extracting
natto |OCaml|int, it is theoretically possible to buildnatvalues that are larger than |OCaml|max_int. It is the user's responsibility to be sure that no overflow or other bad events occur in practice. - Translating an inductive type to an arbitrary ML type does not
magically improve the asymptotic complexity of functions, even if the
ML type is an efficient representation. For instance, when extracting
natto |OCaml|int, the functionNat.mulstays quadratic. It might be interesting to associate this translation with some specific :cmd:`Extract Constant` when primitive counterparts exist.
Typical examples are the following:
.. coqtop:: in Extract Inductive unit => "unit" [ "()" ]. Extract Inductive bool => "bool" [ "true" "false" ]. Extract Inductive sumbool => "bool" [ "true" "false" ].
Note
When extracting to |OCaml|, if an inductive constructor or type has arity 2 and the corresponding string is enclosed by parentheses, and the string meets |OCaml|'s lexical criteria for an infix symbol, then the rest of the string is used as an infix constructor or type.
.. coqtop:: in Extract Inductive list => "list" [ "[]" "(::)" ]. Extract Inductive prod => "(*)" [ "(,)" ].
As an example of translation to a non-inductive datatype, let's turn
nat into |OCaml| int (see caveat above):
.. coqtop:: in Extract Inductive nat => int [ "0" "succ" ] "(fun fO fS n -> if n=0 then fO () else fS (n-1))".
When using :cmd:`Extraction Library`, the names of the extracted files
directly depend on the names of the |Coq| files. It may happen that
these filenames are in conflict with already existing files,
either in the standard library of the target language or in other
code that is meant to be linked with the extracted code.
For instance the module List exists both in |Coq| and in |OCaml|.
It is possible to instruct the extraction not to use particular filenames.
.. cmd:: Extraction Blacklist {+ @ident }
Instruct the extraction to avoid using these names as filenames
for extracted code.
.. cmd:: Print Extraction Blacklist Show the current list of filenames the extraction should avoid.
.. cmd:: Reset Extraction Blacklist Allow the extraction to use any filename.
For |OCaml|, a typical use of these commands is
Extraction Blacklist String List.
.. opt:: Extraction File Comment @string :name: Extraction File Comment Provides a comment that is included at the beginning of the output files.
.. opt:: Extraction Flag @num :name: Extraction Flag Controls which optimizations are used during extraction, providing a finer-grained control than :flag:`Extraction Optimize`. The bits of :token:`num` are used as a bit mask. Keeping an option off keeps the extracted ML more similar to the Coq term. Values are: +-----+-------+----------------------------------------------------------------+ | Bit | Value | Optimization (default is on unless noted otherwise) | +-----+-------+----------------------------------------------------------------+ | 0 | 1 | Remove local dummy variables | +-----+-------+----------------------------------------------------------------+ | 1 | 2 | Use special treatment for fixpoints | +-----+-------+----------------------------------------------------------------+ | 2 | 4 | Simplify case with iota-redux | +-----+-------+----------------------------------------------------------------+ | 3 | 8 | Factor case branches as functions | +-----+-------+----------------------------------------------------------------+ | 4 | 16 | (not available, default false) | +-----+-------+----------------------------------------------------------------+ | 5 | 32 | Simplify case as function of one argument | +-----+-------+----------------------------------------------------------------+ | 6 | 64 | Simplify case by swapping case and lambda | +-----+-------+----------------------------------------------------------------+ | 7 | 128 | Some case optimization | +-----+-------+----------------------------------------------------------------+ | 8 | 256 | Push arguments inside a letin | +-----+-------+----------------------------------------------------------------+ | 9 | 512 | Use linear let reduction (default false) | +-----+-------+----------------------------------------------------------------+ | 10 | 1024 | Use linear beta reduction (default false) | +-----+-------+----------------------------------------------------------------+
.. flag:: Extraction TypeExpand If set, fully expand Coq types in ML. See the Coq source code to learn more.
Differences between |Coq| and ML type systems
Due to differences between |Coq| and ML type systems,
some extracted programs are not directly typable in ML.
We now solve this problem (at least in |OCaml|) by adding
when needed some unsafe casting Obj.magic, which give
a generic type 'a to any term.
First, if some part of the program is very polymorphic, there
may be no ML type for it. In that case the extraction to ML works
alright but the generated code may be refused by the ML
type checker. A very well known example is the distr-pair
function:
.. coqtop:: in
Definition dp {A B:Type}(x:A)(y:B)(f:forall C:Type, C->C) := (f A x, f B y).
In |OCaml|, for instance, the direct extracted term would be:
let dp x y f = Pair((f () x),(f () y))
and would have type:
dp : 'a -> 'a -> (unit -> 'a -> 'b) -> ('b,'b) prod
which is not its original type, but a restriction.
We now produce the following correct version:
let dp x y f = Pair ((Obj.magic f () x), (Obj.magic f () y))
Secondly, some |Coq| definitions may have no counterpart in ML. This happens when there is a quantification over types inside the type of a constructor; for example:
.. coqtop:: in Inductive anything : Type := dummy : forall A:Set, A -> anything.
which corresponds to the definition of an ML dynamic type. In |OCaml|, we must cast any argument of the constructor dummy (no GADT are produced yet by the extraction).
Even with those unsafe castings, you should never get error like
segmentation fault. In fact even if your program may seem
ill-typed to the |OCaml| type checker, it can't go wrong : it comes
from a Coq well-typed terms, so for example inductive types will always
have the correct number of arguments, etc. Of course, when launching
manually some extracted function, you should apply it to arguments
of the right shape (from the |Coq| point-of-view).
More details about the correctness of the extracted programs can be found in :cite:`Let02`.
We have to say, though, that in most "realistic" programs, these problems do not
occur. For example all the programs of Coq library are accepted by the |OCaml|
type checker without any Obj.magic (see examples below).
We present here two examples of extraction, taken from the |Coq| Standard Library. We choose |OCaml| as the target language, but everything, with slight modifications, can also be done in the other languages supported by extraction. We then indicate where to find other examples and tests of extraction.
The file Euclid contains the proof of Euclidean division.
The natural numbers used here are unary, represented by the type nat,
which is defined by two constructors O and S.
This module contains a theorem eucl_dev, whose type is:
forall b:nat, b > 0 -> forall a:nat, diveucl a b
where diveucl is a type for the pair of the quotient and the
modulo, plus some logical assertions that disappear during extraction.
We can now extract this program to |OCaml|:
.. coqtop:: none Reset Initial.
.. coqtop:: all Require Extraction. Require Import Euclid Wf_nat. Extraction Inline gt_wf_rec lt_wf_rec induction_ltof2. Recursive Extraction eucl_dev.
The inlining of gt_wf_rec and others is not
mandatory. It only enhances readability of extracted code.
You can then copy-paste the output to a file euclid.ml or let
|Coq| do it for you with the following command:
Extraction "euclid" eucl_dev.
Let us play the resulting program (in an |OCaml| toplevel):
#use "euclid.ml";; type nat = O | S of nat type sumbool = Left | Right val sub : nat -> nat -> nat = <fun> val le_lt_dec : nat -> nat -> sumbool = <fun> val le_gt_dec : nat -> nat -> sumbool = <fun> type diveucl = Divex of nat * nat val eucl_dev : nat -> nat -> diveucl = <fun> # eucl_dev (S (S O)) (S (S (S (S (S O)))));; - : diveucl = Divex (S (S O), S O)
It is easier to test on |OCaml| integers:
# let rec nat_of_int = function 0 -> O | n -> S (nat_of_int (n-1));; val nat_of_int : int -> nat = <fun> # let rec int_of_nat = function O -> 0 | S p -> 1+(int_of_nat p);; val int_of_nat : nat -> int = <fun> # let div a b = let Divex (q,r) = eucl_dev (nat_of_int b) (nat_of_int a) in (int_of_nat q, int_of_nat r);; val div : int -> int -> int * int = <fun> # div 173 15;; - : int * int = (11, 8)
Note that these nat_of_int and int_of_nat are now
available via a mere Require Import ExtrOcamlIntConv and then
adding these functions to the list of functions to extract. This file
ExtrOcamlIntConv.v and some others in plugins/extraction/
are meant to help building concrete program via extraction.
Some pathological examples of extraction are grouped in the file
test-suite/success/extraction.v of the sources of |Coq|.
Several of the |Coq| Users' Contributions use extraction to produce certified programs. In particular the following ones have an automatic extraction test:
additions: https://github.com/coq-contribs/additionsbdds: https://github.com/coq-contribs/bddscanon-bdds: https://github.com/coq-contribs/canon-bddschinese: https://github.com/coq-contribs/chinesecontinuations: https://github.com/coq-contribs/continuationscoq-in-coq: https://github.com/coq-contribs/coq-in-coqexceptions: https://github.com/coq-contribs/exceptionsfiring-squad: https://github.com/coq-contribs/firing-squadfounify: https://github.com/coq-contribs/founifygraphs: https://github.com/coq-contribs/graphshigman-cf: https://github.com/coq-contribs/higman-cfhigman-nw: https://github.com/coq-contribs/higman-nwhardware: https://github.com/coq-contribs/hardwaremultiplier: https://github.com/coq-contribs/multipliersearch-trees: https://github.com/coq-contribs/search-treesstalmarck: https://github.com/coq-contribs/stalmarck
Note that continuations and multiplier are a bit particular. They are
examples of developments where Obj.magic is needed. This is
probably due to a heavy use of impredicativity. After compilation, those
two examples run nonetheless, thanks to the correction of the
extraction :cite:`Let02`.