The type of the map keys, indexed with the type of the value for such keys.

The type of maps from type 'a key to type 'a.

The empty map.

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).

The function f must be put inside a record of type poly_update to ensure it is polymorphic. It is indeed used in a polymorphic manner by the implementation of update.

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.

The function f must be put inside a record of type poly_merge to ensure it is polymorphic. It is indeed used in a polymorphic manner by the implementation of merge.

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

The function f must be put inside a record of type poly_union to ensure it is polymorphic. It is indeed used in a polymorphic manner by the implementation of union.

'b poly1 is the type of functions that take a key of type 'a key and that return an element of type 'b. They are polymorphic in 'a.

'b poly1 is the type of functions that take a key of type 'a key and one value of type 'a, and that return an element of type 'b. They are polymorphic in 'a.

'b poly2 is the type of functions that take a key of type 'a key and two values of type 'a, and that return an element of type 'b. They are polymorphic in 'a.

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.

Return the number of bindings of a map.

The type of bindings: a key and its associated value

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.

mapo f m returns a map with same domain as m, where the value a associated to a key k in the map m has been replaced by the result of the application of f to k and a. The bindings are passed to f in increasing order with respect to the ordering over the type of the keys.

Iterate on the whole map, in ascending order of keys

Iterate on the whole map, in descending order of keys

to_seq_from k m iterates on a subset of the bindings of m, in ascending order of keys, from key k or above.

Add the given bindings to the map, in order.

Build a map from the given bindings