10. Executing WhyML Programs

This chapter shows how WhyML code can be executed, either by being interpreted or compiled to some existing programming language.

Let us consider the program of Section 3.2 that computes the maximum and the sum of an array of integers.

Let us assume it is contained in a file maxsum.mlw.

10.1. Interpreting WhyML Code

To test function max_sum, we can introduce a WhyML test function in module MaxAndSum

let test () =
  let n = 10 in
  let a = make n 0 in
  a[0] <- 9; a[1] <- 5; a[2] <- 0; a[3] <- 2;  a[4] <- 7;
  a[5] <- 3; a[6] <- 2; a[7] <- 1; a[8] <- 10; a[9] <- 6;
  max_sum a n

and then we use the execute command to interpret this function, as follows:

> why3 execute maxsum.mlw --use=MaxAndSum 'test ()'
result: (int, int) = (45, 10)

We get the expected output, namely the pair (45, 10).

10.2. Compiling WhyML to OCaml

An alternative to interpretation is to compile WhyML to OCaml. We do so using the extract command, as follows:

why3 extract -D ocaml64 maxsum.mlw -o max_sum.ml

The extract command requires the name of a driver, which indicates how theories/modules from the Why3 standard library are translated to OCaml. Here we assume a 64-bit architecture and thus we pass ocaml64. We also specify an output file using option -o, namely max_sum.ml. After this command, the file max_sum.ml contains an OCaml code for function max_sum. To compile it, we create a file main.ml containing a call to max_sum, e.g.,

let a = Array.map Z.of_int [| 9; 5; 0; 2; 7; 3; 2; 1; 10; 6 |]
let s, m = Max_sum.max_sum a (Z.of_int 10)
let () = Format.printf "sum=%s, max=%s@." (Z.to_string s) (Z.to_string m)

It is convenient to use ocamlbuild to compile and link both files max_sum.ml and main.ml:

ocamlbuild -pkg zarith main.native

Since Why3’s type int is translated to OCaml arbitrary precision integers using the ZArith library, we have to pass option -pkg zarith to ocamlbuild. In order to get extracted code that uses OCaml’s native integers instead, one has to use Why3’s types for 63-bit integers from libraries mach.int.Int63 and mach.array.Array63.

10.2.1. Examples

We illustrate different ways of using the extract command through some examples.

Consider the program of Section 3.6.

If we are only interested in extracting function enqueue, we can proceed as follows:

why3 extract -D ocaml64 -L . aqueue.AmortizedQueue.enqueue -o aqueue.ml

Here we assume that file aqueue.mlw contains this program, and that we invoke the extract command from the directory where this file is stored. File aqueue.ml now contains the following OCaml code:

let enqueue (x: 'a) (q: 'a queue) : 'a queue =
  create (q.front) (q.lenf) (x :: (q.rear))
    (Z.add (q.lenr) (Z.of_string "1"))

Choosing a function symbol as the entry point of extraction allows us to focus only on specific parts of the program. However, the generated code cannot be type-checked by the OCaml compiler, as it depends on function create and on type 'a queue, whose definitions are not given. In order to obtain a complete OCaml implementation, we can perform a recursive extraction:

why3 extract --recursive -D ocaml64 -L . aqueue.AmortizedQueue.enqueue -o aqueue.ml

This updates the contents of file aqueue.ml as follows:

type 'a queue = {
  front: 'a list;
  lenf: Z.t;
  rear: 'a list;
  lenr: Z.t;

let create (f: 'a list) (lf: Z.t) (r: 'a list) (lr: Z.t) : 'a queue =
  if Z.geq lf lr
    { front = f; lenf = lf; rear = r; lenr = lr }
    let f1 = List.append f (List.rev r) in
    { front = f1; lenf = Z.add lf lr; rear = []; lenr = (Z.of_string "0") }

let enqueue (x: 'a) (q: 'a queue) : 'a queue =
  create (q.front) (q.lenf) (x :: (q.rear))
    (Z.add (q.lenr) (Z.of_string "1"))

This new version of the code is now accepted by the OCaml compiler (provided the ZArith library is available, as above).

10.2.2. Extraction of Functors

WhyML and OCaml are both dialects of the ML-family, sharing many syntactic and semantics traits. Yet their module systems differ significantly. A WhyML program is a list of modules, a module is a list of top-level declarations, and declarations can be organized within scopes, the WhyML unit for namespaces management. In particular, there is no support for sub-modules in Why3, nor a dedicated syntactic construction for functors. The latter are represented, instead, as modules containing only abstract symbols [FilliatreP20]. One must follow exactly this programming pattern when it comes to extract an OCaml functor from a Why3 proof. Let us consider the following (excerpt) of a WhyML module implementing binary search trees:

module BST
  scope Make
    scope Ord
      type t
      val compare : t -> t -> int

    type elt = Ord.t

    type t = E | N t elt t

    use int.Int

    let rec insert (x: elt) (t: t)
    = match t with
      | E -> N E x E
      | N l y r ->
          if Ord.compare x y > 0 then N l y (insert x r)
          else N (insert x l) y r

For the sake of simplicity, we omit here behavioral specification. Assuming the above example is contained in a file named bst.mlw, one can readily extract it into OCaml, as follows:

> why3 extract -D ocaml64 bst.mlw --modular -o .

This produces the following functorial implementation:

module Make (Ord: sig type t
  val compare : t -> t -> Z.t end) =
  type elt = Ord.t

  type t =
  | E
  | N of t * Ord.t * t

  let rec insert (x: Ord.t) (t: t) : t =
    match t with
    | E -> N (E, x, E)
    | N (l, y, r) ->
        if Z.gt (Ord.compare x y) Z.zero
        then N (l, y, insert x r)
        else N (insert x l, y, r)

The extracted code features the functor Make parameterized with a module containing the abstract type t and function compare. This is similar to the OCaml standard library when it comes to data structures parameterized by an order relation, e.g., the Set and Map modules.

From the result of the extraction, one understands that scope Make is turned into a functor, while the nested scope Ord is extracted as the functor argument. In summary, for a WhyML implementation of the form

module M
  scope A
    scope X ... end
    scope Y ... end
    scope Z ... end

contained in file f.mlw, the Why3 extraction engine produces the following OCaml code:

module A (X: ...) (Y: ...) (Z: ...) = struct

and prints it into file f__M.ml. In order for functor extraction to succeed, scopes X, Y, and Z can only contain non-defined programming symbols, i.e., abstract type declarations, function signatures, and exception declarations. If ever a scope mixes non-defined and defined symbols, or if there is no surrounding scope such as Make, the extraction will complain about the presence of non-defined symbols that cannot be extracted.

It is worth noting that extraction of functors only works for modular extraction (i.e. with command-line option --modular).

10.2.3. Custom Extraction Drivers

Several OCaml drivers can be specified on the command line, using option -D several times. In particular, one can provide a custom driver to map some symbols of a Why3 development to existing OCaml code. Suppose for instance we have a file file.mlw containing a proof parameterized with some type elt and some binary function f:

module M
  type elt
  val f (x y: elt) : elt
  let double (x: elt) : elt = f x x

When it comes to extract this module to OCaml, we may want to instantiate type elt with OCaml’s type int and function f with OCaml’s addition. For this purpose, we provide the following in a file mydriver.drv:

module file.M
  syntax type elt "int"
  syntax val  f   "%1 + %2"

OCaml fragments to be substituted for Why3 symbols are given as arbitrary strings, where %1, %2, etc., will be replaced with actual arguments. Here is the extraction command line and its output:

> why3 extract -D ocaml64 -D mydriver.drv -L . file.M
let double (x: int) : int = x + x

When using such custom drivers, it is not possible to pass Why3 file names on the command line; one has to specify module names to be extracted, as done above.