Chapter 23 Extraction of programs in Objective Caml and Haskell

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 Objective Caml, Haskell and Scheme. In the following, “ML” will be used (abusively) to refer to any of the three.

23.1 Generating ML code

The next two commands are meant to be used for rapid preview of extraction. They both display extracted term(s) inside Coq.

: Extraction qualid.
Extraction of a constant or module in the Coq toplevel.
: Recursive Extraction qualid₁ … qualid_n.
Recursive extraction of all the globals (or modules) qualid₁ … qualid_n 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.

: Extraction "file" qualid₁ … qualid_n.
Recursive extraction of all the globals (or modules) qualid₁ … qualid_n and all their dependencies in one monolithic file file. Global and local identifiers are renamed according to the chosen ML language to fulfill its syntactic conventions, keeping original names as much as possible.
: Extraction Library ident.
Extraction of the whole Coq library ident.v to an ML module ident.ml. In case of name clash, identifiers are here renamed using prefixes coq_ or Coq_ to ensure a session-independent renaming.
: Recursive Extraction Library ident.
Extraction of the Coq library ident.v and all other modules ident.v depends on.
: Separate Extraction qualid₁ … qualid_n.
Recursive extraction of all the globals (or modules) qualid₁ … qualid_n and all their dependencies, just as Extraction "file", 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 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 Extraction Library: identifiers are here renamed using prefixes coq_ or Coq_.

The list of globals qualid_i does not need to be exhaustive: it is automatically completed into a complete and minimal environment.

23.2 Extraction options

23.2.1 Setting the target language

The ability to fix target language is the first and more important of the extraction options. Default is Ocaml.

: Extraction Language Ocaml.
: Extraction Language Haskell.
: Extraction Language Scheme.

23.2.2 Inlining and optimizations

Since Objective Caml 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 want to generate Ocaml programs. 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 constant 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 options:

: Unset Extraction Optimize.
Default is Set. This controls all type-preserving optimizations made on the ML terms (mostly reduction of dummy beta/iota redexes, but also simplifications on Cases, etc). Put this option to Unset if you want a ML term as close as possible to the Coq term.
: Set Extraction Conservative Types.
Default is Unset. This controls the non type-preserving optimizations made on ML terms (which try to avoid function abstraction of dummy types). Turn this option to Set to make sure that e:t implies that e’:t’ where e’ and t’ are the extracted code of e and t respectively.
: Set Extraction KeepSingleton.
Default is Unset. 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 option allows disabling this optimization when one wishes to preserve the inductive structure of types.
: Unset Extraction AutoInline.
Default is Set. 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, do it by Unset.
: Extraction [Inline|NoInline] qualid₁ … qualid_n.
In addition to the automatic inline feature, you can tell to inline some more constants by the Extraction Inline command. Conversely, you can forbid the automatic inlining of some specific constants by the Extraction NoInline command. Those two commands enable a precise control of what is inlined and what is not.
: Print Extraction Inline.
Prints the current state of the table recording the custom inlinings declared by the two previous commands.
: Reset Extraction Inline.
Puts the table recording the custom inlinings back to empty.

Inlining and printing of a constant declaration.

A 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.

23.2.3 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).

: Extraction Implicit qualid [ ident₁ … ident_n ].
This experimental command allows declaring some arguments of qualid as implicit, i.e. useless in extracted code and hence to be removed by extraction. Here qualid can be any function or inductive constructor, and ident_i 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 Extraction Implicit declarations cannot be honored, that is if any of the implicited variables still occurs in the final code. This behavior can be relaxed via the following option:

: Unset Extraction SafeImplicits.
Default is Set. When this option is Unset, a warning is emitted instead of an error if some implicited variables 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 implicited variables). Note that this extracted code might not compile or run properly, depending of the use of these remaining implicited variables.

23.2.4 Realizing axioms

Extraction will fail if it encounters an informative axiom not realized (see Section 23.2.4). 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.

: Extract Constant qualid => string.
Give an ML extraction for the given constant. The string may be an identifier or a quoted string.
: 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 that the Extract Inlined Constant command is sugar for an Extract Constant followed by a Extraction Inline. Hence a Reset Extraction Inline will have an effect on the realized and inlined axiom.

Of course, 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.

Example:

Coq < Axiom X:Set.

Coq < Axiom x:X.

Coq < Extract Constant X => "int".

Coq < 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. The syntax is then:

: Extract Constant qualid string₁ … string_n => string.

The number of type variables is checked by the system.

Example:

Coq < Axiom Y : Set -> Set -> Set.

Coq < Extract Constant Y "'a" "'b" => " 'a*'b ".

Realizing an axiom via Extract Constant is only useful in the case of an informative axiom (of sort Type or Set). A logical axiom have no computational content and hence will not appears 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 Coq one. The syntax is the following:

: Extract Inductive qualid => string [ string … string ] optstring.
Give an ML extraction for the given inductive type. You must specify extractions for the type itself (first string) and all its constructors (between square brackets). If given, the final optional string should contain a function emulating pattern-matching over this inductive type. If this optional string is not given, the ML extraction must be an ML inductive datatype, and the native pattern-matching of the language will be used.

For an inductive type with k constructor, the function used to emulate the match 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 argument 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 nat into int, the code to provide has type: (unit->’a)->(int->’a)->int->’a.

As for Extract Inductive, 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 Ocaml’s int, it is theoretically possible to build nat values that are larger than Ocaml’s 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 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 Ocaml’s int, the function mult stays quadratic. It might be interesting to associate this translation with some specific Extract Constant when primitive counterparts exist.

Example: Typical examples are the following:

Coq < Extract Inductive unit => "unit" [ "()" ].

Coq < Extract Inductive bool => "bool" [ "true" "false" ].

Coq < Extract Inductive sumbool => "bool" [ "true" "false" ].

If an inductive constructor or type has arity 2 and the corresponding string is enclosed by parenthesis, then the rest of the string is used as infix constructor or type.

Coq < Extract Inductive list => "list" [ "[]" "(::)" ].

Coq < Extract Inductive prod => "(*)" [ "(,)" ].

As an example of translation to a non-inductive datatype, let’s turn nat into Ocaml’s int (see caveat above):

Coq < Extract Inductive nat => int [ "0" "succ" ]
"(fun fO fS n -> if n=0 then fO () else fS (n-1))".

23.2.5 Avoiding conflicts with existing filenames

When using Extraction Library, the names of the extracted files directly depends from 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.

: Extraction Blacklist ident … ident.
Instruct the extraction to avoid using these names as filenames for extracted code.
: Print Extraction Blacklist.
Show the current list of filenames the extraction should avoid.
: Reset Extraction Blacklist.
Allow the extraction to use any filename.

For Ocaml, a typical use of these commands is Extraction Blacklist String List.

23.3 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.

For example, here are two kinds of problem that can occur:

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 := 
 fun (A B:Set)(x:A)(y:B)(f:forall C:Set, 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))
```
Some definitions of Coq 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.
```
which corresponds to the definition of an ML dynamic type. In Ocaml, we must cast any argument of the constructor dummy.

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 inductives will always have the correct number of arguments, etc.

More details about the correctness of the extracted programs can be found in [99].

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 Caml type-checker without any Obj.magic (see examples below).

23.4 Some examples

We present here two examples of extractions, taken from the Coq Standard Library. We choose Objective Caml as target language, but all can be done in the other dialects with slight modifications. We then indicate where to find other examples and tests of Extraction.

23.4.1 A detailed example: Euclidean division

The file Euclid contains the proof of Euclidean division (theorem eucl_dev). The natural numbers defined in the example files are unary integers defined by two constructors O and S:

Coq < Inductive nat : Set :=
| O : nat
| S : nat -> nat.

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 Objective Caml:

Coq < Require Import Euclid Wf_nat.

Coq < Extraction Inline gt_wf_rec lt_wf_rec induction_ltof2.

Coq < 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 m0 -> le_lt_dec n0 m0)
(** val le_gt_dec : nat -> nat -> sumbool **)
let le_gt_dec n m =
  le_lt_dec n m
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:

# #use "euclid.ml";;
type nat = O | S of nat
type sumbool = Left | Right
val minus : 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 Objective Caml 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.

23.4.2 Extraction’s horror museum

Some pathological examples of extraction are grouped in the file
test-suite/success/extraction.v of the sources of Coq.

23.4.3 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
bdds
canon-bdds
chinese
continuations
coq-in-coq
exceptions
firing-squad
founify
graphs
higman-cf
higman-nw
hardware
multiplier
search-trees
stalmarck

continuations and multiplier are a bit particular. They are examples of developments where Obj.magic are needed. This is probably due to an heavy use of impredicativity. After compilation, those two examples run nonetheless, thanks to the correction of the extraction [99].