A map of integers to values a.

O(min(n,W)). 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'

Same as difference.

O(1). Is the map empty?

 Data.IntMap.null (empty)           == True
 Data.IntMap.null (singleton 1 'a') == False

O(n). Number of elements in the map.

 size empty                                   == 0
 size (singleton 1 'a')                       == 1
 size (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3

O(min(n,W)). Is the key a member of the map?

 member 5 (fromList [(5,'a'), (3,'b')]) == True
 member 1 (fromList [(5,'a'), (3,'b')]) == False

O(min(n,W)). Is the key not a member of the map?

 notMember 5 (fromList [(5,'a'), (3,'b')]) == False
 notMember 1 (fromList [(5,'a'), (3,'b')]) == True

O(min(n,W)). Lookup the value at a key in the map. See also lookup.

O(min(n,W)). The expression (findWithDefault def k map) returns the value at key k or returns def when the key is not an element of the map.

 findWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x'
 findWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a'

O(log n). Find largest key smaller than the given one and return the corresponding (key, value) pair.

 lookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing
 lookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')

O(log n). Find smallest key greater than the given one and return the corresponding (key, value) pair.

 lookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
 lookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing

O(log n). Find largest key smaller or equal to the given one and return the corresponding (key, value) pair.

 lookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing
 lookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
 lookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')

O(log n). Find smallest key greater or equal to the given one and return the corresponding (key, value) pair.

 lookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
 lookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
 lookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing

O(1). The empty map.

 empty      == fromList []
 size empty == 0

O(1). A map of one element.

 singleton 1 'a'        == fromList [(1, 'a')]
 size (singleton 1 'a') == 1

O(min(n,W)). Insert a new key/value pair in the map. If the key is already present in the map, the associated value is replaced with the supplied value, i.e. 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'

O(min(n,W)). Insert with a combining function. 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 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"

O(min(n,W)). Insert with a combining function. 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 f key new_value old_value.

 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"

O(min(n,W)). 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")])

O(min(n,W)). 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

O(min(n,W)). Adjust a value at a specific key. 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

O(min(n,W)). 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

O(min(n,W)). 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"

O(min(n,W)). The expression (update 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"

O(min(n,W)). Lookup and update. The function returns original value, if it is updated. This is different behavior than updateLookupWithKey. 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 "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")

O(min(n,W)). 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 an IntMap. In short : lookup k (alter f k m) = f (lookup k m).

O(n+m). The (left-biased) union of two maps. It prefers the first map when duplicate keys are encountered, i.e. (union == unionWith const).

 union (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")]

O(n+m). The union with a combining function.

 unionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")]

O(n+m). The union with a combining function.

 let f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value
 unionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")]

The union of a list of maps.

 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")]

The union of a list of maps, with a combining operation.

 unionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
     == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")]

O(n+m). Difference between two maps (based on keys).

 difference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b"

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

 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"

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.

 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"

O(n+m). The (left-biased) intersection of two maps (based on keys).

 intersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a"

O(n+m). The intersection with a combining function.

 intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"

O(n+m). The intersection with a combining function.

 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"

O(n+m). A high-performance universal combining function. Using mergeWithKey, all combining functions can be defined without any loss of efficiency (with exception of union, difference and intersection, where sharing of some nodes is lost with mergeWithKey).

Please make sure you know what is going on when using mergeWithKey, otherwise you can be surprised by unexpected code growth or even corruption of the data structure.

When mergeWithKey is given three arguments, it is inlined to the call site. You should therefore use mergeWithKey only to define your custom combining functions. For example, you could define unionWithKey, differenceWithKey and intersectionWithKey as

 myUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) id id m1 m2
 myDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2
 myIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) (const empty) (const empty) m1 m2

When calling mergeWithKey combine only1 only2, a function combining two IntMaps is created, such that

• if a key is present in both maps, it is passed with both corresponding values to the combine function. Depending on the result, the key is either present in the result with specified value, or is left out;
• a nonempty subtree present only in the first map is passed to only1 and the output is added to the result;
• a nonempty subtree present only in the second map is passed to only2 and the output is added to the result.

The only1 and only2 methods must return a map with a subset (possibly empty) of the keys of the given map. The values can be modified arbitrarily. Most common variants of only1 and only2 are id and const empty, but for example map f or filterWithKey f could be used for any f.

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

 map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]

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")]

O(n). traverseWithKey f s == fromList $ traverse ((k, v) -> (,) k $ f k v) (toList m) That is, behaves exactly like a regular traverse except that the traversing function also has access to the key associated with a value.

 traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
 traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing

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")])

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")])

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

O(n*min(n,W)). 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 greatest of the original 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"

O(n*min(n,W)). 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"

O(n*min(n,W)). 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 slightly better performance than mapKeys.

 mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]

O(n). Fold the values in the map using the given right-associative binary operator, such that foldr f z == foldr f z . elems.

For example,

 elems map = foldr (:) [] map

 let f a len = len + (length a)
 foldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4

O(n). Fold the values in the map using the given left-associative binary operator, such that foldl f z == foldl f z . elems.

For example,

 elems = reverse . foldl (flip (:)) []

 let f len a = len + (length a)
 foldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4

O(n). Fold the keys and values in the map using the given right-associative binary operator, such that foldrWithKey f z == foldr (uncurry f) z . toAscList.

For example,

 keys map = foldrWithKey (\k x ks -> k:ks) [] map

 let f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
 foldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)"

O(n). Fold the keys and values in the map using the given left-associative binary operator, such that foldlWithKey f z == foldl (\z' (kx, x) -> f z' kx x) z . toAscList.

For example,

 keys = reverse . foldlWithKey (\ks k x -> k:ks) []

 let f result k a = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
 foldlWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (3:b)(5:a)"

O(n). A strict version of foldr. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.

O(n). A strict version of foldl. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.

O(n). A strict version of foldrWithKey. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.

O(n). A strict version of foldlWithKey. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.

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

 elems (fromList [(5,"a"), (3,"b")]) == ["b","a"]
 elems empty == []

O(n). Return all keys of the map in ascending order. Subject to list fusion.

 keys (fromList [(5,"a"), (3,"b")]) == [3,5]
 keys empty == []

O(n). An alias for toAscList. Returns all key/value pairs in the map in ascending key order. Subject to list fusion.

 assocs (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
 assocs empty == []

O(n*min(n,W)). The set of all keys of the map.

 keysSet (fromList [(5,"a"), (3,"b")]) == Data.IntSet.fromList [3,5]
 keysSet empty == Data.IntSet.empty

O(n). Build a map from a set of keys and a function which for each key computes its value.

 fromSet (\k -> replicate k 'a') (Data.IntSet.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")]
 fromSet undefined Data.IntSet.empty == empty

O(n). Convert the map to a list of key/value pairs. Subject to list fusion.

 toList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
 toList empty == []

O(n*min(n,W)). Create a map from a list of key/value pairs.

 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")]

O(n*min(n,W)). Create 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,"c")] == fromList [(3, "ab"), (5, "cba")]
 fromListWith (++) [] == empty

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

 let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
 fromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"c")] == fromList [(3, "3:a|b"), (5, "5:c|5:b|a")]
 fromListWithKey f [] == empty

O(n). Convert the map to a list of key/value pairs where the keys are in ascending order. Subject to list fusion.

 toAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]

O(n). Convert the map to a list of key/value pairs where the keys are in descending order. Subject to list fusion.

 toDescList (fromList [(5,"a"), (3,"b")]) == [(5,"a"), (3,"b")]

O(n). Build a map from a list of key/value pairs where the keys are in ascending order.

 fromAscList [(3,"b"), (5,"a")]          == fromList [(3, "b"), (5, "a")]
 fromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")]

O(n). Build a map from a list of key/value pairs where the keys are in ascending order, with a combining function on equal keys. The precondition (input list is ascending) is not checked.

 fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]

O(n). Build a map from a list of key/value pairs where the keys are in ascending order, with a combining function on equal keys. The precondition (input list is ascending) is not checked.

 let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
 fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "5:b|a")]

O(n). Build a map from a list of key/value pairs where the keys are in ascending order and all distinct. The precondition (input list is strictly ascending) is not checked.

 fromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]

O(n). Filter all values that satisfy some 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

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

 filterWithKey (\k _ -> k > 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"

O(n). Partition the map according to some 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")])

O(n). Partition the map according to some 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")])

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"

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"

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")])

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")])

O(min(n,W)). The expression (split k map) is a pair (map1,map2) where all keys in map1 are lower than k and all 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)

O(min(n,W)). Performs a split but also returns whether the pivot key was found in the original 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)

O(n+m). Is this a submap? Defined as (isSubmapOf = isSubmapOfBy (==)).

O(n+m). The expression (isSubmapOfBy f m1 m2) returns True if 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:

 isSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
 isSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
 isSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])

But the following are all False:

 isSubmapOfBy (==) (fromList [(1,2)]) (fromList [(1,1),(2,2)])
 isSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
 isSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])

O(n+m). Is this a proper submap? (ie. a submap but not equal). Defined as (isProperSubmapOf = isProperSubmapOfBy (==)).

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)])

O(min(n,W)). The minimal key of the map.

O(min(n,W)). The maximal key of the map.

O(min(n,W)). Delete the minimal key. An error is thrown if the IntMap is already empty. Note, this is not the same behavior Map.

O(min(n,W)). Delete the maximal key. An error is thrown if the IntMap is already empty. Note, this is not the same behavior Map.

O(min(n,W)). Delete and find the minimal element.

O(min(n,W)). Delete and find the maximal element.

O(min(n,W)). 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"

O(min(n,W)). 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"

O(min(n,W)). 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"

O(min(n,W)). 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"

O(min(n,W)). Retrieves the minimal key of the map, and the map stripped of that element, or Nothing if passed an empty map.

O(min(n,W)). Retrieves the maximal key of the map, and the map stripped of that element, or Nothing if passed an empty map.

O(min(n,W)). Retrieves the minimal (key,value) pair of the map, and the map stripped of that element, or Nothing if passed an empty map.

 minViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((3,"b"), singleton 5 "a")
 minViewWithKey empty == Nothing

O(min(n,W)). Retrieves the maximal (key,value) pair of the map, and the map stripped of that element, or Nothing if passed an empty map.

 maxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b")
 maxViewWithKey empty == Nothing

O(n). Show the tree that implements the map. The tree is shown in a compressed, hanging format.

O(n). The expression (showTreeWith hang wide map) shows the tree that implements the map. 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.