Source file monad.ml

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(** Combinators on monadic computations. *)


(** A definition of monads, each of the combinators is used in the
    [Make] functor. *)
module type Def = sig

  type +'a t
  val return : 'a -> 'a t
  val (>>=) : 'a t -> ('a -> 'b t) -> 'b t
  val (>>) : unit t -> 'a t -> 'a t
  val map : ('a -> 'b) -> 'a t -> 'b t

  (** The monadic laws must hold:
      - [(x>>=f)>>=g] = [x>>=fun x' -> (f x'>>=g)]
      - [return a >>= f] = [f a]
      - [x>>=return] = [x]

      As well as the following identities:
      - [x >> y] = [x >>= fun () -> y]
      - [map f x] = [x >>= fun x' -> f x'] *)

end

module type ListS = sig

  type 'a t

  (** [List.map f l] maps [f] on the elements of [l] in left to right
      order. *)
  val map : ('a -> 'b t) -> 'a list -> 'b list t

  (** [List.map f l] maps [f] on the elements of [l] in right to left
      order. *)
  val map_right : ('a -> 'b t) -> 'a list -> 'b list t

  (** Like the regular [List.fold_right]. The monadic effects are
      threaded right to left.

      Note: many monads behave poorly with right-to-left order. For
      instance a failure monad would still have to traverse the
      whole list in order to fail and failure needs to be propagated
      through the rest of the list in binds which are now
      spurious. It is also the worst case for substitution monads
      (aka free monads), exposing the quadratic behaviour.*)
  val fold_right : ('a -> 'b -> 'b t) -> 'a list -> 'b -> 'b t

  (** Like the regular [List.fold_left]. The monadic effects are
      threaded left to right. It is tail-recursive if the [(>>=)]
      operator calls its second argument in a tail position. *)
  val fold_left : ('a -> 'b -> 'a t) -> 'a -> 'b list -> 'a t

  (** Like the regular [List.iter]. The monadic effects are threaded
      left to right. It is tail-recurisve if the [>>] operator calls
      its second argument in a tail position. *)
  val iter : ('a -> unit t) -> 'a list -> unit t

  (** Like the regular {!CList.map_filter}. The monadic effects are threaded left*)
  val map_filter : ('a -> 'b option t) -> 'a list -> 'b list t


  (** {6 Two-list iterators} *)

  (** [fold_left2 r f s l1 l2] behaves like {!fold_left} but acts
      simultaneously on two lists. Runs [r] (presumably an
      exception-raising computation) if both lists do not have the
      same length. *)
  val fold_left2 : 'a t ->
    ('a -> 'b -> 'c -> 'a t) -> 'a -> 'b list -> 'c list -> 'a t

end

module type S = sig

  include Def

  (** List combinators *)
  module List : ListS with type 'a t := 'a t

end


module Make (M:Def) : S with type +'a t = 'a M.t = struct

  include M

  module List = struct

    (* The combinators are loop-unrolled to spare a some monadic binds
       (it is a common optimisation to treat the last of a list of
       bind specially) and hopefully gain some efficiency using fewer
       jump. *)

    let rec map f = function
      | [] -> return []
      | [a] ->
          M.map (fun a' -> [a']) (f a)
      | a::b::l ->
          f a >>= fun a' ->
          f b >>= fun b' ->
          M.map (fun l' -> a'::b'::l') (map f l)

    let rec map_right f = function
      | [] -> return []
      | [a] ->
          M.map (fun a' -> [a']) (f a)
      | a::b::l ->
          map_right f l >>= fun l' ->
          f b >>= fun b' ->
          M.map (fun a' -> a'::b'::l') (f a)

    let rec fold_right f l x =
      match l with
      | [] -> return x
      | [a] -> f a x
      | a::b::l ->
          fold_right f l x >>= fun acc ->
          f b acc >>= fun acc ->
          f a acc

    let rec fold_left f x = function
      | [] -> return x
      | [a] -> f x a
      | a::b::l ->
          f x a >>= fun x' ->
          f x' b >>= fun x'' ->
          fold_left f x'' l

    let rec iter f = function
      | [] -> return ()
      | [a] -> f a
      | a::b::l -> f a >> f b >> iter f l


    let rec map_filter f = function
      | [] -> return []
      | a::l ->
        f a >>= function
        | None -> map_filter f l
        | Some b ->
          map_filter f l >>= fun filtered ->
          return (b::filtered)

    let rec fold_left2 r f x l1 l2 =
      match l1,l2 with
      | [] , [] -> return x
      | [a] , [b] -> f x a b
      | a1::a2::l1 , b1::b2::l2 ->
          f x a1 b1 >>= fun x' ->
          f x' a2 b2 >>= fun x'' ->
          fold_left2 r f x'' l1 l2
      | _ , _ -> r

  end

end