module type S =`sig`

..`end`

Output signature of the functor `Map.Make`

.

`type `

key

The type of the map keys.

`type ``+'a`

t

The type of maps from type `key`

to type `'a`

.

`val empty : ``'a t`

The empty map.

`val is_empty : ``'a t -> bool`

Test whether a map is empty or not.

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

`mem x m`

returns `true`

if `m`

contains a binding for `x`

,
and `false`

otherwise.

`val add : ``key -> 'a -> 'a t -> 'a t`

`add x y m`

returns a map containing the same bindings as
`m`

, plus a binding of `x`

to `y`

. If `x`

was already bound
in `m`

to a value that is physically equal to `y`

,
`m`

is returned unchanged (the result of the function is
then physically equal to `m`

). Otherwise, the previous binding
of `x`

in `m`

disappears.

**Before 4.03**Physical equality was not ensured.

`val update : ``key -> ('a option -> 'a option) -> 'a t -> 'a t`

`update x f m`

returns a map containing the same bindings as
`m`

, except for the binding of `x`

. Depending on the value of
`y`

where `y`

is `f (find_opt x m)`

, the binding of `x`

is
added, removed or updated. If `y`

is `None`

, the binding is
removed if it exists; otherwise, if `y`

is `Some z`

then `x`

is associated to `z`

in the resulting map. If `x`

was already
bound in `m`

to a value that is physically equal to `z`

, `m`

is returned unchanged (the result of the function is then
physically equal to `m`

).

**Since**4.06.0

`val singleton : ``key -> 'a -> 'a t`

`singleton x y`

returns the one-element map that contains a binding `y`

for `x`

.

**Since**3.12.0

`val remove : ``key -> 'a t -> 'a t`

`remove x m`

returns a map containing the same bindings as
`m`

, except for `x`

which is unbound in the returned map.
If `x`

was not in `m`

, `m`

is returned unchanged
(the result of the function is then physically equal to `m`

).

**Before 4.03**Physical equality was not ensured.

`val merge : ``(key -> 'a option -> 'b option -> 'c option) ->`

'a t -> 'b t -> 'c t

`merge f m1 m2`

computes a map whose keys is a subset of keys of `m1`

and of `m2`

. The presence of each such binding, and the corresponding
value, is determined with the function `f`

.
In terms of the `find_opt`

operation, we have
`find_opt x (merge f m1 m2) = f (find_opt x m1) (find_opt x m2)`

for any key `x`

, provided that `f None None = None`

.

**Since**3.12.0

`val union : ``(key -> 'a -> 'a -> 'a option) ->`

'a t -> 'a t -> 'a t

`union f m1 m2`

computes a map whose keys is the union of keys
of `m1`

and of `m2`

. When the same binding is defined in both
arguments, the function `f`

is used to combine them.
This is a special case of `merge`

: `union f m1 m2`

is equivalent
to `merge f' m1 m2`

, where

`f' None None = None`

`f' (Some v) None = Some v`

`f' None (Some v) = Some v`

`f' (Some v1) (Some v2) = f v1 v2`

**Since**4.03.0

`val compare : ``('a -> 'a -> int) -> 'a t -> 'a t -> int`

Total ordering between maps. The first argument is a total ordering used to compare data associated with equal keys in the two maps.

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

`equal cmp m1 m2`

tests whether the maps `m1`

and `m2`

are
equal, that is, contain equal keys and associate them with
equal data. `cmp`

is the equality predicate used to compare
the data associated with the keys.

`val iter : ``(key -> 'a -> unit) -> 'a t -> unit`

`iter f m`

applies `f`

to all bindings in map `m`

.
`f`

receives the key as first argument, and the associated value
as second argument. The bindings are passed to `f`

in increasing
order with respect to the ordering over the type of the keys.

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

`fold f m a`

computes `(f kN dN ... (f k1 d1 a)...)`

,
where `k1 ... kN`

are the keys of all bindings in `m`

(in increasing order), and `d1 ... dN`

are the associated data.

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

`for_all p m`

checks if all the bindings of the map
satisfy the predicate `p`

.

**Since**3.12.0

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

`exists p m`

checks if at least one binding of the map
satisfies the predicate `p`

.

**Since**3.12.0

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

`filter p m`

returns the map with all the bindings in `m`

that satisfy predicate `p`

. If `p`

satisfies every binding in `m`

,
`m`

is returned unchanged (the result of the function is then
physically equal to `m`

)

**Before 4.03**Physical equality was not ensured.**Since**3.12.0

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

`partition p m`

returns a pair of maps `(m1, m2)`

, where
`m1`

contains all the bindings of `s`

that satisfy the
predicate `p`

, and `m2`

is the map with all the bindings of
`s`

that do not satisfy `p`

.

**Since**3.12.0

`val cardinal : ``'a t -> int`

Return the number of bindings of a map.

**Since**3.12.0

`val bindings : ``'a t -> (key * 'a) list`

Return the list of all bindings of the given map.
The returned list is sorted in increasing order with respect
to the ordering `Ord.compare`

, where `Ord`

is the argument
given to `Map.Make`

.

**Since**3.12.0

`val min_binding : ``'a t -> key * 'a`

Return the smallest binding of the given map
(with respect to the `Ord.compare`

ordering), or raise
`Not_found`

if the map is empty.

**Since**3.12.0

`val min_binding_opt : ``'a t -> (key * 'a) option`

Return the smallest binding of the given map
(with respect to the `Ord.compare`

ordering), or `None`

if the map is empty.

**Since**4.05

`val max_binding : ``'a t -> key * 'a`

Same as `Map.S.min_binding`

, but returns the largest binding
of the given map.

**Since**3.12.0

`val max_binding_opt : ``'a t -> (key * 'a) option`

Same as `Map.S.min_binding_opt`

, but returns the largest binding
of the given map.

**Since**4.05

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

Return one binding of the given map, or raise `Not_found`

if
the map is empty. Which binding is chosen is unspecified,
but equal bindings will be chosen for equal maps.

**Since**3.12.0

`val choose_opt : ``'a t -> (key * 'a) option`

Return one binding of the given map, or `None`

if
the map is empty. Which binding is chosen is unspecified,
but equal bindings will be chosen for equal maps.

**Since**4.05

`val split : ``key -> 'a t -> 'a t * 'a option * 'a t`

`split x m`

returns a triple `(l, data, r)`

, where
`l`

is the map with all the bindings of `m`

whose key
is strictly less than `x`

;
`r`

is the map with all the bindings of `m`

whose key
is strictly greater than `x`

;
`data`

is `None`

if `m`

contains no binding for `x`

,
or `Some v`

if `m`

binds `v`

to `x`

.

**Since**3.12.0

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

`find x m`

returns the current binding of `x`

in `m`

,
or raises `Not_found`

if no such binding exists.

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

`find_opt x m`

returns `Some v`

if the current binding of `x`

in `m`

is `v`

, or `None`

if no such binding exists.

**Since**4.05

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

`find_first f m`

, where `f`

is a monotonically increasing function,
returns the binding of `m`

with the lowest key `k`

such that `f k`

,
or raises `Not_found`

if no such key exists.

For example, `find_first (fun k -> Ord.compare k x >= 0) m`

will return
the first binding `k, v`

of `m`

where `Ord.compare k x >= 0`

(intuitively: `k >= x`

), or raise `Not_found`

if `x`

is greater than any
element of `m`

.

**Since**4.05

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

`find_first_opt f m`

, where `f`

is a monotonically increasing function,
returns an option containing the binding of `m`

with the lowest key `k`

such that `f k`

, or `None`

if no such key exists.

**Since**4.05

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

`find_last f m`

, where `f`

is a monotonically decreasing function,
returns the binding of `m`

with the highest key `k`

such that `f k`

,
or raises `Not_found`

if no such key exists.

**Since**4.05

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

`find_last_opt f m`

, where `f`

is a monotonically decreasing function,
returns an option containing the binding of `m`

with the highest key `k`

such that `f k`

, or `None`

if no such key exists.

**Since**4.05

`val map : ``('a -> 'b) -> 'a t -> 'b t`

`map f m`

returns a map with same domain as `m`

, where the
associated value `a`

of all bindings of `m`

has been
replaced by the result of the application of `f`

to `a`

.
The bindings are passed to `f`

in increasing order
with respect to the ordering over the type of the keys.

`val mapi : ``(key -> 'a -> 'b) -> 'a t -> 'b t`

Same as `Map.S.map`

, but the function receives as arguments both the
key and the associated value for each binding of the map.