Data.Map
 Portability portable Stability provisional Maintainer libraries@haskell.org
 Contents Map type Operators Query Construction Insertion Delete/Update Combine Union Difference Intersection Traversal Map Fold Conversion Lists Ordered lists Filter Submap Indexed Min/Max Debugging
Description

An efficient implementation of maps from keys to values (dictionaries).

Since many function names (but not the type name) clash with Prelude names, this module is usually imported qualified, e.g.

```  import Data.Map (Map)
import qualified Data.Map as Map
```

The implementation of Map is based on size balanced binary trees (or trees of bounded balance) as described by:

• Stephen Adams, "Efficient sets: a balancing act", Journal of Functional Programming 3(4):553-562, October 1993, http://www.swiss.ai.mit.edu/~adams/BB.
• J. Nievergelt and E.M. Reingold, "Binary search trees of bounded balance", SIAM journal of computing 2(1), March 1973.

Note that the implementation is left-biased -- the elements of a first argument are always preferred to the second, for example in union or insert.

Operation comments contain the operation time complexity in the Big-O notation http://en.wikipedia.org/wiki/Big_O_notation.

Synopsis
 data Map k a (!) :: Ord k => Map k a -> k -> a (\\) :: Ord k => Map k a -> Map k b -> Map k a null :: Map k a -> Bool size :: Map k a -> Int member :: Ord k => k -> Map k a -> Bool notMember :: Ord k => k -> Map k a -> Bool lookup :: (Monad m, Ord k) => k -> Map k a -> m a findWithDefault :: Ord k => a -> k -> Map k a -> a empty :: Map k a singleton :: k -> a -> Map k a insert :: Ord k => k -> a -> Map k a -> Map k a insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a) insertWith' :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a insertWithKey' :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a delete :: Ord k => k -> Map k a -> Map k a adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a, Map k a) alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a union :: Ord k => Map k a -> Map k a -> Map k a unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a unions :: Ord k => [Map k a] -> Map k a unionsWith :: Ord k => (a -> a -> a) -> [Map k a] -> Map k a difference :: Ord k => Map k a -> Map k b -> Map k a differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a intersection :: Ord k => Map k a -> Map k b -> Map k a intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c map :: (a -> b) -> Map k a -> Map k b mapWithKey :: (k -> a -> b) -> Map k a -> Map k b mapAccum :: (a -> b -> (a, c)) -> a -> Map k b -> (a, Map k c) mapAccumWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c) mapKeys :: Ord k2 => (k1 -> k2) -> Map k1 a -> Map k2 a mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1 -> k2) -> Map k1 a -> Map k2 a mapKeysMonotonic :: (k1 -> k2) -> Map k1 a -> Map k2 a fold :: (a -> b -> b) -> b -> Map k a -> b foldWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b elems :: Map k a -> [a] keys :: Map k a -> [k] keysSet :: Map k a -> Set k assocs :: Map k a -> [(k, a)] toList :: Map k a -> [(k, a)] fromList :: Ord k => [(k, a)] -> Map k a fromListWith :: Ord k => (a -> a -> a) -> [(k, a)] -> Map k a fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k, a)] -> Map k a toAscList :: Map k a -> [(k, a)] fromAscList :: Eq k => [(k, a)] -> Map k a fromAscListWith :: Eq k => (a -> a -> a) -> [(k, a)] -> Map k a fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k, a)] -> Map k a fromDistinctAscList :: [(k, a)] -> Map k a filter :: Ord k => (a -> Bool) -> Map k a -> Map k a filterWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> Map k a partition :: Ord k => (a -> Bool) -> Map k a -> (Map k a, Map k a) partitionWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> (Map k a, Map k a) mapMaybe :: Ord k => (a -> Maybe b) -> Map k a -> Map k b mapMaybeWithKey :: Ord k => (k -> a -> Maybe b) -> Map k a -> Map k b mapEither :: Ord k => (a -> Either b c) -> Map k a -> (Map k b, Map k c) mapEitherWithKey :: Ord k => (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c) split :: Ord k => k -> Map k a -> (Map k a, Map k a) splitLookup :: Ord k => k -> Map k a -> (Map k a, Maybe a, Map k a) isSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool isSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool isProperSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool lookupIndex :: (Monad m, Ord k) => k -> Map k a -> m Int findIndex :: Ord k => k -> Map k a -> Int elemAt :: Int -> Map k a -> (k, a) updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a deleteAt :: Int -> Map k a -> Map k a findMin :: Map k a -> (k, a) findMax :: Map k a -> (k, a) deleteMin :: Map k a -> Map k a deleteMax :: Map k a -> Map k a deleteFindMin :: Map k a -> ((k, a), Map k a) deleteFindMax :: Map k a -> ((k, a), Map k a) updateMin :: (a -> Maybe a) -> Map k a -> Map k a updateMax :: (a -> Maybe a) -> Map k a -> Map k a updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a minView :: Monad m => Map k a -> m (a, Map k a) maxView :: Monad m => Map k a -> m (a, Map k a) minViewWithKey :: Monad m => Map k a -> m ((k, a), Map k a) maxViewWithKey :: Monad m => Map k a -> m ((k, a), Map k a) showTree :: (Show k, Show a) => Map k a -> String showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String valid :: Ord k => Map k a -> Bool
Map type
data Map k a
A Map from keys k to values a.
Instances
 Typeable2 Map Foldable (Map k) Functor (Map k) Traversable (Map k) (Data k, Data a, Ord k) => Data (Map k a) (Eq k, Eq a) => Eq (Map k a) Ord k => Monoid (Map k v) (Ord k, Ord v) => Ord (Map k v) (Ord k, Read k, Read e) => Read (Map k e) (Show k, Show a) => Show (Map k a)
Operators
(!) :: Ord k => Map k a -> k -> a

O(log n). Find the value at a key. Calls error when the element can not be found.

``` fromList [(5,'a'), (3,'b')] ! 1    Error: element not in the map
fromList [(5,'a'), (3,'b')] ! 5 == 'a'
```
(\\) :: Ord k => Map k a -> Map k b -> Map k a
Same as difference.
Query
null :: Map k a -> Bool

O(1). Is the map empty?

``` Data.Map.null (empty)           == True
Data.Map.null (singleton 1 'a') == False
```
size :: Map k a -> Int

O(1). The number of elements in the map.

``` size empty                                   == 0
size (singleton 1 'a')                       == 1
size (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3
```
member :: Ord k => k -> Map k a -> Bool

O(log n). Is the key a member of the map? See also notMember.

``` member 5 (fromList [(5,'a'), (3,'b')]) == True
member 1 (fromList [(5,'a'), (3,'b')]) == False
```
notMember :: Ord k => k -> Map k a -> Bool

O(log n). Is the key not a member of the map? See also member.

``` notMember 5 (fromList [(5,'a'), (3,'b')]) == False
notMember 1 (fromList [(5,'a'), (3,'b')]) == True
```
lookup :: (Monad m, Ord k) => k -> Map k a -> m a

O(log n). Lookup the value at a key in the map.

The function will return the result in the monad or fail in it the key isn't in the map. Often, the monad to use is Maybe, so you get either (Just result) or Nothing.

``` let m = fromList [(5,'a'), (3,'b'), (7,'c')]
value1 <- Data.Map.lookup 5 m
value1
'a'
value2 <- Data.Map.lookup 1 m
```

An example of using lookup with Maybe monad:

``` import Prelude hiding (lookup)
import Data.Map

employeeDept = fromList([("John","Sales"), ("Bob","IT")])
deptCountry = fromList([("IT","USA"), ("Sales","France")])
countryCurrency = fromList([("USA", "Dollar"), ("France", "Euro")])

employeeCurrency :: String -> Maybe String
employeeCurrency name = do
dept <- lookup name employeeDept
country <- lookup dept deptCountry
lookup country countryCurrency

main = do
putStrLn \$ "John's currency: " ++ (show (employeeCurrency "John"))
putStrLn \$ "Pete's currency: " ++ (show (employeeCurrency "Pete"))
```

The output of this program:

```   John's currency: Just "Euro"
Pete's currency: Nothing
```
findWithDefault :: Ord k => a -> k -> Map k a -> a

O(log n). The expression (findWithDefault def k map) returns the value at key k or returns default value def when the key is not in the map.

``` findWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x'
findWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a'
```
Construction
empty :: Map k a

O(1). The empty map.

``` empty      == fromList []
size empty == 0
```
singleton :: k -> a -> Map k a

O(1). A map with a single element.

``` singleton 1 'a'        == fromList [(1, 'a')]
size (singleton 1 'a') == 1
```
Insertion
insert :: Ord k => k -> a -> Map k a -> Map k a

O(log n). Insert a new key and value in the map. If the key is already present in the map, the associated value is replaced with the supplied value. insert is equivalent to insertWith const.

``` insert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')]
insert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')]
insert 5 'x' empty                         == singleton 5 'x'
```
insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a

O(log n). Insert with a function, combining new value and old value. insertWith f key value mp will insert the pair (key, value) into mp if key does not exist in the map. If the key does exist, the function will insert the pair (key, f new_value old_value).

``` insertWith (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")]
insertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
insertWith (++) 5 "xxx" empty                         == singleton 5 "xxx"
```
insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a

O(log n). Insert with a function, combining key, new value and old value. insertWithKey f key value mp will insert the pair (key, value) into mp if key does not exist in the map. If the key does exist, the function will insert the pair (key,f key new_value old_value). Note that the key passed to f is the same key passed to insertWithKey.

``` let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
insertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")]
insertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
insertWithKey f 5 "xxx" empty                         == singleton 5 "xxx"
```
insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)

O(log n). Combines insert operation with old value retrieval. The expression (insertLookupWithKey f k x map) is a pair where the first element is equal to (lookup k map) and the second element equal to (insertWithKey f k x map).

``` let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
insertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")])
insertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "xxx")])
insertLookupWithKey f 5 "xxx" empty                         == (Nothing,  singleton 5 "xxx")
```

This is how to define insertLookup using insertLookupWithKey:

``` let insertLookup kx x t = insertLookupWithKey (\_ a _ -> a) kx x t
insertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")])
insertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "x")])
```
insertWith' :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
Same as insertWith, but the combining function is applied strictly.
insertWithKey' :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
Same as insertWithKey, but the combining function is applied strictly.
Delete/Update
delete :: Ord k => k -> Map k a -> Map k a

O(log n). Delete a key and its value from the map. When the key is not a member of the map, the original map is returned.

``` delete 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
delete 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
delete 5 empty                         == empty
```
adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a

O(log n). Update a value at a specific key with the result of the provided function. When the key is not a member of the map, the original map is returned.

``` adjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
adjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
adjust ("new " ++) 7 empty                         == empty
```
adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a

O(log n). Adjust a value at a specific key. When the key is not a member of the map, the original map is returned.

``` let f key x = (show key) ++ ":new " ++ x
adjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
adjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
adjustWithKey f 7 empty                         == empty
```
update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a

O(log n). The expression (update f k map) updates the value x at k (if it is in the map). If (f x) is Nothing, the element is deleted. If it is (Just y), the key k is bound to the new value y.

``` let f x = if x == "a" then Just "new a" else Nothing
update f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
update f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
update f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
```
updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a

O(log n). The expression (updateWithKey f k map) updates the value x at k (if it is in the map). If (f k x) is Nothing, the element is deleted. If it is (Just y), the key k is bound to the new value y.

``` let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
updateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
updateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
updateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
```
updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a, Map k a)

O(log n). Lookup and update. See also updateWithKey. The function returns changed value, if it is updated. Returns the original key value if the map entry is deleted.

``` let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
updateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "5:new a", fromList [(3, "b"), (5, "5:new a")])
updateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a")])
updateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a")
```
alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a

O(log n). The expression (alter f k map) alters the value x at k, or absence thereof. alter can be used to insert, delete, or update a value in a Map. In short : lookup k (alter f k m) = f (lookup k m).

``` let f _ = Nothing
alter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
alter f 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"

let f _ = Just "c"
alter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "c")]
alter f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "c")]
```
Combine
Union
union :: Ord k => Map k a -> Map k a -> Map k a

O(n+m). The expression (union t1 t2) takes the left-biased union of t1 and t2. It prefers t1 when duplicate keys are encountered, i.e. (union == unionWith const). The implementation uses the efficient hedge-union algorithm. Hedge-union is more efficient on (bigset `union` smallset).

``` union (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")]
```
unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a

O(n+m). Union with a combining function. The implementation uses the efficient hedge-union algorithm.

``` unionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")]
```
unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a

O(n+m). Union with a combining function. The implementation uses the efficient hedge-union algorithm. Hedge-union is more efficient on (bigset `union` smallset).

``` let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
unionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")]
```
unions :: Ord k => [Map k a] -> Map k a

The union of a list of maps: (unions == foldl union empty).

``` unions [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
== fromList [(3, "b"), (5, "a"), (7, "C")]
unions [(fromList [(5, "A3"), (3, "B3")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "a"), (3, "b")])]
== fromList [(3, "B3"), (5, "A3"), (7, "C")]
```
unionsWith :: Ord k => (a -> a -> a) -> [Map k a] -> Map k a

The union of a list of maps, with a combining operation: (unionsWith f == foldl (unionWith f) empty).

``` unionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
== fromList [(3, "bB3"), (5, "aAA3"), (7, "C")]
```
Difference
difference :: Ord k => Map k a -> Map k b -> Map k a

O(n+m). Difference of two maps. Return elements of the first map not existing in the second map. The implementation uses an efficient hedge algorithm comparable with hedge-union.

``` difference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b"
```
differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a

O(n+m). Difference with a combining function. When two equal keys are encountered, the combining function is applied to the values of these keys. If it returns Nothing, the element is discarded (proper set difference). If it returns (Just y), the element is updated with a new value y. The implementation uses an efficient hedge algorithm comparable with hedge-union.

``` let f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing
differenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")])
== singleton 3 "b:B"
```
differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a

O(n+m). Difference with a combining function. When two equal keys are encountered, the combining function is applied to the key and both values. If it returns Nothing, the element is discarded (proper set difference). If it returns (Just y), the element is updated with a new value y. The implementation uses an efficient hedge algorithm comparable with hedge-union.

``` let f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing
differenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")])
== singleton 3 "3:b|B"
```
Intersection
intersection :: Ord k => Map k a -> Map k b -> Map k a

O(n+m). Intersection of two maps. Return data in the first map for the keys existing in both maps. (intersection m1 m2 == intersectionWith const m1 m2).

``` intersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a"
```
intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c

O(n+m). Intersection with a combining function.

``` intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"
```
intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c

O(n+m). Intersection with a combining function. Intersection is more efficient on (bigset `intersection` smallset).

``` let f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar
intersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A"
```
Traversal
Map
map :: (a -> b) -> Map k a -> Map k b

O(n). Map a function over all values in the map.

``` map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]
```
mapWithKey :: (k -> a -> b) -> Map k a -> Map k b

O(n). Map a function over all values in the map.

``` let f key x = (show key) ++ ":" ++ x
mapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")]
```
mapAccum :: (a -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

O(n). The function mapAccum threads an accumulating argument through the map in ascending order of keys.

``` let f a b = (a ++ b, b ++ "X")
mapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")])
```
mapAccumWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

O(n). The function mapAccumWithKey threads an accumulating argument through the map in ascending order of keys.

``` let f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")
mapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])
```
mapKeys :: Ord k2 => (k1 -> k2) -> Map k1 a -> Map k2 a

O(n*log n). mapKeys f s is the map obtained by applying f to each key of s.

The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case the value at the smallest of these keys is retained.

``` mapKeys (+ 1) (fromList [(5,"a"), (3,"b")])                        == fromList [(4, "b"), (6, "a")]
mapKeys (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
mapKeys (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"
```
mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1 -> k2) -> Map k1 a -> Map k2 a

O(n*log n). mapKeysWith c f s is the map obtained by applying f to each key of s.

The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case the associated values will be combined using c.

``` mapKeysWith (++) (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
mapKeysWith (++) (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"
```
mapKeysMonotonic :: (k1 -> k2) -> Map k1 a -> Map k2 a

O(n). mapKeysMonotonic f s == mapKeys f s, but works only when f is strictly monotonic. That is, for any values x and y, if x < y then f x < f y. The precondition is not checked. Semi-formally, we have:

``` and [x < y ==> f x < f y | x <- ls, y <- ls]
==> mapKeysMonotonic f s == mapKeys f s
where ls = keys s
```

This means that f maps distinct original keys to distinct resulting keys. This function has better performance than mapKeys.

``` mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]
valid (mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")])) == True
valid (mapKeysMonotonic (\ _ -> 1)     (fromList [(5,"a"), (3,"b")])) == False
```
Fold
fold :: (a -> b -> b) -> b -> Map k a -> b

O(n). Fold the values in the map, such that fold f z == foldr f z . elems. For example,

``` elems map = fold (:) [] map
```
``` let f a len = len + (length a)
fold f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
```
foldWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b

O(n). Fold the keys and values in the map, such that foldWithKey f z == foldr (uncurry f) z . toAscList. For example,

``` keys map = foldWithKey (\k x ks -> k:ks) [] map
```
``` let f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
foldWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)"
```
Conversion
elems :: Map k a -> [a]

O(n). Return all elements of the map in the ascending order of their keys.

``` elems (fromList [(5,"a"), (3,"b")]) == ["b","a"]
elems empty == []
```
keys :: Map k a -> [k]

O(n). Return all keys of the map in ascending order.

``` keys (fromList [(5,"a"), (3,"b")]) == [3,5]
keys empty == []
```
keysSet :: Map k a -> Set k

O(n). The set of all keys of the map.

``` keysSet (fromList [(5,"a"), (3,"b")]) == Data.Set.fromList [3,5]
keysSet empty == Data.Set.empty
```
assocs :: Map k a -> [(k, a)]

O(n). Return all key/value pairs in the map in ascending key order.

``` assocs (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
assocs empty == []
```
Lists
toList :: Map k a -> [(k, a)]

O(n). Convert to a list of key/value pairs.

``` toList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
toList empty == []
```
fromList :: Ord k => [(k, a)] -> Map k a

O(n*log n). Build a map from a list of key/value pairs. See also fromAscList. If the list contains more than one value for the same key, the last value for the key is retained.

``` fromList [] == empty
fromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")]
fromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")]
```
fromListWith :: Ord k => (a -> a -> a) -> [(k, a)] -> Map k a

O(n*log n). Build a map from a list of key/value pairs with a combining function. See also fromAscListWith.

``` fromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")]
fromListWith (++) [] == empty
```
fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

O(n*log n). Build a map from a list of key/value pairs with a combining function. See also fromAscListWithKey.

``` let f k a1 a2 = (show k) ++ a1 ++ a2
fromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "3ab"), (5, "5a5ba")]
fromListWithKey f [] == empty
```
Ordered lists
toAscList :: Map k a -> [(k, a)]

O(n). Convert to an ascending list.

``` toAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
```
fromAscList :: Eq k => [(k, a)] -> Map k a

O(n). Build a map from an ascending list in linear time. The precondition (input list is ascending) is not checked.

``` fromAscList [(3,"b"), (5,"a")]          == fromList [(3, "b"), (5, "a")]
fromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")]
valid (fromAscList [(3,"b"), (5,"a"), (5,"b")]) == True
valid (fromAscList [(5,"a"), (3,"b"), (5,"b")]) == False
```
fromAscListWith :: Eq k => (a -> a -> a) -> [(k, a)] -> Map k a

O(n). Build a map from an ascending list in linear time with a combining function for equal keys. The precondition (input list is ascending) is not checked.

``` fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]
valid (fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")]) == True
valid (fromAscListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False
```
fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

O(n). Build a map from an ascending list in linear time with a combining function for equal keys. The precondition (input list is ascending) is not checked.

``` let f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2
fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")] == fromList [(3, "b"), (5, "5:b5:ba")]
valid (fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")]) == True
valid (fromAscListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False
```
fromDistinctAscList :: [(k, a)] -> Map k a

O(n). Build a map from an ascending list of distinct elements in linear time. The precondition is not checked.

``` fromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]
valid (fromDistinctAscList [(3,"b"), (5,"a")])          == True
valid (fromDistinctAscList [(3,"b"), (5,"a"), (5,"b")]) == False
```
Filter
filter :: Ord k => (a -> Bool) -> Map k a -> Map k a

O(n). Filter all values that satisfy the predicate.

``` filter (> "a") (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
filter (> "x") (fromList [(5,"a"), (3,"b")]) == empty
filter (< "a") (fromList [(5,"a"), (3,"b")]) == empty
```
filterWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> Map k a

O(n). Filter all keys/values that satisfy the predicate.

``` filterWithKey (\k _ -> k > 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
```
partition :: Ord k => (a -> Bool) -> Map k a -> (Map k a, Map k a)

O(n). Partition the map according to a predicate. The first map contains all elements that satisfy the predicate, the second all elements that fail the predicate. See also split.

``` partition (> "a") (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
partition (< "x") (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
partition (> "x") (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
```
partitionWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> (Map k a, Map k a)

O(n). Partition the map according to a predicate. The first map contains all elements that satisfy the predicate, the second all elements that fail the predicate. See also split.

``` partitionWithKey (\ k _ -> k > 3) (fromList [(5,"a"), (3,"b")]) == (singleton 5 "a", singleton 3 "b")
partitionWithKey (\ k _ -> k < 7) (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
partitionWithKey (\ k _ -> k > 7) (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
```
mapMaybe :: Ord k => (a -> Maybe b) -> Map k a -> Map k b

O(n). Map values and collect the Just results.

``` let f x = if x == "a" then Just "new a" else Nothing
mapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a"
```
mapMaybeWithKey :: Ord k => (k -> a -> Maybe b) -> Map k a -> Map k b

O(n). Map keys/values and collect the Just results.

``` let f k _ = if k < 5 then Just ("key : " ++ (show k)) else Nothing
mapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3"
```
mapEither :: Ord k => (a -> Either b c) -> Map k a -> (Map k b, Map k c)

O(n). Map values and separate the Left and Right results.

``` let f a = if a < "c" then Left a else Right a
mapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
== (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])

mapEither (\ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
== (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
```
mapEitherWithKey :: Ord k => (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c)

O(n). Map keys/values and separate the Left and Right results.

``` let f k a = if k < 5 then Left (k * 2) else Right (a ++ a)
mapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
== (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])

mapEitherWithKey (\_ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
== (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])
```
split :: Ord k => k -> Map k a -> (Map k a, Map k a)

O(log n). The expression (split k map) is a pair (map1,map2) where the keys in map1 are smaller than k and the keys in map2 larger than k. Any key equal to k is found in neither map1 nor map2.

``` split 2 (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3,"b"), (5,"a")])
split 3 (fromList [(5,"a"), (3,"b")]) == (empty, singleton 5 "a")
split 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
split 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", empty)
split 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], empty)
```
splitLookup :: Ord k => k -> Map k a -> (Map k a, Maybe a, Map k a)

O(log n). The expression (splitLookup k map) splits a map just like split but also returns lookup k map.

``` splitLookup 2 (fromList [(5,"a"), (3,"b")]) == (empty, Nothing, fromList [(3,"b"), (5,"a")])
splitLookup 3 (fromList [(5,"a"), (3,"b")]) == (empty, Just "b", singleton 5 "a")
splitLookup 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Nothing, singleton 5 "a")
splitLookup 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Just "a", empty)
splitLookup 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], Nothing, empty)
```
Submap
isSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool
O(n+m). This function is defined as (isSubmapOf = isSubmapOfBy (==)).
isSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool

O(n+m). The expression (isSubmapOfBy f t1 t2) returns True if all keys in t1 are in tree t2, and when f returns True when applied to their respective values. For example, the following expressions are all True:

``` isSubmapOfBy (==) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
isSubmapOfBy (<=) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1),('b',2)])
```

But the following are all False:

``` isSubmapOfBy (==) (fromList [('a',2)]) (fromList [('a',1),('b',2)])
isSubmapOfBy (<)  (fromList [('a',1)]) (fromList [('a',1),('b',2)])
isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1)])
```
isProperSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool
O(n+m). Is this a proper submap? (ie. a submap but not equal). Defined as (isProperSubmapOf = isProperSubmapOfBy (==)).
isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool

O(n+m). Is this a proper submap? (ie. a submap but not equal). The expression (isProperSubmapOfBy f m1 m2) returns True when m1 and m2 are not equal, all keys in m1 are in m2, and when f returns True when applied to their respective values. For example, the following expressions are all True:

``` isProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
isProperSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
```

But the following are all False:

``` isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
isProperSubmapOfBy (<)  (fromList [(1,1)])       (fromList [(1,1),(2,2)])
```
Indexed
lookupIndex :: (Monad m, Ord k) => k -> Map k a -> m Int

O(log n). Lookup the index of a key. The index is a number from 0 up to, but not including, the size of the map.

``` isJust (lookupIndex 2 (fromList [(5,"a"), (3,"b")]))   == False
fromJust (lookupIndex 3 (fromList [(5,"a"), (3,"b")])) == 0
fromJust (lookupIndex 5 (fromList [(5,"a"), (3,"b")])) == 1
isJust (lookupIndex 6 (fromList [(5,"a"), (3,"b")]))   == False
```
findIndex :: Ord k => k -> Map k a -> Int

O(log n). Return the index of a key. The index is a number from 0 up to, but not including, the size of the map. Calls error when the key is not a member of the map.

``` findIndex 2 (fromList [(5,"a"), (3,"b")])    Error: element is not in the map
findIndex 3 (fromList [(5,"a"), (3,"b")]) == 0
findIndex 5 (fromList [(5,"a"), (3,"b")]) == 1
findIndex 6 (fromList [(5,"a"), (3,"b")])    Error: element is not in the map
```
elemAt :: Int -> Map k a -> (k, a)

O(log n). Retrieve an element by index. Calls error when an invalid index is used.

``` elemAt 0 (fromList [(5,"a"), (3,"b")]) == (3,"b")
elemAt 1 (fromList [(5,"a"), (3,"b")]) == (5, "a")
elemAt 2 (fromList [(5,"a"), (3,"b")])    Error: index out of range
```
updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a

O(log n). Update the element at index. Calls error when an invalid index is used.

``` updateAt (\ _ _ -> Just "x") 0    (fromList [(5,"a"), (3,"b")]) == fromList [(3, "x"), (5, "a")]
updateAt (\ _ _ -> Just "x") 1    (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "x")]
updateAt (\ _ _ -> Just "x") 2    (fromList [(5,"a"), (3,"b")])    Error: index out of range
updateAt (\ _ _ -> Just "x") (-1) (fromList [(5,"a"), (3,"b")])    Error: index out of range
updateAt (\_ _  -> Nothing)  0    (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
updateAt (\_ _  -> Nothing)  1    (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
updateAt (\_ _  -> Nothing)  2    (fromList [(5,"a"), (3,"b")])    Error: index out of range
updateAt (\_ _  -> Nothing)  (-1) (fromList [(5,"a"), (3,"b")])    Error: index out of range
```
deleteAt :: Int -> Map k a -> Map k a

O(log n). Delete the element at index. Defined as (deleteAt i map = updateAt (k x -> Nothing) i map).

``` deleteAt 0  (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
deleteAt 1  (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
deleteAt 2 (fromList [(5,"a"), (3,"b")])     Error: index out of range
deleteAt (-1) (fromList [(5,"a"), (3,"b")])  Error: index out of range
```
Min/Max
findMin :: Map k a -> (k, a)

O(log n). The minimal key of the map. Calls error is the map is empty.

``` findMin (fromList [(5,"a"), (3,"b")]) == (3,"b")
findMin empty                            Error: empty map has no minimal element
```
findMax :: Map k a -> (k, a)

O(log n). The maximal key of the map. Calls error is the map is empty.

``` findMax (fromList [(5,"a"), (3,"b")]) == (5,"a")
findMax empty                            Error: empty map has no maximal element
```
deleteMin :: Map k a -> Map k a

O(log n). Delete the minimal key. Returns an empty map if the map is empty.

``` deleteMin (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(5,"a"), (7,"c")]
deleteMin empty == empty
```
deleteMax :: Map k a -> Map k a

O(log n). Delete the maximal key. Returns an empty map if the map is empty.

``` deleteMax (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(3,"b"), (5,"a")]
deleteMax empty == empty
```
deleteFindMin :: Map k a -> ((k, a), Map k a)

O(log n). Delete and find the minimal element.

``` deleteFindMin (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((3,"b"), fromList[(5,"a"), (10,"c")])
deleteFindMin                                            Error: can not return the minimal element of an empty map
```
deleteFindMax :: Map k a -> ((k, a), Map k a)

O(log n). Delete and find the maximal element.

``` deleteFindMax (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((10,"c"), fromList [(3,"b"), (5,"a")])
deleteFindMax empty                                      Error: can not return the maximal element of an empty map
```
updateMin :: (a -> Maybe a) -> Map k a -> Map k a

O(log n). Update the value at the minimal key.

``` updateMin (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")]
updateMin (\ _ -> Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
```
updateMax :: (a -> Maybe a) -> Map k a -> Map k a

O(log n). Update the value at the maximal key.

``` updateMax (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")]
updateMax (\ _ -> Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
```
updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a

O(log n). Update the value at the minimal key.

``` updateMinWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")]
updateMinWithKey (\ _ _ -> Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
```
updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a

O(log n). Update the value at the maximal key.

``` updateMaxWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")]
updateMaxWithKey (\ _ _ -> Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
```
minView :: Monad m => Map k a -> m (a, Map k a)

O(log n). Retrieves the minimal key's value of the map, and the map stripped from that element fails (in the monad) when passed an empty map.

``` v <- minView (fromList [(5,"a"), (3,"b")])
v == ("b", singleton 5 "a")
minView empty                     Error: empty map
```
maxView :: Monad m => Map k a -> m (a, Map k a)

O(log n). Retrieves the maximal key's value of the map, and the map stripped from that element fails (in the monad) when passed an empty map.

``` v <- maxView (fromList [(5,"a"), (3,"b")])
v == ("a", singleton 3 "b")
maxView empty                     Error: empty map
```
minViewWithKey :: Monad m => Map k a -> m ((k, a), Map k a)

O(log n). Retrieves the minimal (key,value) pair of the map, and the map stripped from that element fails (in the monad) when passed an empty map.

``` v <- minViewWithKey (fromList [(5,"a"), (3,"b")])
v ==  ((3,"b"), singleton 5 "a")
minViewWithKey empty              Error: empty map
```
maxViewWithKey :: Monad m => Map k a -> m ((k, a), Map k a)

O(log n). Retrieves the maximal (key,value) pair of the map, and the map stripped from that element fails (in the monad) when passed an empty map.

``` v <- maxViewWithKey (fromList [(5,"a"), (3,"b")])
v == ((5,"a"), singleton 3 "b")
maxViewWithKey empty              Error: empty map
```
Debugging
showTree :: (Show k, Show a) => Map k a -> String
O(n). Show the tree that implements the map. The tree is shown in a compressed, hanging format. See showTreeWith.
showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String

O(n). The expression (showTreeWith showelem hang wide map) shows the tree that implements the map. Elements are shown using the showElem function. If hang is True, a hanging tree is shown otherwise a rotated tree is shown. If wide is True, an extra wide version is shown.

```  Map> let t = fromDistinctAscList [(x,()) | x <- [1..5]]
Map> putStrLn \$ showTreeWith (\k x -> show (k,x)) True False t
(4,())
+--(2,())
|  +--(1,())
|  +--(3,())
+--(5,())

Map> putStrLn \$ showTreeWith (\k x -> show (k,x)) True True t
(4,())
|
+--(2,())
|  |
|  +--(1,())
|  |
|  +--(3,())
|
+--(5,())

Map> putStrLn \$ showTreeWith (\k x -> show (k,x)) False True t
+--(5,())
|
(4,())
|
|  +--(3,())
|  |
+--(2,())
|
+--(1,())
```
valid :: Ord k => Map k a -> Bool

O(n). Test if the internal map structure is valid.

``` valid (fromAscList [(3,"b"), (5,"a")]) == True
valid (fromAscList [(5,"a"), (3,"b")]) == False
```