* Applicatives model computations in which values computed by subcomputations can not 
     affect what subsequent computations will take place . 

     Relative to monads , this restriction takes power away from the user of the interface 
     and gives it to the implementation .   In particular , because the structure of the 
     entire computation is known , one can augment its definition with some description of 
     that structure . 

     For more information , see : 

     { v 
       Applicative Programming with Effects . 
         and   . 
       Journal of Functional Programming 18:1 ( 2008 ) , pages 1 - 13 . 
       /~ross/papers/Applicative.pdf 
     v }
    affect what subsequent computations will take place.

    Relative to monads, this restriction takes power away from the user of the interface
    and gives it to the implementation.  In particular, because the structure of the
    entire computation is known, one can augment its definition with some description of
    that structure.

    For more information, see:

    {v
      Applicative Programming with Effects.
      Conor McBride and Ross Paterson.
      Journal of Functional Programming 18:1 (2008), pages 1-13.
      /~ross/papers/Applicative.pdf
    v} *)

open! Import

module type Basic = sig
  type 'a t

  val return : 'a -> 'a t
  val apply : ('a -> 'b) t -> 'a t -> 'b t

* The following identities ought to hold for every Applicative ( for some value of = ): 

       - identity :      [ return Fn.id < * > t = t ] 
       - composition :   [ return Fn.compose < * > tf < * > tg < * > tx = tf < * > ( tg < * >  ) ] 
       - homomorphism : [ return f < * > return x = return ( f x ) ] 
       - interchange :   [ tf < * > return x = return ( fun f - > f x ) < * > tf ] 

       Note : < * > is the infix notation for apply .

      - identity:     [return Fn.id <*> t = t]
      - composition:  [return Fn.compose <*> tf <*> tg <*> tx = tf <*> (tg <*> tx)]
      - homomorphism: [return f <*> return x = return (f x)]
      - interchange:  [tf <*> return x = return (fun f -> f x) <*> tf]

      Note: <*> is the infix notation for apply. *)

  (** The [map] argument to [Applicative.Make] says how to implement the applicative's
      [map] function.  [`Define_using_apply] means to define [map t ~f = return f <*> t].
      [`Custom] overrides the default implementation, presumably with something more
      efficient.

      Some other functions returned by [Applicative.Make] are defined in terms of [map],
      so passing in a more efficient [map] will improve their efficiency as well. *)
  val map : [ `Define_using_apply | `Custom of 'a t -> f:('a -> 'b) -> 'b t ]
end

(** Similar to [Basic], with the same laws, and the additional requirement that ['a t]
    can be mapped with a local function. *)
module type Basic_local = sig
  type 'a t

  val return : 'a -> 'a t
  val apply : ('a -> 'b) t -> 'a t -> 'b t
  val map : 'a t -> f:(('a -> 'b)[@local]) -> 'b t
end

module type Basic_using_map2 = sig
  type 'a t

  val return : 'a -> 'a t
  val map2 : 'a t -> 'b t -> f:('a -> 'b -> 'c) -> 'c t
  val map : [ `Define_using_map2 | `Custom of 'a t -> f:('a -> 'b) -> 'b t ]
end

module type Basic_using_map2_local = sig
  type 'a t

  val return : 'a -> 'a t
  val map2 : 'a t -> 'b t -> f:(('a -> 'b -> 'c)[@local]) -> 'c t
  val map : [ `Define_using_map2 | `Custom of 'a t -> f:(('a -> 'b)[@local]) -> 'b t ]
end

module type Applicative_infix_gen = sig
  type 'a t
  type ('a, 'b) fn


  (** same as [apply] *)
  val ( <*> ) : ('a -> 'b) t -> 'a t -> 'b t

  val ( <* ) : 'a t -> unit t -> 'a t
  val ( *> ) : unit t -> 'a t -> 'a t
  val ( >>| ) : 'a t -> ('a -> 'b, 'b t) fn
end

module type Applicative_infix = Applicative_infix_gen with type ('a, 'b) fn := 'a -> 'b

module type Applicative_infix_local =
  Applicative_infix_gen with type ('a, 'b) fn := ('a[@local]) -> 'b

module type For_let_syntax_gen = sig
  type 'a t
  type ('a, 'b) fn
  type ('a, 'b) f_labeled_fn

  val return : 'a -> 'a t
  val map : 'a t -> ('a -> 'b, 'b t) f_labeled_fn
  val both : 'a t -> 'b t -> ('a * 'b) t

  include Applicative_infix_gen with type 'a t := 'a t and type ('a, 'b) fn := ('a, 'b) fn
end

module type For_let_syntax =
  For_let_syntax_gen
  with type ('a, 'b) fn := 'a -> 'b
   and type ('a, 'b) f_labeled_fn := f:'a -> 'b

module type For_let_syntax_local =
  For_let_syntax_gen
  with type ('a, 'b) fn := ('a[@local]) -> 'b
   and type ('a, 'b) f_labeled_fn := f:('a[@local]) -> 'b

module type S_gen = sig
  include For_let_syntax_gen

  type ('a, 'b, 'c) fun2
  type ('a, 'b, 'c, 'd) fun3

  val apply : ('a -> 'b) t -> 'a t -> 'b t
  val map2 : 'a t -> 'b t -> (('a, 'b, 'c) fun2, 'c t) f_labeled_fn
  val map3 : 'a t -> 'b t -> 'c t -> (('a, 'b, 'c, 'd) fun3, 'd t) f_labeled_fn
  val all : 'a t list -> 'a list t
  val all_unit : unit t list -> unit t

  module Applicative_infix :
    Applicative_infix_gen with type 'a t := 'a t and type ('a, 'b) fn := ('a, 'b) fn
end

module type S =
  S_gen
  with type ('a, 'b) fn := 'a -> 'b
   and type ('a, 'b) f_labeled_fn := f:'a -> 'b
   and type ('a, 'b, 'c) fun2 := 'a -> 'b -> 'c
   and type ('a, 'b, 'c, 'd) fun3 := 'a -> 'b -> 'c -> 'd

module type S_local =
  S_gen
  with type ('a, 'b) fn := ('a[@local]) -> 'b
   and type ('a, 'b) f_labeled_fn := f:('a[@local]) -> 'b
   and type ('a, 'b, 'c) fun2 := 'a -> (('b -> 'c)[@local])
   and type ('a, 'b, 'c, 'd) fun3 := 'a -> (('b -> (('c -> 'd)[@local]))[@local])

module type Let_syntax = sig
  type 'a t

  module Open_on_rhs_intf : sig
    module type S
  end

  module Let_syntax : sig
    val return : 'a -> 'a t

    include Applicative_infix with type 'a t := 'a t

    module Let_syntax : sig
      val return : 'a -> 'a t
      val map : 'a t -> f:('a -> 'b) -> 'b t
      val both : 'a t -> 'b t -> ('a * 'b) t

      module Open_on_rhs : Open_on_rhs_intf.S
    end
  end
end

module type Basic2 = sig
  type ('a, 'e) t

  val return : 'a -> ('a, _) t
  val apply : ('a -> 'b, 'e) t -> ('a, 'e) t -> ('b, 'e) t
  val map : [ `Define_using_apply | `Custom of ('a, 'e) t -> f:('a -> 'b) -> ('b, 'e) t ]
end

module type Basic2_local = sig
  type ('a, 'e) t

  val return : 'a -> ('a, _) t
  val apply : ('a -> 'b, 'e) t -> ('a, 'e) t -> ('b, 'e) t
  val map : ('a, 'e) t -> f:(('a -> 'b)[@local]) -> ('b, 'e) t
end

module type Basic2_using_map2 = sig
  type ('a, 'e) t

  val return : 'a -> ('a, _) t
  val map2 : ('a, 'e) t -> ('b, 'e) t -> f:('a -> 'b -> 'c) -> ('c, 'e) t
  val map : [ `Define_using_map2 | `Custom of ('a, 'e) t -> f:('a -> 'b) -> ('b, 'e) t ]
end

module type Basic2_using_map2_local = sig
  type ('a, 'e) t

  val return : 'a -> ('a, _) t
  val map2 : ('a, 'e) t -> ('b, 'e) t -> f:(('a -> 'b -> 'c)[@local]) -> ('c, 'e) t

  val map
    : [ `Define_using_map2
      | `Custom of ('a, 'e) t -> f:(('a -> 'b)[@local]) -> ('b, 'e) t
      ]
end

module type Applicative_infix2_gen = sig
  type ('a, 'e) t
  type ('a, 'b) fn

  val ( <*> ) : ('a -> 'b, 'e) t -> ('a, 'e) t -> ('b, 'e) t
  val ( <* ) : ('a, 'e) t -> (unit, 'e) t -> ('a, 'e) t
  val ( *> ) : (unit, 'e) t -> ('a, 'e) t -> ('a, 'e) t
  val ( >>| ) : ('a, 'e) t -> ('a -> 'b, ('b, 'e) t) fn
end

module type Applicative_infix2 = Applicative_infix2_gen with type ('a, 'b) fn := 'a -> 'b

module type Applicative_infix2_local =
  Applicative_infix2_gen with type ('a, 'b) fn := ('a[@local]) -> 'b

module type For_let_syntax2_gen = sig
  type ('a, 'e) t
  type ('a, 'b) fn
  type ('a, 'b) f_labeled_fn

  val return : 'a -> ('a, _) t
  val map : ('a, 'e) t -> ('a -> 'b, ('b, 'e) t) f_labeled_fn
  val both : ('a, 'e) t -> ('b, 'e) t -> ('a * 'b, 'e) t

  include
    Applicative_infix2_gen
    with type ('a, 'e) t := ('a, 'e) t
     and type ('a, 'b) fn := ('a, 'b) fn
end

module type For_let_syntax2 =
  For_let_syntax2_gen
  with type ('a, 'b) fn := 'a -> 'b
   and type ('a, 'b) f_labeled_fn := f:'a -> 'b

module type For_let_syntax2_local =
  For_let_syntax2_gen
  with type ('a, 'b) fn := ('a[@local]) -> 'b
   and type ('a, 'b) f_labeled_fn := f:('a[@local]) -> 'b

module type S2_gen = sig
  include For_let_syntax2_gen

  type ('a, 'b, 'c) fun2
  type ('a, 'b, 'c, 'd) fun3

  val apply : ('a -> 'b, 'e) t -> ('a, 'e) t -> ('b, 'e) t
  val map2 : ('a, 'e) t -> ('b, 'e) t -> (('a, 'b, 'c) fun2, ('c, 'e) t) f_labeled_fn

  val map3
    :  ('a, 'e) t
    -> ('b, 'e) t
    -> ('c, 'e) t
    -> (('a, 'b, 'c, 'd) fun3, ('d, 'e) t) f_labeled_fn

  val all : ('a, 'e) t list -> ('a list, 'e) t
  val all_unit : (unit, 'e) t list -> (unit, 'e) t

  module Applicative_infix :
    Applicative_infix2_gen
    with type ('a, 'e) t := ('a, 'e) t
     and type ('a, 'b) fn := ('a, 'b) fn
end

module type S2 =
  S2_gen
  with type ('a, 'b) fn := 'a -> 'b
   and type ('a, 'b) f_labeled_fn := f:'a -> 'b
   and type ('a, 'b, 'c) fun2 := 'a -> 'b -> 'c
   and type ('a, 'b, 'c, 'd) fun3 := 'a -> 'b -> 'c -> 'd

module type S2_local =
  S2_gen
  with type ('a, 'b) fn := ('a[@local]) -> 'b
   and type ('a, 'b) f_labeled_fn := f:('a[@local]) -> 'b
   and type ('a, 'b, 'c) fun2 := 'a -> (('b -> 'c)[@local])
   and type ('a, 'b, 'c, 'd) fun3 := 'a -> (('b -> (('c -> 'd)[@local]))[@local])

module type Let_syntax2 = sig
  type ('a, 'e) t

  module Open_on_rhs_intf : sig
    module type S
  end

  module Let_syntax : sig
    val return : 'a -> ('a, _) t

    include Applicative_infix2 with type ('a, 'e) t := ('a, 'e) t

    module Let_syntax : sig
      val return : 'a -> ('a, _) t
      val map : ('a, 'e) t -> f:('a -> 'b) -> ('b, 'e) t
      val both : ('a, 'e) t -> ('b, 'e) t -> ('a * 'b, 'e) t

      module Open_on_rhs : Open_on_rhs_intf.S
    end
  end
end

module type Basic3 = sig
  type ('a, 'd, 'e) t

  val return : 'a -> ('a, _, _) t
  val apply : ('a -> 'b, 'd, 'e) t -> ('a, 'd, 'e) t -> ('b, 'd, 'e) t

  val map
    : [ `Define_using_apply
      | `Custom of ('a, 'd, 'e) t -> f:('a -> 'b) -> ('b, 'd, 'e) t
      ]
end

module type Basic3_using_map2 = sig
  type ('a, 'd, 'e) t

  val return : 'a -> ('a, _, _) t
  val map2 : ('a, 'd, 'e) t -> ('b, 'd, 'e) t -> f:('a -> 'b -> 'c) -> ('c, 'd, 'e) t

  val map
    : [ `Define_using_map2 | `Custom of ('a, 'd, 'e) t -> f:('a -> 'b) -> ('b, 'd, 'e) t ]
end

module type Basic3_using_map2_local = sig
  type ('a, 'd, 'e) t

  val return : 'a -> ('a, _, _) t

  val map2
    :  ('a, 'd, 'e) t
    -> ('b, 'd, 'e) t
    -> f:(('a -> 'b -> 'c)[@local])
    -> ('c, 'd, 'e) t

  val map
    : [ `Define_using_map2
      | `Custom of ('a, 'd, 'e) t -> f:(('a -> 'b)[@local]) -> ('b, 'd, 'e) t
      ]
end

module type Applicative_infix3_gen = sig
  type ('a, 'd, 'e) t
  type ('a, 'b) fn

  val ( <*> ) : ('a -> 'b, 'd, 'e) t -> ('a, 'd, 'e) t -> ('b, 'd, 'e) t
  val ( <* ) : ('a, 'd, 'e) t -> (unit, 'd, 'e) t -> ('a, 'd, 'e) t
  val ( *> ) : (unit, 'd, 'e) t -> ('a, 'd, 'e) t -> ('a, 'd, 'e) t
  val ( >>| ) : ('a, 'd, 'e) t -> ('a -> 'b, ('b, 'd, 'e) t) fn
end

module type Applicative_infix3 = Applicative_infix3_gen with type ('a, 'b) fn := 'a -> 'b

module type Applicative_infix3_local =
  Applicative_infix3_gen with type ('a, 'b) fn := ('a[@local]) -> 'b

module type For_let_syntax3_gen = sig
  type ('a, 'd, 'e) t
  type ('a, 'b) fn
  type ('a, 'b) f_labeled_fn

  val return : 'a -> ('a, _, _) t
  val map : ('a, 'd, 'e) t -> ('a -> 'b, ('b, 'd, 'e) t) f_labeled_fn
  val both : ('a, 'd, 'e) t -> ('b, 'd, 'e) t -> ('a * 'b, 'd, 'e) t

  include
    Applicative_infix3_gen
    with type ('a, 'd, 'e) t := ('a, 'd, 'e) t
     and type ('a, 'b) fn := ('a, 'b) fn
end

module type For_let_syntax3 =
  For_let_syntax3_gen
  with type ('a, 'b) fn := 'a -> 'b
   and type ('a, 'b) f_labeled_fn := f:'a -> 'b

module type For_let_syntax3_local =
  For_let_syntax3_gen
  with type ('a, 'b) fn := ('a[@local]) -> 'b
   and type ('a, 'b) f_labeled_fn := f:('a[@local]) -> 'b

module type S3_gen = sig
  include For_let_syntax3_gen

  type ('a, 'b, 'c) fun2
  type ('a, 'b, 'c, 'd) fun3

  val apply : ('a -> 'b, 'd, 'e) t -> ('a, 'd, 'e) t -> ('b, 'd, 'e) t

  val map2
    :  ('a, 'd, 'e) t
    -> ('b, 'd, 'e) t
    -> (('a, 'b, 'c) fun2, ('c, 'd, 'e) t) f_labeled_fn

  val map3
    :  ('a, 'd, 'e) t
    -> ('b, 'd, 'e) t
    -> ('c, 'd, 'e) t
    -> (('a, 'b, 'c, 'result) fun3, ('result, 'd, 'e) t) f_labeled_fn

  val all : ('a, 'd, 'e) t list -> ('a list, 'd, 'e) t
  val all_unit : (unit, 'd, 'e) t list -> (unit, 'd, 'e) t

  module Applicative_infix :
    Applicative_infix3_gen
    with type ('a, 'd, 'e) t := ('a, 'd, 'e) t
     and type ('a, 'b) fn := ('a, 'b) fn
end

module type S3 =
  S3_gen
  with type ('a, 'b) fn := 'a -> 'b
   and type ('a, 'b) f_labeled_fn := f:'a -> 'b
   and type ('a, 'b, 'c) fun2 := 'a -> 'b -> 'c
   and type ('a, 'b, 'c, 'd) fun3 := 'a -> 'b -> 'c -> 'd

module type S3_local =
  S3_gen
  with type ('a, 'b) fn := ('a[@local]) -> 'b
   and type ('a, 'b) f_labeled_fn := f:('a[@local]) -> 'b
   and type ('a, 'b, 'c) fun2 := 'a -> (('b -> 'c)[@local])
   and type ('a, 'b, 'c, 'd) fun3 := 'a -> (('b -> (('c -> 'd)[@local]))[@local])

module type Let_syntax3 = sig
  type ('a, 'd, 'e) t

  module Open_on_rhs_intf : sig
    module type S
  end

  module Let_syntax : sig
    val return : 'a -> ('a, _, _) t

    include Applicative_infix3 with type ('a, 'd, 'e) t := ('a, 'd, 'e) t

    module Let_syntax : sig
      val return : 'a -> ('a, _, _) t
      val map : ('a, 'd, 'e) t -> f:('a -> 'b) -> ('b, 'd, 'e) t
      val both : ('a, 'd, 'e) t -> ('b, 'd, 'e) t -> ('a * 'b, 'd, 'e) t

      module Open_on_rhs : Open_on_rhs_intf.S
    end
  end
end

(** [Lazy_applicative] is an applicative whose structure may be computed on-demand,
    instead of being constructed up-front. This is useful when implementing traversals
    over large data structures, where otherwise we have to pay O(n) up-front cost both
    in time and in memory. *)
module type Lazy_applicative = sig
  include S

  val of_thunk : (unit -> 'a t) -> 'a t
end

module type Applicative = sig
  module type Applicative_infix = Applicative_infix
  module type Applicative_infix2 = Applicative_infix2
  module type Applicative_infix3 = Applicative_infix3
  module type Applicative_infix_local = Applicative_infix_local
  module type Applicative_infix2_local = Applicative_infix2_local
  module type Basic = Basic
  module type Basic2 = Basic2
  module type Basic3 = Basic3
  module type Basic_local = Basic_local
  module type Basic2_local = Basic2_local
  module type Basic_using_map2 = Basic_using_map2
  module type Basic2_using_map2 = Basic2_using_map2
  module type Basic3_using_map2 = Basic3_using_map2
  module type Basic_using_map2_local = Basic_using_map2_local
  module type Basic2_using_map2_local = Basic2_using_map2_local
  module type Basic3_using_map2_local = Basic3_using_map2_local
  module type Let_syntax = Let_syntax
  module type Let_syntax2 = Let_syntax2
  module type Let_syntax3 = Let_syntax3
  module type S = S
  module type S2 = S2
  module type S3 = S3
  module type Lazy_applicative = Lazy_applicative
  module type S_local = S_local
  module type S2_local = S2_local

  module S2_to_S (T : T.T) (X : S2) : S with type 'a t = ('a, T.t) X.t
  module S_to_S2 (X : S) : S2 with type ('a, 'e) t = 'a X.t
  module S3_to_S2 (T : T.T) (X : S3) : S2 with type ('a, 'd) t = ('a, 'd, T.t) X.t
  module S3_to_S (T1 : T.T) (T2 : T.T) (X : S3) : S with type 'a t = ('a, T1.t, T2.t) X.t
  module S2_to_S3 (X : S2) : S3 with type ('a, 'd, 'e) t = ('a, 'd) X.t
  module Make (X : Basic) : S with type 'a t := 'a X.t
  module Make2 (X : Basic2) : S2 with type ('a, 'e) t := ('a, 'e) X.t
  module Make3 (X : Basic3) : S3 with type ('a, 'd, 'e) t := ('a, 'd, 'e) X.t

  module Make_let_syntax
      (X : For_let_syntax) (Intf : sig
                              module type S
                            end)
      (Impl : Intf.S) :
    Let_syntax with type 'a t := 'a X.t with module Open_on_rhs_intf := Intf

  module Make_let_syntax2
      (X : For_let_syntax2) (Intf : sig
                               module type S
                             end)
      (Impl : Intf.S) :
    Let_syntax2 with type ('a, 'e) t := ('a, 'e) X.t with module Open_on_rhs_intf := Intf

  module Make_let_syntax3
      (X : For_let_syntax3) (Intf : sig
                               module type S
                             end)
      (Impl : Intf.S) :
    Let_syntax3
    with type ('a, 'd, 'e) t := ('a, 'd, 'e) X.t
    with module Open_on_rhs_intf := Intf

  module Make_using_map2 (X : Basic_using_map2) : S with type 'a t := 'a X.t

  module Make2_using_map2 (X : Basic2_using_map2) :
    S2 with type ('a, 'e) t := ('a, 'e) X.t

  module Make3_using_map2 (X : Basic3_using_map2) :
    S3 with type ('a, 'd, 'e) t := ('a, 'd, 'e) X.t

  module Make_using_map2_local (X : Basic_using_map2_local) :
    S_local with type 'a t := 'a X.t

  module Make2_using_map2_local (X : Basic2_using_map2_local) :
    S2_local with type ('a, 'e) t := ('a, 'e) X.t

  module Make3_using_map2_local (X : Basic3_using_map2_local) :
    S3_local with type ('a, 'd, 'e) t := ('a, 'd, 'e) X.t

  (** The following functors give a sense of what Applicatives one can define.

      Of these, [Of_monad] is likely the most useful.  The others are mostly didactic. *)

  (** Every monad is Applicative via:

      {[
        let apply mf mx =
          mf >>= fun f ->
          mx >>| fun x ->
          f x
      ]} *)
  module Of_monad (M : Monad.S) : S with type 'a t := 'a M.t

  module Of_monad2 (M : Monad.S2) : S2 with type ('a, 'e) t := ('a, 'e) M.t
  module Compose (F : S) (G : S) : S with type 'a t = 'a F.t G.t
  module Pair (F : S) (G : S) : S with type 'a t = 'a F.t * 'a G.t
end

* The [map] argument to [Applicative.Make] says how to implement the applicative's
      [map] function.  [`Define_using_apply] means to define [map t ~f = return f <*> t].
      [`Custom] overrides the default implementation, presumably with something more
      efficient.

      Some other functions returned by [Applicative.Make] are defined in terms of [map],
      so passing in a more efficient [map] will improve their efficiency as well. 
* Similar to [Basic], with the same laws, and the additional requirement that ['a t]
    can be mapped with a local function. 
* same as [apply] 
* [Lazy_applicative] is an applicative whose structure may be computed on-demand,
    instead of being constructed up-front. This is useful when implementing traversals
    over large data structures, where otherwise we have to pay O(n) up-front cost both
    in time and in memory. 
* The following functors give a sense of what Applicatives one can define.

      Of these, [Of_monad] is likely the most useful.  The others are mostly didactic. 
* Every monad is Applicative via:

      {[
        let apply mf mx =
          mf >>= fun f ->
          mx >>| fun x ->
          f x
      ]} 
* Applicatives model computations in which values computed by subcomputations can not 
     affect what subsequent computations will take place . 

     Relative to monads , this restriction takes power away from the user of the interface 
     and gives it to the implementation .   In particular , because the structure of the 
     entire computation is known , one can augment its definition with some description of 
     that structure . 

     For more information , see : 

     { v 
       Applicative Programming with Effects . 
         and   . 
       Journal of Functional Programming 18:1 ( 2008 ) , pages 1 - 13 . 
       /~ross/papers/Applicative.pdf 
     v }
    affect what subsequent computations will take place.

    Relative to monads, this restriction takes power away from the user of the interface
    and gives it to the implementation.  In particular, because the structure of the
    entire computation is known, one can augment its definition with some description of
    that structure.

    For more information, see:

    {v
      Applicative Programming with Effects.
      Conor McBride and Ross Paterson.
      Journal of Functional Programming 18:1 (2008), pages 1-13.
      /~ross/papers/Applicative.pdf
    v} *)

open! Import

module type Basic = sig
  type 'a t

  val return : 'a -> 'a t
  val apply : ('a -> 'b) t -> 'a t -> 'b t

* The following identities ought to hold for every Applicative ( for some value of = ): 

       - identity :      [ return Fn.id < * > t = t ] 
       - composition :   [ return Fn.compose < * > tf < * > tg < * > tx = tf < * > ( tg < * >  ) ] 
       - homomorphism : [ return f < * > return x = return ( f x ) ] 
       - interchange :   [ tf < * > return x = return ( fun f - > f x ) < * > tf ] 

       Note : < * > is the infix notation for apply .

      - identity:     [return Fn.id <*> t = t]
      - composition:  [return Fn.compose <*> tf <*> tg <*> tx = tf <*> (tg <*> tx)]
      - homomorphism: [return f <*> return x = return (f x)]
      - interchange:  [tf <*> return x = return (fun f -> f x) <*> tf]

      Note: <*> is the infix notation for apply. *)

  val map : [ `Define_using_apply | `Custom of 'a t -> f:('a -> 'b) -> 'b t ]
end

module type Basic_local = sig
  type 'a t

  val return : 'a -> 'a t
  val apply : ('a -> 'b) t -> 'a t -> 'b t
  val map : 'a t -> f:(('a -> 'b)[@local]) -> 'b t
end

module type Basic_using_map2 = sig
  type 'a t

  val return : 'a -> 'a t
  val map2 : 'a t -> 'b t -> f:('a -> 'b -> 'c) -> 'c t
  val map : [ `Define_using_map2 | `Custom of 'a t -> f:('a -> 'b) -> 'b t ]
end

module type Basic_using_map2_local = sig
  type 'a t

  val return : 'a -> 'a t
  val map2 : 'a t -> 'b t -> f:(('a -> 'b -> 'c)[@local]) -> 'c t
  val map : [ `Define_using_map2 | `Custom of 'a t -> f:(('a -> 'b)[@local]) -> 'b t ]
end

module type Applicative_infix_gen = sig
  type 'a t
  type ('a, 'b) fn


  val ( <*> ) : ('a -> 'b) t -> 'a t -> 'b t

  val ( <* ) : 'a t -> unit t -> 'a t
  val ( *> ) : unit t -> 'a t -> 'a t
  val ( >>| ) : 'a t -> ('a -> 'b, 'b t) fn
end

module type Applicative_infix = Applicative_infix_gen with type ('a, 'b) fn := 'a -> 'b

module type Applicative_infix_local =
  Applicative_infix_gen with type ('a, 'b) fn := ('a[@local]) -> 'b

module type For_let_syntax_gen = sig
  type 'a t
  type ('a, 'b) fn
  type ('a, 'b) f_labeled_fn

  val return : 'a -> 'a t
  val map : 'a t -> ('a -> 'b, 'b t) f_labeled_fn
  val both : 'a t -> 'b t -> ('a * 'b) t

  include Applicative_infix_gen with type 'a t := 'a t and type ('a, 'b) fn := ('a, 'b) fn
end

module type For_let_syntax =
  For_let_syntax_gen
  with type ('a, 'b) fn := 'a -> 'b
   and type ('a, 'b) f_labeled_fn := f:'a -> 'b

module type For_let_syntax_local =
  For_let_syntax_gen
  with type ('a, 'b) fn := ('a[@local]) -> 'b
   and type ('a, 'b) f_labeled_fn := f:('a[@local]) -> 'b

module type S_gen = sig
  include For_let_syntax_gen

  type ('a, 'b, 'c) fun2
  type ('a, 'b, 'c, 'd) fun3

  val apply : ('a -> 'b) t -> 'a t -> 'b t
  val map2 : 'a t -> 'b t -> (('a, 'b, 'c) fun2, 'c t) f_labeled_fn
  val map3 : 'a t -> 'b t -> 'c t -> (('a, 'b, 'c, 'd) fun3, 'd t) f_labeled_fn
  val all : 'a t list -> 'a list t
  val all_unit : unit t list -> unit t

  module Applicative_infix :
    Applicative_infix_gen with type 'a t := 'a t and type ('a, 'b) fn := ('a, 'b) fn
end

module type S =
  S_gen
  with type ('a, 'b) fn := 'a -> 'b
   and type ('a, 'b) f_labeled_fn := f:'a -> 'b
   and type ('a, 'b, 'c) fun2 := 'a -> 'b -> 'c
   and type ('a, 'b, 'c, 'd) fun3 := 'a -> 'b -> 'c -> 'd

module type S_local =
  S_gen
  with type ('a, 'b) fn := ('a[@local]) -> 'b
   and type ('a, 'b) f_labeled_fn := f:('a[@local]) -> 'b
   and type ('a, 'b, 'c) fun2 := 'a -> (('b -> 'c)[@local])
   and type ('a, 'b, 'c, 'd) fun3 := 'a -> (('b -> (('c -> 'd)[@local]))[@local])

module type Let_syntax = sig
  type 'a t

  module Open_on_rhs_intf : sig
    module type S
  end

  module Let_syntax : sig
    val return : 'a -> 'a t

    include Applicative_infix with type 'a t := 'a t

    module Let_syntax : sig
      val return : 'a -> 'a t
      val map : 'a t -> f:('a -> 'b) -> 'b t
      val both : 'a t -> 'b t -> ('a * 'b) t

      module Open_on_rhs : Open_on_rhs_intf.S
    end
  end
end

module type Basic2 = sig
  type ('a, 'e) t

  val return : 'a -> ('a, _) t
  val apply : ('a -> 'b, 'e) t -> ('a, 'e) t -> ('b, 'e) t
  val map : [ `Define_using_apply | `Custom of ('a, 'e) t -> f:('a -> 'b) -> ('b, 'e) t ]
end

module type Basic2_local = sig
  type ('a, 'e) t

  val return : 'a -> ('a, _) t
  val apply : ('a -> 'b, 'e) t -> ('a, 'e) t -> ('b, 'e) t
  val map : ('a, 'e) t -> f:(('a -> 'b)[@local]) -> ('b, 'e) t
end

module type Basic2_using_map2 = sig
  type ('a, 'e) t

  val return : 'a -> ('a, _) t
  val map2 : ('a, 'e) t -> ('b, 'e) t -> f:('a -> 'b -> 'c) -> ('c, 'e) t
  val map : [ `Define_using_map2 | `Custom of ('a, 'e) t -> f:('a -> 'b) -> ('b, 'e) t ]
end

module type Basic2_using_map2_local = sig
  type ('a, 'e) t

  val return : 'a -> ('a, _) t
  val map2 : ('a, 'e) t -> ('b, 'e) t -> f:(('a -> 'b -> 'c)[@local]) -> ('c, 'e) t

  val map
    : [ `Define_using_map2
      | `Custom of ('a, 'e) t -> f:(('a -> 'b)[@local]) -> ('b, 'e) t
      ]
end

module type Applicative_infix2_gen = sig
  type ('a, 'e) t
  type ('a, 'b) fn

  val ( <*> ) : ('a -> 'b, 'e) t -> ('a, 'e) t -> ('b, 'e) t
  val ( <* ) : ('a, 'e) t -> (unit, 'e) t -> ('a, 'e) t
  val ( *> ) : (unit, 'e) t -> ('a, 'e) t -> ('a, 'e) t
  val ( >>| ) : ('a, 'e) t -> ('a -> 'b, ('b, 'e) t) fn
end

module type Applicative_infix2 = Applicative_infix2_gen with type ('a, 'b) fn := 'a -> 'b

module type Applicative_infix2_local =
  Applicative_infix2_gen with type ('a, 'b) fn := ('a[@local]) -> 'b

module type For_let_syntax2_gen = sig
  type ('a, 'e) t
  type ('a, 'b) fn
  type ('a, 'b) f_labeled_fn

  val return : 'a -> ('a, _) t
  val map : ('a, 'e) t -> ('a -> 'b, ('b, 'e) t) f_labeled_fn
  val both : ('a, 'e) t -> ('b, 'e) t -> ('a * 'b, 'e) t

  include
    Applicative_infix2_gen
    with type ('a, 'e) t := ('a, 'e) t
     and type ('a, 'b) fn := ('a, 'b) fn
end

module type For_let_syntax2 =
  For_let_syntax2_gen
  with type ('a, 'b) fn := 'a -> 'b
   and type ('a, 'b) f_labeled_fn := f:'a -> 'b

module type For_let_syntax2_local =
  For_let_syntax2_gen
  with type ('a, 'b) fn := ('a[@local]) -> 'b
   and type ('a, 'b) f_labeled_fn := f:('a[@local]) -> 'b

module type S2_gen = sig
  include For_let_syntax2_gen

  type ('a, 'b, 'c) fun2
  type ('a, 'b, 'c, 'd) fun3

  val apply : ('a -> 'b, 'e) t -> ('a, 'e) t -> ('b, 'e) t
  val map2 : ('a, 'e) t -> ('b, 'e) t -> (('a, 'b, 'c) fun2, ('c, 'e) t) f_labeled_fn

  val map3
    :  ('a, 'e) t
    -> ('b, 'e) t
    -> ('c, 'e) t
    -> (('a, 'b, 'c, 'd) fun3, ('d, 'e) t) f_labeled_fn

  val all : ('a, 'e) t list -> ('a list, 'e) t
  val all_unit : (unit, 'e) t list -> (unit, 'e) t

  module Applicative_infix :
    Applicative_infix2_gen
    with type ('a, 'e) t := ('a, 'e) t
     and type ('a, 'b) fn := ('a, 'b) fn
end

module type S2 =
  S2_gen
  with type ('a, 'b) fn := 'a -> 'b
   and type ('a, 'b) f_labeled_fn := f:'a -> 'b
   and type ('a, 'b, 'c) fun2 := 'a -> 'b -> 'c
   and type ('a, 'b, 'c, 'd) fun3 := 'a -> 'b -> 'c -> 'd

module type S2_local =
  S2_gen
  with type ('a, 'b) fn := ('a[@local]) -> 'b
   and type ('a, 'b) f_labeled_fn := f:('a[@local]) -> 'b
   and type ('a, 'b, 'c) fun2 := 'a -> (('b -> 'c)[@local])
   and type ('a, 'b, 'c, 'd) fun3 := 'a -> (('b -> (('c -> 'd)[@local]))[@local])

module type Let_syntax2 = sig
  type ('a, 'e) t

  module Open_on_rhs_intf : sig
    module type S
  end

  module Let_syntax : sig
    val return : 'a -> ('a, _) t

    include Applicative_infix2 with type ('a, 'e) t := ('a, 'e) t

    module Let_syntax : sig
      val return : 'a -> ('a, _) t
      val map : ('a, 'e) t -> f:('a -> 'b) -> ('b, 'e) t
      val both : ('a, 'e) t -> ('b, 'e) t -> ('a * 'b, 'e) t

      module Open_on_rhs : Open_on_rhs_intf.S
    end
  end
end

module type Basic3 = sig
  type ('a, 'd, 'e) t

  val return : 'a -> ('a, _, _) t
  val apply : ('a -> 'b, 'd, 'e) t -> ('a, 'd, 'e) t -> ('b, 'd, 'e) t

  val map
    : [ `Define_using_apply
      | `Custom of ('a, 'd, 'e) t -> f:('a -> 'b) -> ('b, 'd, 'e) t
      ]
end

module type Basic3_using_map2 = sig
  type ('a, 'd, 'e) t

  val return : 'a -> ('a, _, _) t
  val map2 : ('a, 'd, 'e) t -> ('b, 'd, 'e) t -> f:('a -> 'b -> 'c) -> ('c, 'd, 'e) t

  val map
    : [ `Define_using_map2 | `Custom of ('a, 'd, 'e) t -> f:('a -> 'b) -> ('b, 'd, 'e) t ]
end

module type Basic3_using_map2_local = sig
  type ('a, 'd, 'e) t

  val return : 'a -> ('a, _, _) t

  val map2
    :  ('a, 'd, 'e) t
    -> ('b, 'd, 'e) t
    -> f:(('a -> 'b -> 'c)[@local])
    -> ('c, 'd, 'e) t

  val map
    : [ `Define_using_map2
      | `Custom of ('a, 'd, 'e) t -> f:(('a -> 'b)[@local]) -> ('b, 'd, 'e) t
      ]
end

module type Applicative_infix3_gen = sig
  type ('a, 'd, 'e) t
  type ('a, 'b) fn

  val ( <*> ) : ('a -> 'b, 'd, 'e) t -> ('a, 'd, 'e) t -> ('b, 'd, 'e) t
  val ( <* ) : ('a, 'd, 'e) t -> (unit, 'd, 'e) t -> ('a, 'd, 'e) t
  val ( *> ) : (unit, 'd, 'e) t -> ('a, 'd, 'e) t -> ('a, 'd, 'e) t
  val ( >>| ) : ('a, 'd, 'e) t -> ('a -> 'b, ('b, 'd, 'e) t) fn
end

module type Applicative_infix3 = Applicative_infix3_gen with type ('a, 'b) fn := 'a -> 'b

module type Applicative_infix3_local =
  Applicative_infix3_gen with type ('a, 'b) fn := ('a[@local]) -> 'b

module type For_let_syntax3_gen = sig
  type ('a, 'd, 'e) t
  type ('a, 'b) fn
  type ('a, 'b) f_labeled_fn

  val return : 'a -> ('a, _, _) t
  val map : ('a, 'd, 'e) t -> ('a -> 'b, ('b, 'd, 'e) t) f_labeled_fn
  val both : ('a, 'd, 'e) t -> ('b, 'd, 'e) t -> ('a * 'b, 'd, 'e) t

  include
    Applicative_infix3_gen
    with type ('a, 'd, 'e) t := ('a, 'd, 'e) t
     and type ('a, 'b) fn := ('a, 'b) fn
end

module type For_let_syntax3 =
  For_let_syntax3_gen
  with type ('a, 'b) fn := 'a -> 'b
   and type ('a, 'b) f_labeled_fn := f:'a -> 'b

module type For_let_syntax3_local =
  For_let_syntax3_gen
  with type ('a, 'b) fn := ('a[@local]) -> 'b
   and type ('a, 'b) f_labeled_fn := f:('a[@local]) -> 'b

module type S3_gen = sig
  include For_let_syntax3_gen

  type ('a, 'b, 'c) fun2
  type ('a, 'b, 'c, 'd) fun3

  val apply : ('a -> 'b, 'd, 'e) t -> ('a, 'd, 'e) t -> ('b, 'd, 'e) t

  val map2
    :  ('a, 'd, 'e) t
    -> ('b, 'd, 'e) t
    -> (('a, 'b, 'c) fun2, ('c, 'd, 'e) t) f_labeled_fn

  val map3
    :  ('a, 'd, 'e) t
    -> ('b, 'd, 'e) t
    -> ('c, 'd, 'e) t
    -> (('a, 'b, 'c, 'result) fun3, ('result, 'd, 'e) t) f_labeled_fn

  val all : ('a, 'd, 'e) t list -> ('a list, 'd, 'e) t
  val all_unit : (unit, 'd, 'e) t list -> (unit, 'd, 'e) t

  module Applicative_infix :
    Applicative_infix3_gen
    with type ('a, 'd, 'e) t := ('a, 'd, 'e) t
     and type ('a, 'b) fn := ('a, 'b) fn
end

module type S3 =
  S3_gen
  with type ('a, 'b) fn := 'a -> 'b
   and type ('a, 'b) f_labeled_fn := f:'a -> 'b
   and type ('a, 'b, 'c) fun2 := 'a -> 'b -> 'c
   and type ('a, 'b, 'c, 'd) fun3 := 'a -> 'b -> 'c -> 'd

module type S3_local =
  S3_gen
  with type ('a, 'b) fn := ('a[@local]) -> 'b
   and type ('a, 'b) f_labeled_fn := f:('a[@local]) -> 'b
   and type ('a, 'b, 'c) fun2 := 'a -> (('b -> 'c)[@local])
   and type ('a, 'b, 'c, 'd) fun3 := 'a -> (('b -> (('c -> 'd)[@local]))[@local])

module type Let_syntax3 = sig
  type ('a, 'd, 'e) t

  module Open_on_rhs_intf : sig
    module type S
  end

  module Let_syntax : sig
    val return : 'a -> ('a, _, _) t

    include Applicative_infix3 with type ('a, 'd, 'e) t := ('a, 'd, 'e) t

    module Let_syntax : sig
      val return : 'a -> ('a, _, _) t
      val map : ('a, 'd, 'e) t -> f:('a -> 'b) -> ('b, 'd, 'e) t
      val both : ('a, 'd, 'e) t -> ('b, 'd, 'e) t -> ('a * 'b, 'd, 'e) t

      module Open_on_rhs : Open_on_rhs_intf.S
    end
  end
end

module type Lazy_applicative = sig
  include S

  val of_thunk : (unit -> 'a t) -> 'a t
end

module type Applicative = sig
  module type Applicative_infix = Applicative_infix
  module type Applicative_infix2 = Applicative_infix2
  module type Applicative_infix3 = Applicative_infix3
  module type Applicative_infix_local = Applicative_infix_local
  module type Applicative_infix2_local = Applicative_infix2_local
  module type Basic = Basic
  module type Basic2 = Basic2
  module type Basic3 = Basic3
  module type Basic_local = Basic_local
  module type Basic2_local = Basic2_local
  module type Basic_using_map2 = Basic_using_map2
  module type Basic2_using_map2 = Basic2_using_map2
  module type Basic3_using_map2 = Basic3_using_map2
  module type Basic_using_map2_local = Basic_using_map2_local
  module type Basic2_using_map2_local = Basic2_using_map2_local
  module type Basic3_using_map2_local = Basic3_using_map2_local
  module type Let_syntax = Let_syntax
  module type Let_syntax2 = Let_syntax2
  module type Let_syntax3 = Let_syntax3
  module type S = S
  module type S2 = S2
  module type S3 = S3
  module type Lazy_applicative = Lazy_applicative
  module type S_local = S_local
  module type S2_local = S2_local

  module S2_to_S (T : T.T) (X : S2) : S with type 'a t = ('a, T.t) X.t
  module S_to_S2 (X : S) : S2 with type ('a, 'e) t = 'a X.t
  module S3_to_S2 (T : T.T) (X : S3) : S2 with type ('a, 'd) t = ('a, 'd, T.t) X.t
  module S3_to_S (T1 : T.T) (T2 : T.T) (X : S3) : S with type 'a t = ('a, T1.t, T2.t) X.t
  module S2_to_S3 (X : S2) : S3 with type ('a, 'd, 'e) t = ('a, 'd) X.t
  module Make (X : Basic) : S with type 'a t := 'a X.t
  module Make2 (X : Basic2) : S2 with type ('a, 'e) t := ('a, 'e) X.t
  module Make3 (X : Basic3) : S3 with type ('a, 'd, 'e) t := ('a, 'd, 'e) X.t

  module Make_let_syntax
      (X : For_let_syntax) (Intf : sig
                              module type S
                            end)
      (Impl : Intf.S) :
    Let_syntax with type 'a t := 'a X.t with module Open_on_rhs_intf := Intf

  module Make_let_syntax2
      (X : For_let_syntax2) (Intf : sig
                               module type S
                             end)
      (Impl : Intf.S) :
    Let_syntax2 with type ('a, 'e) t := ('a, 'e) X.t with module Open_on_rhs_intf := Intf

  module Make_let_syntax3
      (X : For_let_syntax3) (Intf : sig
                               module type S
                             end)
      (Impl : Intf.S) :
    Let_syntax3
    with type ('a, 'd, 'e) t := ('a, 'd, 'e) X.t
    with module Open_on_rhs_intf := Intf

  module Make_using_map2 (X : Basic_using_map2) : S with type 'a t := 'a X.t

  module Make2_using_map2 (X : Basic2_using_map2) :
    S2 with type ('a, 'e) t := ('a, 'e) X.t

  module Make3_using_map2 (X : Basic3_using_map2) :
    S3 with type ('a, 'd, 'e) t := ('a, 'd, 'e) X.t

  module Make_using_map2_local (X : Basic_using_map2_local) :
    S_local with type 'a t := 'a X.t

  module Make2_using_map2_local (X : Basic2_using_map2_local) :
    S2_local with type ('a, 'e) t := ('a, 'e) X.t

  module Make3_using_map2_local (X : Basic3_using_map2_local) :
    S3_local with type ('a, 'd, 'e) t := ('a, 'd, 'e) X.t


  module Of_monad (M : Monad.S) : S with type 'a t := 'a M.t

  module Of_monad2 (M : Monad.S2) : S2 with type ('a, 'e) t := ('a, 'e) M.t
  module Compose (F : S) (G : S) : S with type 'a t = 'a F.t G.t
  module Pair (F : S) (G : S) : S with type 'a t = 'a F.t * 'a G.t
end
