Source file seq.ml

(**************************************************************************)
(*                                                                        *)
(*                                 OCaml                                  *)
(*                                                                        *)
(*                 Simon Cruanes                                          *)
(*                                                                        *)
(*   Copyright 2017 Institut National de Recherche en Informatique et     *)
(*     en Automatique.                                                    *)
(*                                                                        *)
(*   All rights reserved.  This file is distributed under the terms of    *)
(*   the GNU Lesser General Public License version 2.1, with the          *)
(*   special exception on linking described in the file LICENSE.          *)
(*                                                                        *)
(**************************************************************************)

(* Module [Seq]: functional iterators *)

type +'a node =
  | Nil
  | Cons of 'a * 'a t

and 'a t = unit -> 'a node

let empty () = Nil

let return x () = Cons (x, empty)

let cons x next () = Cons (x, next)

let rec append seq1 seq2 () =
  match seq1() with
  | Nil -> seq2()
  | Cons (x, next) -> Cons (x, append next seq2)

let rec map f seq () = match seq() with
  | Nil -> Nil
  | Cons (x, next) -> Cons (f x, map f next)

let rec filter_map f seq () = match seq() with
  | Nil -> Nil
  | Cons (x, next) ->
      match f x with
        | None -> filter_map f next ()
        | Some y -> Cons (y, filter_map f next)

let rec filter f seq () = match seq() with
  | Nil -> Nil
  | Cons (x, next) ->
      if f x
      then Cons (x, filter f next)
      else filter f next ()

let rec concat seq () = match seq () with
  | Nil -> Nil
  | Cons (x, next) ->
     append x (concat next) ()

let rec flat_map f seq () = match seq () with
  | Nil -> Nil
  | Cons (x, next) ->
    append (f x) (flat_map f next) ()

let concat_map = flat_map

let rec fold_left f acc seq =
  match seq () with
    | Nil -> acc
    | Cons (x, next) ->
        let acc = f acc x in
        fold_left f acc next

let rec iter f seq =
  match seq () with
    | Nil -> ()
    | Cons (x, next) ->
        f x;
        iter f next

let rec unfold f u () =
  match f u with
  | None -> Nil
  | Some (x, u') -> Cons (x, unfold f u')

let is_empty xs =
  match xs() with
  | Nil ->
      true
  | Cons (_, _) ->
      false

let uncons xs =
  match xs() with
  | Cons (x, xs) ->
      Some (x, xs)
  | Nil ->
      None



let rec length_aux accu xs =
  match xs() with
  | Nil ->
      accu
  | Cons (_, xs) ->
      length_aux (accu + 1) xs

let[@inline] length xs =
  length_aux 0 xs

let rec iteri_aux f i xs =
  match xs() with
  | Nil ->
      ()
  | Cons (x, xs) ->
      f i x;
      iteri_aux f (i+1) xs

let[@inline] iteri f xs =
  iteri_aux f 0 xs

let rec fold_lefti_aux f accu i xs =
  match xs() with
  | Nil ->
      accu
  | Cons (x, xs) ->
      let accu = f accu i x in
      fold_lefti_aux f accu (i+1) xs

let[@inline] fold_lefti f accu xs =
  fold_lefti_aux f accu 0 xs

let rec for_all p xs =
  match xs() with
  | Nil ->
      true
  | Cons (x, xs) ->
      p x && for_all p xs

let rec exists p xs =
  match xs() with
  | Nil ->
      false
  | Cons (x, xs) ->
      p x || exists p xs

let rec find p xs =
  match xs() with
  | Nil ->
      None
  | Cons (x, xs) ->
      if p x then Some x else find p xs

let find_index p xs =
  let rec aux i xs = match xs() with
    | Nil ->
        None
    | Cons (x, xs) ->
        if p x then Some i else aux (i+1) xs in
  aux 0 xs

let rec find_map f xs =
  match xs() with
  | Nil ->
      None
  | Cons (x, xs) ->
      match f x with
      | None ->
          find_map f xs
      | Some _ as result ->
          result

let find_mapi f xs =
  let rec aux i xs = match xs() with
    | Nil ->
        None
    | Cons (x, xs) ->
        match f i x with
        | None ->
            aux (i+1) xs
        | Some _ as result ->
            result in
  aux 0 xs

(* [iter2], [fold_left2], [for_all2], [exists2], [map2], [zip] work also in
   the case where the two sequences have different lengths. They stop as soon
   as one sequence is exhausted. Their behavior is slightly asymmetric: when
   [xs] is empty, they do not force [ys]; however, when [ys] is empty, [xs] is
   forced, even though the result of the function application [xs()] turns out
   to be useless. *)

let rec iter2 f xs ys =
  match xs() with
  | Nil ->
      ()
  | Cons (x, xs) ->
      match ys() with
      | Nil ->
          ()
      | Cons (y, ys) ->
          f x y;
          iter2 f xs ys

let rec fold_left2 f accu xs ys =
  match xs() with
  | Nil ->
      accu
  | Cons (x, xs) ->
      match ys() with
      | Nil ->
          accu
      | Cons (y, ys) ->
          let accu = f accu x y in
          fold_left2 f accu xs ys

let rec for_all2 f xs ys =
  match xs() with
  | Nil ->
      true
  | Cons (x, xs) ->
      match ys() with
      | Nil ->
          true
      | Cons (y, ys) ->
          f x y && for_all2 f xs ys

let rec exists2 f xs ys =
  match xs() with
  | Nil ->
      false
  | Cons (x, xs) ->
      match ys() with
      | Nil ->
          false
      | Cons (y, ys) ->
          f x y || exists2 f xs ys

let rec equal eq xs ys =
  match xs(), ys() with
  | Nil, Nil ->
      true
  | Cons (x, xs), Cons (y, ys) ->
      eq x y && equal eq xs ys
  | Nil, Cons (_, _)
  | Cons (_, _), Nil ->
      false

let rec compare cmp xs ys =
  match xs(), ys() with
  | Nil, Nil ->
      0
  | Cons (x, xs), Cons (y, ys) ->
      let c = cmp x y in
      if c <> 0 then c else compare cmp xs ys
  | Nil, Cons (_, _) ->
      -1
  | Cons (_, _), Nil ->
      +1



(* [init_aux f i j] is the sequence [f i, ..., f (j-1)]. *)

let rec init_aux f i j () =
  if i < j then begin
    Cons (f i, init_aux f (i + 1) j)
  end
  else
    Nil

let init n f =
  if n < 0 then
    invalid_arg "Seq.init"
  else
    init_aux f 0 n

let rec repeat x () =
  Cons (x, repeat x)

let rec forever f () =
  Cons (f(), forever f)

(* This preliminary definition of [cycle] requires the sequence [xs]
   to be nonempty. Applying it to an empty sequence would produce a
   sequence that diverges when it is forced. *)

let rec cycle_nonempty xs () =
  append xs (cycle_nonempty xs) ()

(* [cycle xs] checks whether [xs] is empty and, if so, returns an empty
   sequence. Otherwise, [cycle xs] produces one copy of [xs] followed
   with the infinite sequence [cycle_nonempty xs]. Thus, the nonemptiness
   check is performed just once. *)

let cycle xs () =
  match xs() with
  | Nil ->
      Nil
  | Cons (x, xs') ->
      Cons (x, append xs' (cycle_nonempty xs))

(* [iterate1 f x] is the sequence [f x, f (f x), ...].
   It is equivalent to [tail (iterate f x)].
   [iterate1] is used as a building block in the definition of [iterate]. *)

let rec iterate1 f x () =
  let y = f x in
  Cons (y, iterate1 f y)

(* [iterate f x] is the sequence [x, f x, ...]. *)

(* The reason why we give this slightly indirect definition of [iterate],
   as opposed to the more naive definition that may come to mind, is that
   we are careful to avoid evaluating [f x] until this function call is
   actually necessary. The naive definition (not shown here) computes the
   second argument of the sequence, [f x], when the first argument is
   requested by the user. *)

let iterate f x =
  cons x (iterate1 f x)



let rec mapi_aux f i xs () =
  match xs() with
  | Nil ->
      Nil
  | Cons (x, xs) ->
      Cons (f i x, mapi_aux f (i+1) xs)

let[@inline] mapi f xs =
  mapi_aux f 0 xs

(* [tail_scan f s xs] is equivalent to [tail (scan f s xs)].
   [tail_scan] is used as a building block in the definition of [scan]. *)

(* This slightly indirect definition of [scan] is meant to avoid computing
   elements too early; see the above comment about [iterate1] and [iterate]. *)

let rec tail_scan f s xs () =
  match xs() with
  | Nil ->
      Nil
  | Cons (x, xs) ->
      let s = f s x in
      Cons (s, tail_scan f s xs)

let scan f s xs =
  cons s (tail_scan f s xs)

(* [take] is defined in such a way that [take 0 xs] returns [empty]
   immediately, without allocating any memory. *)

let rec take_aux n xs =
  if n = 0 then
    empty
  else
    fun () ->
      match xs() with
      | Nil ->
          Nil
      | Cons (x, xs) ->
          Cons (x, take_aux (n-1) xs)

let take n xs =
  if n < 0 then invalid_arg "Seq.take";
  take_aux n xs

(* [force_drop n xs] is equivalent to [drop n xs ()].
   [force_drop n xs] requires [n > 0].
   [force_drop] is used as a building block in the definition of [drop]. *)

let rec force_drop n xs =
  match xs() with
  | Nil ->
      Nil
  | Cons (_, xs) ->
      let n = n - 1 in
      if n = 0 then
        xs()
      else
        force_drop n xs

(* [drop] is defined in such a way that [drop 0 xs] returns [xs] immediately,
   without allocating any memory. *)

let drop n xs =
  if n < 0 then invalid_arg "Seq.drop"
  else if n = 0 then
    xs
  else
    fun () ->
      force_drop n xs

let rec take_while p xs () =
  match xs() with
  | Nil ->
      Nil
  | Cons (x, xs) ->
      if p x then Cons (x, take_while p xs) else Nil

let rec drop_while p xs () =
  match xs() with
  | Nil ->
      Nil
  | Cons (x, xs) as node ->
      if p x then drop_while p xs () else node

let rec group eq xs () =
  match xs() with
  | Nil ->
      Nil
  | Cons (x, xs) ->
      Cons (cons x (take_while (eq x) xs), group eq (drop_while (eq x) xs))

exception Forced_twice

module Suspension = struct

  type 'a suspension =
    unit -> 'a

  (* Conversions. *)

  let to_lazy : 'a suspension -> 'a Lazy.t =
    Lazy.from_fun
    (* fun s -> lazy (s()) *)

  let from_lazy (s : 'a Lazy.t) : 'a suspension =
    fun () -> Lazy.force s

  (* [memoize] turns an arbitrary suspension into a persistent suspension. *)

  let memoize (s : 'a suspension) : 'a suspension =
    from_lazy (to_lazy s)

  (* [failure] is a suspension that fails when forced. *)

  let failure : _ suspension =
    fun () ->
      (* A suspension created by [once] has been forced twice. *)
      raise Forced_twice

  (* If [f] is a suspension, then [once f] is a suspension that can be forced
     at most once. If it is forced more than once, then [Forced_twice] is
     raised. *)

  let once (f : 'a suspension) : 'a suspension =
    let action = Atomic.make f in
    fun () ->
      (* Get the function currently stored in [action], and write the
         function [failure] in its place, so the next access will result
         in a call to [failure()]. *)
      let f = Atomic.exchange action failure in
      f()

end (* Suspension *)

let rec memoize xs =
  Suspension.memoize (fun () ->
    match xs() with
    | Nil ->
        Nil
    | Cons (x, xs) ->
        Cons (x, memoize xs)
  )

let rec once xs =
  Suspension.once (fun () ->
    match xs() with
    | Nil ->
        Nil
    | Cons (x, xs) ->
        Cons (x, once xs)
  )


let rec zip xs ys () =
  match xs() with
  | Nil ->
      Nil
  | Cons (x, xs) ->
      match ys() with
      | Nil ->
          Nil
      | Cons (y, ys) ->
          Cons ((x, y), zip xs ys)

let rec map2 f xs ys () =
  match xs() with
  | Nil ->
      Nil
  | Cons (x, xs) ->
      match ys() with
      | Nil ->
          Nil
      | Cons (y, ys) ->
          Cons (f x y, map2 f xs ys)

let rec interleave xs ys () =
  match xs() with
  | Nil ->
      ys()
  | Cons (x, xs) ->
      Cons (x, interleave ys xs)

(* [sorted_merge1l cmp x xs ys] is equivalent to
     [sorted_merge cmp (cons x xs) ys].

   [sorted_merge1r cmp xs y ys] is equivalent to
     [sorted_merge cmp xs (cons y ys)].

   [sorted_merge1 cmp x xs y ys] is equivalent to
     [sorted_merge cmp (cons x xs) (cons y ys)].

   These three functions are used as building blocks in the definition
   of [sorted_merge]. *)

let rec sorted_merge1l cmp x xs ys () =
  match ys() with
  | Nil ->
      Cons (x, xs)
  | Cons (y, ys) ->
      sorted_merge1 cmp x xs y ys

and sorted_merge1r cmp xs y ys () =
  match xs() with
  | Nil ->
      Cons (y, ys)
  | Cons (x, xs) ->
      sorted_merge1 cmp x xs y ys

and sorted_merge1 cmp x xs y ys =
  if cmp x y <= 0 then
    Cons (x, sorted_merge1r cmp xs y ys)
  else
    Cons (y, sorted_merge1l cmp x xs ys)

let sorted_merge cmp xs ys () =
  match xs(), ys() with
    | Nil, Nil ->
        Nil
    | Nil, c
    | c, Nil ->
        c
    | Cons (x, xs), Cons (y, ys) ->
        sorted_merge1 cmp x xs y ys


let rec map_fst xys () =
  match xys() with
  | Nil ->
      Nil
  | Cons ((x, _), xys) ->
      Cons (x, map_fst xys)

let rec map_snd xys () =
  match xys() with
  | Nil ->
      Nil
  | Cons ((_, y), xys) ->
      Cons (y, map_snd xys)

let unzip xys =
  map_fst xys, map_snd xys

let split =
  unzip

(* [filter_map_find_left_map f xs] is equivalent to
   [filter_map Either.find_left (map f xs)]. *)

let rec filter_map_find_left_map f xs () =
  match xs() with
  | Nil ->
      Nil
  | Cons (x, xs) ->
      match f x with
      | Either.Left y ->
          Cons (y, filter_map_find_left_map f xs)
      | Either.Right _ ->
          filter_map_find_left_map f xs ()

let rec filter_map_find_right_map f xs () =
  match xs() with
  | Nil ->
      Nil
  | Cons (x, xs) ->
      match f x with
      | Either.Left _ ->
          filter_map_find_right_map f xs ()
      | Either.Right z ->
          Cons (z, filter_map_find_right_map f xs)

let partition_map f xs =
  filter_map_find_left_map f xs,
  filter_map_find_right_map f xs

let partition p xs =
  filter p xs, filter (fun x -> not (p x)) xs

(* If [xss] is a matrix (a sequence of rows), then [peel xss] is a pair of
   the first column (a sequence of elements) and of the remainder of the
   matrix (a sequence of shorter rows). These two sequences have the same
   length. The rows of the matrix [xss] are not required to have the same
   length. An empty row is ignored. *)

(* Because [peel] uses [unzip], its argument must be persistent. The same
   remark applies to [transpose], [diagonals], [product], etc. *)

let peel xss =
  unzip (filter_map uncons xss)

let rec transpose xss () =
  let heads, tails = peel xss in
  if is_empty heads then begin
    assert (is_empty tails);
    Nil
  end
  else
    Cons (heads, transpose tails)

(* The internal function [diagonals] takes an extra argument, [remainders],
   which contains the remainders of the rows that have already been
   discovered. *)

let rec diagonals remainders xss () =
  match xss() with
  | Cons (xs, xss) ->
      begin match xs() with
      | Cons (x, xs) ->
          (* We discover a new nonempty row [x :: xs]. Thus, the next diagonal
             is [x :: heads]: this diagonal begins with [x] and continues with
             the first element of every row in [remainders]. In the recursive
             call, the argument [remainders] is instantiated with [xs ::
             tails], which means that we have one more remaining row, [xs],
             and that we keep the tails of the pre-existing remaining rows. *)
          let heads, tails = peel remainders in
          Cons (cons x heads, diagonals (cons xs tails) xss)
      | Nil ->
          (* We discover a new empty row. In this case, the new diagonal is
             just [heads], and [remainders] is instantiated with just [tails],
             as we do not have one more remaining row. *)
          let heads, tails = peel remainders in
          Cons (heads, diagonals tails xss)
      end
  | Nil ->
      (* There are no more rows to be discovered. There remains to exhaust
         the remaining rows. *)
      transpose remainders ()

(* If [xss] is a matrix (a sequence of rows), then [diagonals xss] is
   the sequence of its diagonals.

   The first diagonal contains just the first element of the
   first row. The second diagonal contains the first element of the
   second row and the second element of the first row; and so on.
   This kind of diagonal is in fact sometimes known as an antidiagonal.

   - Every diagonal is a finite sequence.
   - The rows of the matrix [xss] are not required to have the same length.
   - The matrix [xss] is not required to be finite (in either direction).
   - The matrix [xss] must be persistent. *)

let diagonals xss =
  diagonals empty xss

let map_product f xs ys =
  concat (diagonals (
    map (fun x ->
      map (fun y ->
        f x y
      ) ys
    ) xs
  ))

let product xs ys =
  map_product (fun x y -> (x, y)) xs ys

let of_dispenser it =
  let rec c () =
    match it() with
    | None ->
        Nil
    | Some x ->
        Cons (x, c)
  in
  c

let to_dispenser xs =
  let s = ref xs in
  fun () ->
    match (!s)() with
    | Nil ->
        None
    | Cons (x, xs) ->
        s := xs;
        Some x



let rec ints i () =
  Cons (i, ints (i + 1))