(***********************************************************************)
(*                                                                     *)
(*                                OCaml                                *)
(*                                                                     *)
(*            Xavier Leroy, projet Cristal, INRIA Rocquencourt         *)
(*                                                                     *)
(*  Copyright 1996 Institut National de Recherche en Informatique et   *)
(*  en Automatique.  All rights reserved.  This file is distributed    *)
(*  under the terms of the GNU Library General Public License, with    *)
(*  the special exception on linking described in file ../LICENSE.     *)
(*                                                                     *)
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(** Sets over ordered types.

   This module implements the set data structure, given a total ordering
   function over the set elements. All operations over sets
   are purely applicative (no side-effects).
   The implementation uses balanced binary trees, and is therefore
   reasonably efficient: insertion and membership take time
   logarithmic in the size of the set, for instance.

   The [Make] functor constructs implementations for any type, given a
   [compare] function.
   For instance:
   {[
     module IntPairs =
       struct
         type t = int * int
         let compare (x0,y0) (x1,y1) =
           match Pervasives.compare x0 x1 with
               0 -> Pervasives.compare y0 y1
             | c -> c
       end

     module PairsSet = Set.Make(IntPairs)

     let m = PairsSet.(empty |> add (2,3) |> add (5,7) |> add (11,13))
   ]}

   This creates a new module [PairsSet], with a new type [PairsSet.t]
   of sets of [int * int].
*)

module type OrderedType =
  sig
    type t
      (** The type of the set elements. *)
    val compare : t -> t -> int
      (** A total ordering function over the set elements.
          This is a two-argument function [f] such that
          [f e1 e2] is zero if the elements [e1] and [e2] are equal,
          [f e1 e2] is strictly negative if [e1] is smaller than [e2],
          and [f e1 e2] is strictly positive if [e1] is greater than [e2].
          Example: a suitable ordering function is the generic structural
          comparison function {!Pervasives.compare}. *)
  end
(** Input signature of the functor {!Set.Make}. *)

module type S =
  sig
    type elt
    (** The type of the set elements. *)

    type t
    (** The type of sets. *)

    val empty: t
    (** The empty set. *)

    val is_empty: t -> bool
    (** Test whether a set is empty or not. *)

    val mem: elt -> t -> bool
    (** [mem x s] tests whether [x] belongs to the set [s]. *)

    val add: elt -> t -> t
    (** [add x s] returns a set containing all elements of [s],
       plus [x]. If [x] was already in [s], [s] is returned unchanged. *)

    val singleton: elt -> t
    (** [singleton x] returns the one-element set containing only [x]. *)

    val remove: elt -> t -> t
    (** [remove x s] returns a set containing all elements of [s],
       except [x]. If [x] was not in [s], [s] is returned unchanged. *)

    val union: t -> t -> t
    (** Set union. *)

    val inter: t -> t -> t
    (** Set intersection. *)

    val diff: t -> t -> t
    (** Set difference. *)

    val compare: t -> t -> int
    (** Total ordering between sets. Can be used as the ordering function
       for doing sets of sets. *)

    val equal: t -> t -> bool
    (** [equal s1 s2] tests whether the sets [s1] and [s2] are
       equal, that is, contain equal elements. *)

    val subset: t -> t -> bool
    (** [subset s1 s2] tests whether the set [s1] is a subset of
       the set [s2]. *)

    val iter: (elt -> unit) -> t -> unit
    (** [iter f s] applies [f] in turn to all elements of [s].
       The elements of [s] are presented to [f] in increasing order
       with respect to the ordering over the type of the elements. *)

    val fold: (elt -> 'a -> 'a) -> t -> 'a -> 'a
    (** [fold f s a] computes [(f xN ... (f x2 (f x1 a))...)],
       where [x1 ... xN] are the elements of [s], in increasing order. *)

    val for_all: (elt -> bool) -> t -> bool
    (** [for_all p s] checks if all elements of the set
       satisfy the predicate [p]. *)

    val exists: (elt -> bool) -> t -> bool
    (** [exists p s] checks if at least one element of
       the set satisfies the predicate [p]. *)

    val filter: (elt -> bool) -> t -> t
    (** [filter p s] returns the set of all elements in [s]
       that satisfy predicate [p]. *)

    val partition: (elt -> bool) -> t -> t * t
    (** [partition p s] returns a pair of sets [(s1, s2)], where
       [s1] is the set of all the elements of [s] that satisfy the
       predicate [p], and [s2] is the set of all the elements of
       [s] that do not satisfy [p]. *)

    val cardinal: t -> int
    (** Return the number of elements of a set. *)

    val elements: t -> elt list
    (** Return the list of all elements of the given set.
       The returned list is sorted in increasing order with respect
       to the ordering [Ord.compare], where [Ord] is the argument
       given to {!Set.Make}. *)

    val min_elt: t -> elt
    (** Return the smallest element of the given set
       (with respect to the [Ord.compare] ordering), or raise
       [Not_found] if the set is empty. *)

    val max_elt: t -> elt
    (** Same as {!Set.S.min_elt}, but returns the largest element of the
       given set. *)

    val choose: t -> elt
    (** Return one element of the given set, or raise [Not_found] if
       the set is empty. Which element is chosen is unspecified,
       but equal elements will be chosen for equal sets. *)

    val split: elt -> t -> t * bool * t
    (** [split x s] returns a triple [(l, present, r)], where
          [l] is the set of elements of [s] that are
          strictly less than [x];
          [r] is the set of elements of [s] that are
          strictly greater than [x];
          [present] is [false] if [s] contains no element equal to [x],
          or [true] if [s] contains an element equal to [x]. *)

    val find: elt -> t -> elt
    (** [find x s] returns the element of [s] equal to [x] (according
        to [Ord.compare]), or raise [Not_found] if no such element
        exists.
        @since 4.01.0 *)

    val of_list: elt list -> t
    (** [of_list l] creates a set from a list of elements.
        This is usually more efficient than folding [add] over the list,
        except perhaps for lists with many duplicated elements.
        @since 4.02.0 *)
  end
(** Output signature of the functor {!Set.Make}. *)

module Make (Ord : OrderedType) : S with type elt = Ord.t
(** Functor building an implementation of the set structure
   given a totally ordered type. *)
