# Module BatSet

    module BatSet: sig .. end

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.

Note OCaml, Batteries Included, provides two implementations of sets: polymorphic sets and functorized sets. Functorized sets (see `BatSet.S` and `BatSet.Make`) are slightly more complex to use but offer stronger type-safety. Polymorphic sets make it easier to shoot yourself in the foot. In case of doubt, you should use functorized sets.

The functorized set implementation is built upon Stdlib's Set module, but provides the complete interface.
Author(s): Xavier Leroy, Nicolas Cannasse, Markus Mottl, David Rajchenbach-Teller

#### Functorized Sets

    module type OrderedType = BatInterfaces.OrderedType

Input signature of the functor `Set.Make`.

    module type S = sig .. end

Output signature of the functor `Set.Make`.

    module Make: functor (Ord : OrderedType) -> S  with type elt = Ord.t

Functor building an implementation of the set structure given a totally ordered type.

###### Polymorphic sets

The definitions below describe the polymorphic set interface.

They are similar in functionality to the functorized `BatSet.Make` module, but the compiler cannot ensure that sets using different element ordering have different types: the responsibility of not mixing non-sensical comparison functions together is to the programmer. If in doubt, you should rather use the `BatSet.Make` functor for additional safety.

    type 'a t

The type of sets.

    include BatEnum.Enumerable

    include BatInterfaces.Mappable

    val empty : 'a t

The empty set, using `compare` as comparison function

    val is_empty : 'a t -> bool

Test whether a set is empty or not.

    val singleton : 'a -> 'a t

Creates a new set with the single given element in it.

    val mem : 'a -> 'a t -> bool

`mem x s` tests whether `x` belongs to the set `s`.

    val find : 'a -> 'a t -> 'a

`find x s` returns the set element that compares equal to `x`.
Since 2.1
Raises `Not_found` if no such element exists

    val add : 'a -> 'a t -> 'a 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 remove : 'a -> 'a t -> 'a 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 : 'a t -> 'a t -> 'a t

`union s t` returns the union of `s` and `t` - the set containing all elements in either `s` and `t`. The returned set uses `t`'s comparison function. The current implementation works better for small `s`.

    val intersect : 'a t -> 'a t -> 'a t

`intersect s t` returns a new set of those elements that are in both `s` and `t`. The returned set uses `s`'s comparison function.

    val diff : 'a t -> 'a t -> 'a t

`diff s t` returns the set of all elements in `s` but not in `t`. The returned set uses `s`'s comparison function.

    val sym_diff : 'a t -> 'a t -> 'a t

`sym_diff s t` returns the set of all elements in `s` or `t` but not both, also known as the symmetric difference. This is the same as `diff (union s t) (inter s t)`. The returned set uses `s`'s comparison function.

    val compare : 'a t -> 'a t -> int

Total ordering between sets. Can be used as the ordering function for doing sets of sets.

    val equal : 'a t -> 'a t -> bool

`equal s1 s2` tests whether the sets `s1` and `s2` are equal, that is, contain equal elements.

    val subset : 'a t -> 'a t -> bool

`subset a b` returns true if `a` is a subset of `b`. O(|a|).

    val disjoint : 'a t -> 'a t -> bool

`disjoint s1 s2` tests whether the sets `s1` and `s2` contain no shared elements. (i.e. `inter s1 s2` is empty.)

    val iter : ('a -> unit) -> 'a 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 map : ('a -> 'b) -> 'a t -> 'b t

`map f x` creates a new set with elements `f a0`, `f a1`... `f aN`, where `a0`, `a1`, ..., `aN` are the elements of `x`.

This function places no restriction on `f`; it can map multiple input values to the same output value, in which case the resulting set will have smaller cardinality than the input. `f` does not need to be order preserving, although if it is, then `Incubator.op_map` may be more efficient.

    val filter : ('a -> bool) -> 'a t -> 'a t

`filter p s` returns the set of all elements in `s` that satisfy predicate `p`.

    val filter_map : ('a -> 'b option) -> 'a t -> 'b t

`filter_map f m` combines the features of `filter` and `map`. It calls calls `f a0`, `f a1`, `f aN` where `a0,a1..an` are the elements of `m` and returns the set of pairs `bi` such as `f ai = Some bi` (when `f` returns `None`, the corresponding element of `m` is discarded).

The resulting map uses the polymorphic `compare` function to order elements.

    val fold : ('a -> 'b -> 'b) -> 'a t -> 'b -> 'b

`fold f s a` computes `(f xN ... (f x1 (f x0 a))...)`, where `x0,x1..xN` are the elements of `s`, in increasing order.

    val exists : ('a -> bool) -> 'a t -> bool

`exists p s` checks if at least one element of the set satisfies the predicate `p`.

    val for_all : ('a -> bool) -> 'a t -> bool

Returns whether the given predicate applies to all elements in the set

    val partition : ('a -> bool) -> 'a t -> 'a t * 'a t

returns two disjoint subsets, those that satisfy the given predicate and those that don't

    val split : 'a -> 'a t -> 'a t * bool * 'a 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 split_opt : 'a -> 'a t -> 'a t * 'a option * 'a t

`split_opt x s` returns a triple `(l, maybe_v, 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`; `maybe_v` is `None` if `s` contains no element equal to `x`, or `Some v` if `s` contains an element `v` that compares equal to `x`.
Since 2.2.0

    val split_lt : 'a -> 'a t -> 'a t * 'a t

`split_lt x s` returns a pair of sets `(l, r)`, such that `l` is the subset of `s` with elements < `x`; `r` is the subset of `s` with elements >= `x`.
Since 2.2.0

    val split_le : 'a -> 'a t -> 'a t * 'a t

`split_le x s` returns a pair of sets `(l, r)`, such that `l` is the subset of `s` with elements <= `x`; `r` is the subset of `s` with elements > `x`.
Since 2.2.0

    val cardinal : 'a t -> int

Return the number of elements of a set.

    val elements : 'a t -> 'a list

Return the list of all elements of the given set. The returned list is sorted in increasing order with respect to the ordering of the given set.

    val to_list : 'a t -> 'a list

Alias for `elements`.
Since 2.2.0

    val min_elt : 'a t -> 'a

returns the smallest element of the set.
Raises `Invalid_argument` if given an empty set.

    val max_elt : 'a t -> 'a

returns the largest element of the set.
Raises `Invalid_argument` if given an empty set.

    val choose : 'a t -> 'a

returns an arbitrary (but deterministic) element of the given set.
Raises `Invalid_argument` if given an empty set.

    val pop : 'a t -> 'a * 'a t

returns one element of the set and the set without that element.
Raises `Not_found` if given an empty set

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

cartesian product of the two sets
Since 2.2.0

    val enum : 'a t -> 'a BatEnum.t

Return an enumeration of all elements of the given set. The returned enumeration is sorted in increasing order with respect to the ordering of this set.

    val of_enum : 'a BatEnum.t -> 'a t

    val backwards : 'a t -> 'a BatEnum.t

Return an enumeration of all elements of the given set. The returned enumeration is sorted in decreasing order with respect to the ordering `Pervasives.compare`.

    val of_list : 'a list -> 'a t

builds a set from the given list, using the default comparison function

###### Boilerplate code

Printing

    val print : ?first:string ->
           ?last:string ->
           ?sep:string ->
           ('a BatInnerIO.output -> 'c -> unit) ->
           'a BatInnerIO.output -> 'c t -> unit

    module Incubator: sig .. end

Incubator

    module PSet: sig .. end