Program extraction¶
- 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
.
- Require Extraction.
- [Loading ML file extraction_plugin.cmxs (using legacy method) ... done]
Generating ML Code¶
Note
In the following, a qualified identifier qualid
can be used to refer to any kind of Coq global "object" : constant,
inductive type, inductive constructor or module name.
- Command Extraction qualid¶
- Command Recursive Extraction qualid+¶
- Command Extraction string qualid+
The first two forms display the extracted term(s) in Coq as a convenient preview of the extracted term(s):
the first form extracts
qualid
and displays the resulting term;the second form extracts the listed
qualid
s and all their dependencies, and displays the resulting terms.
The third form produces a single extraction file named
string
for all the specified objects and all of their dependencies.Global and local identifiers are renamed as needed to fulfill the syntactic requirements of the target language, keeping original names as much as possible.
The following commands also generate file(s).
- Command Extraction Library ident¶
Extraction of the whole Coq library
ident.v
to an ML moduleident.ml
. In case of name clash, identifiers are here renamed using prefixescoq_
orCoq_
to ensure a session-independent renaming.
- Command Recursive Extraction Library ident¶
Extraction of the Coq library
ident.v
and all other modulesident.v
depends on.
- Command Separate Extraction qualid+¶
Recursive extraction of all the mentioned objects and all their dependencies, just as
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 toRecursive 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 asExtraction Library
: identifiers are here renamed using prefixescoq_
orCoq_
.
The following command is meant to help automatic testing of
the extraction, see for instance the test-suite
directory
in the Coq sources.
- Command 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.
- Command Show Extraction¶
Extraction Options¶
Setting the target language¶
- Command Extraction Language language¶
- language
::=
OCaml|
Haskell|
Scheme|
JSONThe ability to fix target language is the first and most important of the extraction options. Default is
OCaml
.The JSON output is mostly for development or debugging: it contains the raw ML term produced as an intermediary target.
Inlining and optimizations¶
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 flag 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 flag 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 thate':t'
wheree'
andt'
are the extracted code ofe
andt
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.
- Command Extraction Inline qualid+¶
In addition to the automatic inline feature, the constants mentioned by this command will always be inlined during extraction.
- Command Extraction NoInline qualid+¶
Conversely, the constants mentioned by this command will never be inlined during extraction.
- Command Print Extraction Inline¶
Prints the current state of the table recording the custom inlinings declared by the two previous commands.
- Command 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:
Extra elimination of useless arguments¶
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).
- Command Extraction Implicit qualid [ identinteger* ]¶
Declares some arguments of
qualid
as implicit, meaning that they are useless in extracted code. The extracted code will omit these arguments. Herequalid
can be any function or inductive constructor, and theident
s are the names of the useless arguments. Arguments can can also be identified positionally byinteger
s starting from 1.
When an actual extraction takes place, an error is normally raised if the
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.
Realizing axioms¶
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.
- Command Extract Constant qualid string_{tv}* => identstring¶
Give an ML extraction for the given constant.
string_{tv}
If the type scheme axiom is an arity (a sequence of products followed by a sort), then some type variables have to be given (as quoted strings).
The number of type variables is checked by the system. For example:
- Axiom Y : Set -> Set -> Set.
- Y is declared
- Extract Constant Y "'a" "'b" => " 'a * 'b ".
- Command Extract Inlined Constant qualid => identstring¶
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
Extract Constant
followed by aExtraction Inline
. Hence aReset 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:
- Axiom X:Set.
- X is declared
- Axiom x:X.
- x is declared
- Extract Constant X => "int".
- Extract Constant x => "0".
Realizing an axiom via 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.
Realizing inductive types¶
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:
- Command Extract Inductive qualid => identstring [ identstring* ] string_{match}?¶
Give an ML extraction for the given inductive type. You must specify extractions for the type itself (the initial
identstring
) and all its constructors (the[ identstring* ]
). In this form, the ML extraction must be an ML inductive datatype, and the native pattern matching of the language will be used.When the initial
identstring
matches the name of the type of characters or strings (char
andstring
for OCaml,Prelude.Char
andPrelude.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 byqualid
to be registered as the standard char or string type, using theRegister
command.string_{match}
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 \(k\) constructors, the function used to emulate the pattern matching should expect \(k+1\) arguments, first the \(k\) 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 extractingnat
into OCamlint
, the code to be provided has type:(unit->'a)->(int->'a)->int->'a
.
Caution
As for 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
nat
to OCamlint
, it is theoretically possible to buildnat
values that are larger than OCamlmax_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
nat
to OCamlint
, the functionNat.mul
stays quadratic. It might be interesting to associate this translation with some specificExtract Constant
when primitive counterparts exist.
Typical examples are the following:
- 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.
- 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):
- Extract Inductive nat => int [ "0" "succ" ] "(fun fO fS n -> if n=0 then fO () else fS (n-1))".
Avoiding conflicts with existing filenames¶
When using 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.
- Command Extraction Blacklist ident+¶
Instruct the extraction to avoid using these names as filenames for extracted code.
- Command Print Extraction Blacklist¶
Show the current list of filenames the extraction should avoid.
- Command Reset Extraction Blacklist¶
Allow the extraction to use any filename.
For OCaml, a typical use of these commands is
Extraction Blacklist String List
.
Additional settings¶
- Option Extraction File Comment string¶
This option provides a comment that is included at the beginning of the output files.
- Option Extraction Flag natural¶
This option controls which optimizations are used during extraction, providing a finer-grained control than
Extraction Optimize
. The bits ofnatural
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)
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:
- Definition dp {A B:Type}(x:A)(y:B)(f:forall C:Type, C->C) := (f A x, f B y).
- dp is defined
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:
- Inductive anything : Type := dummy : forall A:Set, A -> anything.
- anything is defined anything_rect is defined anything_ind is defined anything_rec is defined anything_sind is defined
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 [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).
Some examples¶
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.
A detailed example: Euclidean division¶
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:
- Require Extraction.
- Require Import Euclid Wf_nat.
- Extraction Inline gt_wf_rec lt_wf_rec induction_ltof2.
- Recursive Extraction eucl_dev.
- type nat = | O | S of nat type sumbool = | Left | Right (** val sub : nat -> nat -> nat **) let rec sub n m = match n with | O -> n | S k -> (match m with | O -> n | S l -> sub k l) (** val le_lt_dec : nat -> nat -> sumbool **) let rec le_lt_dec n m = match n with | O -> Left | S n0 -> (match m with | O -> Right | S n1 -> le_lt_dec n0 n1) (** val le_gt_dec : nat -> nat -> sumbool **) let le_gt_dec = le_lt_dec type diveucl = | Divex of nat * nat (** val eucl_dev : nat -> nat -> diveucl **) let rec eucl_dev n m = let s = le_gt_dec n m in (match s with | Left -> let d = let y = sub m n in eucl_dev n y in let Divex (q, r) = d in Divex ((S q), r) | Right -> Divex (O, m))
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.
Extraction's horror museum¶
Some pathological examples of extraction are grouped in the file
test-suite/success/extraction.v
of the sources of Coq.
Users' Contributions¶
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/additions
canon-bdds
: https://github.com/coq-contribs/canon-bdds
chinese
: https://github.com/coq-contribs/chinese
continuations
: https://github.com/coq-contribs/continuations
coq-in-coq
: https://github.com/coq-contribs/coq-in-coq
exceptions
: https://github.com/coq-contribs/exceptions
firing-squad
: https://github.com/coq-contribs/firing-squad
founify
: https://github.com/coq-contribs/founify
higman-cf
: https://github.com/coq-contribs/higman-cf
higman-nw
: https://github.com/coq-contribs/higman-nw
hardware
: https://github.com/coq-contribs/hardware
multiplier
: https://github.com/coq-contribs/multiplier
search-trees
: https://github.com/coq-contribs/search-trees
stalmarck
: 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 [Let02].