##### 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 (Ord : OrderedType) : S with type elt = Ord.t and type t = Stdlib.Set.Make(Ord).t`

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

`module Make2 (O1 : OrderedType) (O2 : OrderedType) : sig ... end`

## Common instantiations

`module Int : S with type elt = int`
`module Int32 : S with type elt = int32`
`module Int64 : S with type elt = int64`
`module Nativeint : S with type elt = nativeint`
`module Float : S with type elt = float`
`module Char : S with type elt = char`
`module String : S with type elt = string`

##### Polymorphic sets

The definitions below describe the polymorphic set interface.

They are similar in functionality to the functorized `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 `Make` functor for additional safety.

- author Nicolas Cannasse

- author Markus Mottl

- author David Rajchenbach-Teller

`type +'a t`

The type of sets.

`include BatEnum.Enumerable with type 'a enumerable = 'a t`
`type 'a enumerable = 'a t`

The data structure, e.g. `'a List.t`

`include BatInterfaces.Mappable with type 'a mappable = 'a t`
`type 'a mappable = 'a t`

The data structure, e.g. `'a List.t`

`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 is_singleton : 'a t -> bool`

Test if the set is a singleton.

`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`.

- raises `Not_found`

  if no such element exists

- since 2.1

`val find_opt : 'a -> 'a t -> 'a option`

`find_opt x s` returns `Some k` for the element `k` in `s` that tests equal to `x` under its comparison function. If no element is equal, return `None`

- since 3.3.0

`val find_first : ('a -> bool) -> 'a t -> 'a`

`find_first f m` returns the first element `e` for which `f e` is true or raises `Not_found` if there is no such element. `f` must be monotonically increasing, i.e. if `k1 < k2 && f k1` is true then `f k2` must also be true.

- since 3.3.0

`val find_first_opt : ('a -> bool) -> 'a t -> 'a option`

`find_first_opt f m` returns `Some e` for the first element `e` for which `f e` is true or returns `None` if there is no such element. `f` must be monotonically increasing, i.e. if `k1 < k2 && f k1` is true then `f k2` must also be true.

- since 3.3.0

`val find_last : ('a -> bool) -> 'a t -> 'a`

`find_last f m` returns the last element `e` for which `f e` is true or raises `Not_found` if there is no such element. `f` must be monotonically decreasing, i.e. if `k1 < k2 && f k2` is true then `f k1` must also be true.

- since 3.3.0

`val find_last_opt : ('a -> bool) -> 'a t -> 'a option`

`find_last_opt f m` returns `Some e` for the last element `e` for which `f e` is true or returns `None` if there is no such element. `f` must be monotonically decreasing, i.e. if `k1 < k2 && f k2` is true then `f k1` must also be true.

- since 3.3.0

`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.

- before 3.3.0

  or OCaml 4.03 Physical equality was not ensured.

`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.

- before 3.3.0

  or OCaml 4.03 Physical equality was not ensured.

`val remove_exn : 'a -> 'a t -> 'a t`

`remove_exn x s` behaves like `remove x s` except that it raises an exception if `x` is not in `s`.

- raises `Not_found`

  if `x` is not in `s`.

- since 3.2.0

`val update : 'a -> 'a -> 'a t -> 'a t`

`update x y s` replace `x` by `y` in `s`. `update` is faster when `x` compares equal to `y` according to the comparison function used by your set. When `x` and `y` are physically equal, `m` is returned unchanged.

- raises `Not_found`

  if `x` is not in `s`.

- since 2.4

- before 3.3.0

  Physical equality was not ensured.

`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 at_rank_exn : int -> 'a t -> 'a`

`at_rank_exn i s` returns element at rank `i` in `s`, that is the `i`-th element in increasing order (the `0`-th element being the smallest element of `s`).

- raises `Not_found`

  if `s = empty`.

- raises `Invalid_argument`

  error_message if `i < 0 || i >= cardinal s`

- since 2.4

`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 map_endo : ('a -> 'a) -> 'a t -> 'a t`

`map_endo 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` (beyond the type signature being more restricted than for `map` above); 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.

This version of map will result in a physically equal map if `f` returns physically equal keys.

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

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

if `p` returns `true` for all elements then `s` is returned unmodified.

- before 3.3.0

  or OCaml 4.03 Physical equality was not ensured.

`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 filter_map_endo : ('a -> 'a option) -> 'a t -> 'a t`

`filter_map_endo 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.

If the filter function `f` returns `true` for all elements in `m`, the resulting map is physically equal to `m`.

`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 to_array : 'a t -> 'a array`

Same as `to_list` but with an array instead of a list.

- since 2.4

`val min_elt : 'a t -> 'a`

returns the smallest element of the set.

- raises `Not_found`

  if given an empty set.

`val min_elt_opt : 'a t -> 'a option`

Return `Some e` for the smallest element `e` of the given set (with respect to the `Ord.compare` ordering). Return None if the set is empty.

- since 3.3.0

`val pop_min : 'a t -> 'a * 'a t`

Returns the smallest element of the given set along with the rest of the set. Semantically equivalent and faster than

`let mini = min_elt s in (mini, remove mini s)`

- raises `Not_found`

  if the set is empty.

- since 2.4

`val pop_max : 'a t -> 'a * 'a t`

Returns the biggest element of the given set along with the rest of the set. Semantically equivalent and faster than

`let maxi = max_elt s in (maxi, remove maxi s)`

- raises `Not_found`

  if the set is empty.

- since 2.4

`val max_elt : 'a t -> 'a`

returns the largest element of the set.

- raises `Not_found`

  if given an empty set.

`val max_elt_opt : 'a t -> 'a option`

Same as `Set.S.min_elt_opt`, but for the largest element of the given set.

- since 3.3.0

`val choose : 'a t -> 'a`

returns an arbitrary (but deterministic) element of the given set.

- raises `Not_found`

  if given an empty set.

`val choose_opt : 'a t -> 'a option`

Return `Some e` for one element `e` of the given set. Which element is chosen is unspecified, but equal elements will be chosen for equal sets. Return `None` if the set is empty.

- since 3.3.0

`val any : 'a t -> 'a`

Return one element of the given set. The difference with choose is that there is no guarantee that equals elements will be picked for equal sets. This merely returns the quickest element to get (O(1)).

- raises `Not_found`

  if the set is empty.

`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

`val of_array : 'a array -> 'a t`

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

`val to_seq : 'a t -> 'a BatSeq.t`

Iterate on the whole set, in ascending order.

- since 3.3.0

`val to_rev_seq : 'a t -> 'a BatSeq.t`

Iterate on the whole set, in descending order.

- since 3.3.0

`val to_seq_from : 'a -> 'a t -> 'a BatSeq.t`

`to_seq_from x s` iterates on a subset of the elements in `s`, namely those greater or equal to `x`, in ascending order.

- since 3.3.0

`val add_seq : 'a BatSeq.t -> 'a t -> 'a t`

add the given elements to the set, in order.

- since 3.3.0

`val of_seq : 'a BatSeq.t -> 'a t`

build a set from the given elements

- since 3.3.0

## 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`
`module PSet : sig ... end`