#### iri

`type key = t`

The type of the map keys.

`val empty : 'a t`

The empty map.

`val is_empty : 'a t -> bool`

Test whether a map is empty or not.

`mem x m`

returns `true`

if `m`

contains a binding for `x`

, and `false`

otherwise.

`add key data m`

returns a map containing the same bindings as `m`

, plus a binding of `key`

to `data`

. If `key`

was already bound in `m`

to a value that is physically equal to `data`

, `m`

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

). Otherwise, the previous binding of `key`

in `m`

disappears.

`update key f m`

returns a map containing the same bindings as `m`

, except for the binding of `key`

. Depending on the value of `y`

where `y`

is `f (find_opt key m)`

, the binding of `key`

is added, removed or updated. If `y`

is `None`

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

is `Some z`

then `key`

is associated to `z`

in the resulting map. If `key`

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`

).

`singleton x y`

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

for `x`

.

`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`

).

`merge f m1 m2`

computes a map whose keys are a subset of the 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 x (find_opt x m1) (find_opt x m2)`

for any key `x`

, provided that `f x None None = None`

.

`union f m1 m2`

computes a map whose keys are a subset of the 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' _key None None = None`

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

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

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

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

`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.

`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.

`fold f m init`

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

, where `k1 ... kN`

are the keys of all bindings in `m`

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

are the associated data.

`for_all f m`

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

.

`exists f m`

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

.

`filter f m`

returns the map with all the bindings in `m`

that satisfy predicate `p`

. If every binding in `m`

satisfies `f`

, `m`

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

)

`filter_map f m`

applies the function `f`

to every binding of `m`

, and builds a map from the results. For each binding `(k, v)`

in the input map:

- if
`f k v`

is`None`

then`k`

is not in the result, - if
`f k v`

is`Some v'`

then the binding`(k, v')`

is in the output map.

For example, the following function on maps whose values are lists

```
filter_map
(fun _k li -> match li with [] -> None | _::tl -> Some tl)
m
```

drops all bindings of `m`

whose value is an empty list, and pops the first element of each value that is non-empty.

`partition f m`

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

, where `m1`

contains all the bindings of `m`

that satisfy the predicate `f`

, and `m2`

is the map with all the bindings of `m`

that do not satisfy `f`

.

`val cardinal : 'a t -> int`

Return the number of bindings of a map.

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

, where `Ord`

is the argument given to `Make`

.

Return the binding with the smallest key in a given map (with respect to the `Ord.compare`

ordering), or raise `Not_found`

if the map is empty.

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

ordering), or `None`

if the map is empty.

Same as `S`

.min_binding, but returns the binding with the largest key in the given map.

Same as `S`

.min_binding_opt, but returns the binding with the largest key in the given map.

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.

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.

`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`

.

`find x m`

returns the current value of `x`

in `m`

, or raises `Not_found`

if no binding for `x`

exists.

`find_opt x m`

returns `Some v`

if the current value of `x`

in `m`

is `v`

, or `None`

if no binding for `x`

exists.

`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`

.

`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.

`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.

`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.

`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.

Same as `S`

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

## Iterators

`to_seq_from k m`

iterates on a subset of the bindings of `m`

, in ascending order of keys, from key `k`

or above.