-- Hoogle documentation, generated by Haddock -- See Hoogle, http://www.haskell.org/hoogle/ -- | Assorted concrete container types -- -- This package contains efficient general-purpose implementations of -- various basic immutable container types. The declared cost of each -- operation is either worst-case or amortized, but remains valid even if -- structures are shared. @package containers @version 0.4.0.0 -- | An efficient implementation of sets. -- -- Since many function names (but not the type name) clash with -- Prelude names, this module is usually imported -- qualified, e.g. -- -- 
--   import Data.Set (Set)
--   import qualified Data.Set as Set
--
-- -- The implementation of Set is based on size balanced -- binary trees (or trees of bounded balance) as described by: -- -- -- -- 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. Of course, left-biasing can only be -- observed when equality is an equivalence relation instead of -- structural equality. module Data.Set -- | A set of values a. data Set a -- | O(n+m). See difference. (\\) :: Ord a => Set a -> Set a -> Set a -- | O(1). Is this the empty set? null :: Set a -> Bool -- | O(1). The number of elements in the set. size :: Set a -> Int -- | O(log n). Is the element in the set? member :: Ord a => a -> Set a -> Bool -- | O(log n). Is the element not in the set? notMember :: Ord a => a -> Set a -> Bool -- | O(n+m). Is this a subset? (s1 isSubsetOf s2) -- tells whether s1 is a subset of s2. isSubsetOf :: Ord a => Set a -> Set a -> Bool -- | O(n+m). Is this a proper subset? (ie. a subset but not equal). isProperSubsetOf :: Ord a => Set a -> Set a -> Bool -- | O(1). The empty set. empty :: Set a -- | O(1). Create a singleton set. singleton :: a -> Set a -- | O(log n). Insert an element in a set. If the set already -- contains an element equal to the given value, it is replaced with the -- new value. insert :: Ord a => a -> Set a -> Set a -- | O(log n). Delete an element from a set. delete :: Ord a => a -> Set a -> Set a -- | O(n+m). The union of two sets, preferring the first set when -- equal elements are encountered. The implementation uses the efficient -- hedge-union algorithm. Hedge-union is more efficient on (bigset -- union smallset). union :: Ord a => Set a -> Set a -> Set a -- | The union of a list of sets: (unions == foldl -- union empty). unions :: Ord a => [Set a] -> Set a -- | O(n+m). Difference of two sets. The implementation uses an -- efficient hedge algorithm comparable with hedge-union. difference :: Ord a => Set a -> Set a -> Set a -- | O(n+m). The intersection of two sets. Elements of the result -- come from the first set, so for example -- -- 
--   import qualified Data.Set as S
--   data AB = A | B deriving Show
--   instance Ord AB where compare _ _ = EQ
--   instance Eq AB where _ == _ = True
--   main = print (S.singleton A `S.intersection` S.singleton B,
--                 S.singleton B `S.intersection` S.singleton A)
--
-- -- prints (fromList [A],fromList [B]). intersection :: Ord a => Set a -> Set a -> Set a -- | O(n). Filter all elements that satisfy the predicate. filter :: Ord a => (a -> Bool) -> Set a -> Set a -- | O(n). Partition the set into two sets, one with all elements -- that satisfy the predicate and one with all elements that don't -- satisfy the predicate. See also split. partition :: Ord a => (a -> Bool) -> Set a -> (Set a, Set a) -- | O(log n). The expression (split x set) is a -- pair (set1,set2) where set1 comprises the elements -- of set less than x and set2 comprises the -- elements of set greater than x. split :: Ord a => a -> Set a -> (Set a, Set a) -- | O(log n). Performs a split but also returns whether the -- pivot element was found in the original set. splitMember :: Ord a => a -> Set a -> (Set a, Bool, Set a) -- | O(n*log n). map f s is the set obtained by -- applying f to each element of s. -- -- It's worth noting that the size of the result may be smaller if, for -- some (x,y), x /= y && f x == f y map :: (Ord a, Ord b) => (a -> b) -> Set a -> Set b -- | O(n). The -- -- mapMonotonic f s == map f s, but works only -- when f is monotonic. The precondition is not checked. -- Semi-formally, we have: -- -- 
--   and [x < y ==> f x < f y | x <- ls, y <- ls] 
--                       ==> mapMonotonic f s == map f s
--       where ls = toList s
--
mapMonotonic :: (a -> b) -> Set a -> Set b -- | O(n). Fold over the elements of a set in an unspecified order. fold :: (a -> b -> b) -> b -> Set a -> b -- | O(log n). The minimal element of a set. findMin :: Set a -> a -- | O(log n). The maximal element of a set. findMax :: Set a -> a -- | O(log n). Delete the minimal element. deleteMin :: Set a -> Set a -- | O(log n). Delete the maximal element. deleteMax :: Set a -> Set a -- | O(log n). Delete and find the minimal element. -- -- 
--   deleteFindMin set = (findMin set, deleteMin set)
--
deleteFindMin :: Set a -> (a, Set a) -- | O(log n). Delete and find the maximal element. -- -- 
--   deleteFindMax set = (findMax set, deleteMax set)
--
deleteFindMax :: Set a -> (a, Set a) -- | O(log n). Retrieves the maximal key of the set, and the set -- stripped of that element, or Nothing if passed an empty set. maxView :: Set a -> Maybe (a, Set a) -- | O(log n). Retrieves the minimal key of the set, and the set -- stripped of that element, or Nothing if passed an empty set. minView :: Set a -> Maybe (a, Set a) -- | O(n). The elements of a set. elems :: Set a -> [a] -- | O(n). Convert the set to a list of elements. toList :: Set a -> [a] -- | O(n*log n). Create a set from a list of elements. fromList :: Ord a => [a] -> Set a -- | O(n). Convert the set to an ascending list of elements. toAscList :: Set a -> [a] -- | O(n). Build a set from an ascending list in linear time. The -- precondition (input list is ascending) is not checked. fromAscList :: Eq a => [a] -> Set a -- | O(n). Build a set from an ascending list of distinct elements -- in linear time. The precondition (input list is strictly ascending) -- is not checked. fromDistinctAscList :: [a] -> Set a -- | O(n). Show the tree that implements the set. The tree is shown -- in a compressed, hanging format. showTree :: Show a => Set a -> String -- | O(n). The expression (showTreeWith hang wide map) -- shows the tree that implements the set. 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. -- -- 
--   Set> putStrLn $ showTreeWith True False $ fromDistinctAscList [1..5]
--   4
--   +--2
--   |  +--1
--   |  +--3
--   +--5
--   
--   Set> putStrLn $ showTreeWith True True $ fromDistinctAscList [1..5]
--   4
--   |
--   +--2
--   |  |
--   |  +--1
--   |  |
--   |  +--3
--   |
--   +--5
--   
--   Set> putStrLn $ showTreeWith False True $ fromDistinctAscList [1..5]
--   +--5
--   |
--   4
--   |
--   |  +--3
--   |  |
--   +--2
--      |
--      +--1
--
showTreeWith :: Show a => Bool -> Bool -> Set a -> String -- | O(n). Test if the internal set structure is valid. valid :: Ord a => Set a -> Bool instance Typeable1 Set instance (Read a, Ord a) => Read (Set a) instance Show a => Show (Set a) instance Ord a => Ord (Set a) instance Eq a => Eq (Set a) instance (Data a, Ord a) => Data (Set a) instance Foldable Set instance Ord a => Monoid (Set a) -- | 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: -- -- -- -- 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. module Data.Map -- | A Map from keys k to values a. data Map 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 -> k -> a -- | Same as difference. (\\) :: Ord k => Map k a -> Map k b -> Map k a -- | O(1). Is the map empty? -- -- 
--   Data.Map.null (empty)           == True
--   Data.Map.null (singleton 1 'a') == False
--
null :: Map k a -> Bool -- | 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
--
size :: Map k a -> Int -- | 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
--
member :: 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
--
notMember :: Ord k => k -> Map k a -> Bool -- | O(log n). Lookup the value at a key in the map. -- -- The function will return the corresponding value as (Just -- value), or Nothing if the key isn't in the map. -- -- An example of using lookup: -- -- 
--   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
--
lookup :: Ord k => k -> Map k a -> Maybe 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'
--
findWithDefault :: Ord k => a -> k -> Map k a -> a -- | O(1). The empty map. -- -- 
--   empty      == fromList []
--   size empty == 0
--
empty :: Map k a -- | O(1). A map with a single element. -- -- 
--   singleton 1 'a'        == fromList [(1, 'a')]
--   size (singleton 1 'a') == 1
--
singleton :: 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'
--
insert :: Ord k => 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"
--
insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a -- | Same as insertWith, but the combining function is applied -- strictly. This is often the most desirable behavior. -- -- For example, to update a counter: -- -- 
--   insertWith' (+) k 1 m
--
insertWith' :: Ord 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"
--
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. insertWithKey' :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k 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")])
--
insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a) -- | O(log n). A strict version of insertLookupWithKey. insertLookupWithKey' :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe 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
--
delete :: Ord k => 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
--
adjust :: Ord 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
--
adjustWithKey :: Ord k => (k -> a -> 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"
--
update :: Ord 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"
--
updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k 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")
--
updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe 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")]
--
alter :: Ord k => (Maybe a -> Maybe a) -> k -> 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")]
--
union :: Ord k => 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")]
--
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. Hedge-union is more -- efficient on (bigset `union` smallset). -- -- 
--   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")]
--
unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> 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")]
--
unions :: Ord k => [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")]
--
unionsWith :: Ord k => (a -> a -> a) -> [Map k a] -> 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"
--
difference :: Ord k => 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"
--
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 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"
--
differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> 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"
--
intersection :: Ord k => Map k a -> Map k b -> Map k a -- | O(n+m). Intersection with a combining function. -- -- 
--   intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"
--
intersectionWith :: Ord 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"
--
intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c -- | O(n). Map a function over all values in the map. -- -- 
--   map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]
--
map :: (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")]
--
mapWithKey :: (k -> a -> b) -> Map k a -> Map k b -- | 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")])
--
mapAccum :: (a -> 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")])
--
mapAccumWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c) -- | O(n). The function mapAccumR threads an accumulating -- argument through the map in descending order of keys. mapAccumRWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c) -- | 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"
--
mapKeys :: Ord k2 => (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"
--
mapKeysWith :: Ord k2 => (a -> a -> a) -> (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
--
mapKeysMonotonic :: (k1 -> k2) -> Map k1 a -> Map k2 a -- | 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
--
fold :: (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)"
--
-- -- This is identical to foldrWithKey, and you should use that one -- instead of this one. This name is kept for backward compatibility. foldWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b -- | O(n). Post-order fold. The function will be applied from the -- lowest value to the highest. foldrWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b -- | O(n). Pre-order fold. The function will be applied from the -- highest value to the lowest. foldlWithKey :: (b -> k -> a -> b) -> b -> Map k a -> b -- | 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 == []
--
elems :: Map k a -> [a] -- | O(n). Return all keys of the map in ascending order. -- -- 
--   keys (fromList [(5,"a"), (3,"b")]) == [3,5]
--   keys empty == []
--
keys :: Map k a -> [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
--
keysSet :: Map k a -> Set k -- | 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 == []
--
assocs :: 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 == []
--
toList :: Map k a -> [(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")]
--
fromList :: Ord k => [(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
--
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 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
--
fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k, a)] -> Map k a -- | O(n). Convert to an ascending list. -- -- 
--   toAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
--
toAscList :: Map k a -> [(k, a)] -- | O(n). Convert to a descending list. toDescList :: Map k a -> [(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
--
fromAscList :: Eq k => [(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
--
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. -- -- 
--   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
--
fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(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
--
fromDistinctAscList :: [(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
--
filter :: Ord 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"
--
filterWithKey :: Ord k => (k -> a -> Bool) -> 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")])
--
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. -- -- 
--   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")])
--
partitionWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> (Map k a, Map k 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"
--
mapMaybe :: Ord 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"
--
mapMaybeWithKey :: Ord k => (k -> a -> Maybe b) -> Map k a -> Map k b -- | 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")])
--
mapEither :: Ord 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")])
--
mapEitherWithKey :: Ord k => (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c) -- | 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)
--
split :: Ord k => k -> Map k a -> (Map k 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)
--
splitLookup :: Ord k => k -> Map k a -> (Map k a, Maybe a, Map k a) -- | O(n+m). This function is defined as (isSubmapOf = -- isSubmapOfBy (==)). isSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> 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)])
--
isSubmapOfBy :: 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). -- Defined as (isProperSubmapOf = isProperSubmapOfBy -- (==)). 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). -- 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)])
--
isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool -- | 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
--
lookupIndex :: Ord k => k -> Map k a -> Maybe 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
--
findIndex :: Ord k => k -> Map k a -> Int -- | 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
--
elemAt :: Int -> Map k a -> (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
--
updateAt :: (k -> a -> Maybe a) -> 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
--
deleteAt :: Int -> Map k a -> Map 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
--
findMin :: 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
--
findMax :: Map k a -> (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
--
deleteMin :: 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
--
deleteMax :: Map 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
--
deleteFindMin :: 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
--
deleteFindMax :: Map k a -> ((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"
--
updateMin :: (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"
--
updateMax :: (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"
--
updateMinWithKey :: (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"
--
updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a -- | O(log n). Retrieves the value associated with minimal key of -- the map, and the map stripped of that element, or Nothing if -- passed an empty map. -- -- 
--   minView (fromList [(5,"a"), (3,"b")]) == Just ("b", singleton 5 "a")
--   minView empty == Nothing
--
minView :: Map k a -> Maybe (a, Map k a) -- | O(log n). Retrieves the value associated with maximal key of -- the map, and the map stripped of that element, or Nothing if -- passed an -- -- 
--   maxView (fromList [(5,"a"), (3,"b")]) == Just ("a", singleton 3 "b")
--   maxView empty == Nothing
--
maxView :: Map k a -> Maybe (a, Map k a) -- | O(log n). 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
--
minViewWithKey :: Map k a -> Maybe ((k, a), Map k a) -- | O(log n). 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
--
maxViewWithKey :: Map k a -> Maybe ((k, a), Map k a) -- | O(n). Show the tree that implements the map. The tree is shown -- in a compressed, hanging format. See showTreeWith. showTree :: (Show k, Show a) => 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,())
--
showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String -- | O(n). Test if the internal map structure is valid. -- -- 
--   valid (fromAscList [(3,"b"), (5,"a")]) == True
--   valid (fromAscList [(5,"a"), (3,"b")]) == False
--
valid :: Ord k => Map k a -> Bool instance Typeable2 Map instance (Show k, Show a) => Show (Map k a) instance (Ord k, Read k, Read e) => Read (Map k e) instance Foldable (Map k) instance Traversable (Map k) instance Functor (Map k) instance (Ord k, Ord v) => Ord (Map k v) instance (Eq k, Eq a) => Eq (Map k a) instance (Data k, Data a, Ord k) => Data (Map k a) instance Ord k => Monoid (Map k v) -- | An efficient implementation of integer sets. -- -- Since many function names (but not the type name) clash with -- Prelude names, this module is usually imported -- qualified, e.g. -- -- 
--   import Data.IntSet (IntSet)
--   import qualified Data.IntSet as IntSet
--
-- -- The implementation is based on big-endian patricia trees. This -- data structure performs especially well on binary operations like -- union and intersection. However, my benchmarks show that -- it is also (much) faster on insertions and deletions when compared to -- a generic size-balanced set implementation (see Data.Set). -- -- -- -- Many operations have a worst-case complexity of O(min(n,W)). -- This means that the operation can become linear in the number of -- elements with a maximum of W -- the number of bits in an -- Int (32 or 64). module Data.IntSet -- | A set of integers. data IntSet -- | O(n+m). See difference. (\\) :: IntSet -> IntSet -> IntSet -- | O(1). Is the set empty? null :: IntSet -> Bool -- | O(n). Cardinality of the set. size :: IntSet -> Int -- | O(min(n,W)). Is the value a member of the set? member :: Int -> IntSet -> Bool -- | O(min(n,W)). Is the element not in the set? notMember :: Int -> IntSet -> Bool -- | O(n+m). Is this a subset? (s1 isSubsetOf s2) -- tells whether s1 is a subset of s2. isSubsetOf :: IntSet -> IntSet -> Bool -- | O(n+m). Is this a proper subset? (ie. a subset but not equal). isProperSubsetOf :: IntSet -> IntSet -> Bool -- | O(1). The empty set. empty :: IntSet -- | O(1). A set of one element. singleton :: Int -> IntSet -- | O(min(n,W)). Add a value to the set. When the value is already -- an element of the set, it is replaced by the new one, ie. -- insert is left-biased. insert :: Int -> IntSet -> IntSet -- | O(min(n,W)). Delete a value in the set. Returns the original -- set when the value was not present. delete :: Int -> IntSet -> IntSet -- | O(n+m). The union of two sets. union :: IntSet -> IntSet -> IntSet -- | The union of a list of sets. unions :: [IntSet] -> IntSet -- | O(n+m). Difference between two sets. difference :: IntSet -> IntSet -> IntSet -- | O(n+m). The intersection of two sets. intersection :: IntSet -> IntSet -> IntSet -- | O(n). Filter all elements that satisfy some predicate. filter :: (Int -> Bool) -> IntSet -> IntSet -- | O(n). partition the set according to some predicate. partition :: (Int -> Bool) -> IntSet -> (IntSet, IntSet) -- | O(min(n,W)). The expression (split x set) is a -- pair (set1,set2) where set1 comprises the elements -- of set less than x and set2 comprises the -- elements of set greater than x. -- -- 
--   split 3 (fromList [1..5]) == (fromList [1,2], fromList [4,5])
--
split :: Int -> IntSet -> (IntSet, IntSet) -- | O(min(n,W)). Performs a split but also returns whether -- the pivot element was found in the original set. splitMember :: Int -> IntSet -> (IntSet, Bool, IntSet) -- | O(min(n,W)). The minimal element of the set. findMin :: IntSet -> Int -- | O(min(n,W)). The maximal element of a set. findMax :: IntSet -> Int -- | O(min(n,W)). Delete the minimal element. deleteMin :: IntSet -> IntSet -- | O(min(n,W)). Delete the maximal element. deleteMax :: IntSet -> IntSet -- | O(min(n,W)). Delete and find the minimal element. -- -- 
--   deleteFindMin set = (findMin set, deleteMin set)
--
deleteFindMin :: IntSet -> (Int, IntSet) -- | O(min(n,W)). Delete and find the maximal element. -- -- 
--   deleteFindMax set = (findMax set, deleteMax set)
--
deleteFindMax :: IntSet -> (Int, IntSet) -- | O(min(n,W)). Retrieves the maximal key of the set, and the set -- stripped of that element, or Nothing if passed an empty set. maxView :: IntSet -> Maybe (Int, IntSet) -- | O(min(n,W)). Retrieves the minimal key of the set, and the set -- stripped of that element, or Nothing if passed an empty set. minView :: IntSet -> Maybe (Int, IntSet) -- | O(n*min(n,W)). map f s is the set obtained by -- applying f to each element of s. -- -- It's worth noting that the size of the result may be smaller if, for -- some (x,y), x /= y && f x == f y map :: (Int -> Int) -> IntSet -> IntSet -- | O(n). Fold over the elements of a set in an unspecified order. -- -- 
--   sum set   == fold (+) 0 set
--   elems set == fold (:) [] set
--
fold :: (Int -> b -> b) -> b -> IntSet -> b -- | O(n). The elements of a set. (For sets, this is equivalent to -- toList) elems :: IntSet -> [Int] -- | O(n). Convert the set to a list of elements. toList :: IntSet -> [Int] -- | O(n*min(n,W)). Create a set from a list of integers. fromList :: [Int] -> IntSet -- | O(n). Convert the set to an ascending list of elements. toAscList :: IntSet -> [Int] -- | O(n). Build a set from an ascending list of elements. The -- precondition (input list is ascending) is not checked. fromAscList :: [Int] -> IntSet -- | O(n). Build a set from an ascending list of distinct elements. -- The precondition (input list is strictly ascending) is not -- checked. fromDistinctAscList :: [Int] -> IntSet -- | O(n). Show the tree that implements the set. The tree is shown -- in a compressed, hanging format. showTree :: IntSet -> String -- | O(n). The expression (showTreeWith hang wide -- map) shows the tree that implements the set. 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. showTreeWith :: Bool -> Bool -> IntSet -> String instance Typeable IntSet instance Read IntSet instance Show IntSet instance Ord IntSet instance Eq IntSet instance Data IntSet instance Monoid IntSet -- | An efficient implementation of maps from integer keys to values. -- -- Since many function names (but not the type name) clash with -- Prelude names, this module is usually imported -- qualified, e.g. -- -- 
--   import Data.IntMap (IntMap)
--   import qualified Data.IntMap as IntMap
--
-- -- The implementation is based on big-endian patricia trees. This -- data structure performs especially well on binary operations like -- union and intersection. However, my benchmarks show that -- it is also (much) faster on insertions and deletions when compared to -- a generic size-balanced map implementation (see Data.Map). -- -- -- -- Operation comments contain the operation time complexity in the Big-O -- notation http://en.wikipedia.org/wiki/Big_O_notation. Many -- operations have a worst-case complexity of O(min(n,W)). This -- means that the operation can become linear in the number of elements -- with a maximum of W -- the number of bits in an Int (32 -- or 64). module Data.IntMap -- | A map of integers to values a. data IntMap a type Key = Int -- | 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'
--
(!) :: IntMap a -> Key -> a -- | Same as difference. (\\) :: IntMap a -> IntMap b -> IntMap a -- | O(1). Is the map empty? -- -- 
--   Data.IntMap.null (empty)           == True
--   Data.IntMap.null (singleton 1 'a') == False
--
null :: IntMap a -> Bool -- | 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
--
size :: IntMap a -> Int -- | 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
--
member :: Key -> IntMap a -> Bool -- | O(log n). 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
--
notMember :: Key -> IntMap a -> Bool -- | O(min(n,W)). Lookup the value at a key in the map. See also -- Data.Map.lookup. lookup :: Key -> IntMap a -> Maybe a -- | 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'
--
findWithDefault :: a -> Key -> IntMap a -> a -- | O(1). The empty map. -- -- 
--   empty      == fromList []
--   size empty == 0
--
empty :: IntMap a -- | O(1). A map of one element. -- -- 
--   singleton 1 'a'        == fromList [(1, 'a')]
--   size (singleton 1 'a') == 1
--
singleton :: Key -> a -> IntMap a -- | 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'
--
insert :: Key -> a -> IntMap a -> IntMap a -- | 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"
--
insertWith :: (a -> a -> a) -> Key -> a -> IntMap a -> IntMap a -- | 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"
--
insertWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> IntMap a -- | 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")])
--
insertLookupWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> (Maybe a, IntMap a) -- | 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
--
delete :: Key -> IntMap a -> IntMap a -- | 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
--
adjust :: (a -> a) -> Key -> IntMap a -> IntMap a -- | 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
--
adjustWithKey :: (Key -> a -> a) -> Key -> IntMap a -> IntMap 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 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"
--
update :: (a -> Maybe a) -> Key -> IntMap a -> IntMap 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"
--
updateWithKey :: (Key -> a -> Maybe a) -> Key -> IntMap a -> IntMap a -- | O(min(n,W)). Lookup and update. The function returns original -- value, if it is updated. This is different behavior than -- Data.Map.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")
--
updateLookupWithKey :: (Key -> a -> Maybe a) -> Key -> IntMap a -> (Maybe a, IntMap 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 an IntMap. -- In short : lookup k (alter f k m) = f (lookup -- k m). alter :: (Maybe a -> Maybe a) -> Int -> IntMap a -> IntMap a -- | 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")]
--
union :: IntMap a -> IntMap a -> IntMap a -- | 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")]
--
unionWith :: (a -> a -> a) -> IntMap a -> IntMap a -> IntMap a -- | 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")]
--
unionWithKey :: (Key -> a -> a -> a) -> IntMap a -> IntMap a -> IntMap a -- | 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")]
--
unions :: [IntMap a] -> IntMap a -- | 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")]
--
unionsWith :: (a -> a -> a) -> [IntMap a] -> IntMap a -- | O(n+m). Difference between two maps (based on keys). -- -- 
--   difference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b"
--
difference :: IntMap a -> IntMap b -> IntMap a -- | 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"
--
differenceWith :: (a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap 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. -- -- 
--   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"
--
differenceWithKey :: (Key -> a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap a -- | 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"
--
intersection :: IntMap a -> IntMap b -> IntMap a -- | O(n+m). The intersection with a combining function. -- -- 
--   intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"
--
intersectionWith :: (a -> b -> c) -> IntMap a -> IntMap b -> IntMap c -- | 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"
--
intersectionWithKey :: (Key -> a -> b -> c) -> IntMap a -> IntMap b -> IntMap c -- | O(n). Map a function over all values in the map. -- -- 
--   map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]
--
map :: (a -> b) -> IntMap a -> IntMap 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")]
--
mapWithKey :: (Key -> a -> b) -> IntMap a -> IntMap b -- | 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")])
--
mapAccum :: (a -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap 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")])
--
mapAccumWithKey :: (a -> Key -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap c) -- | O(n). The function mapAccumR threads an -- accumulating argument through the map in descending order of keys. mapAccumRWithKey :: (a -> Key -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap c) -- | O(n). Fold the values in the map, such that fold f z -- == Prelude.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
--
fold :: (a -> b -> b) -> b -> IntMap a -> b -- | O(n). Fold the keys and values in the map, such that -- foldWithKey f z == Prelude.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)"
--
foldWithKey :: (Key -> a -> b -> b) -> b -> IntMap a -> b -- | 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 == []
--
elems :: IntMap a -> [a] -- | O(n). Return all keys of the map in ascending order. -- -- 
--   keys (fromList [(5,"a"), (3,"b")]) == [3,5]
--   keys empty == []
--
keys :: IntMap a -> [Key] -- | 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
--
keysSet :: IntMap a -> IntSet -- | 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 == []
--
assocs :: IntMap a -> [(Key, a)] -- | O(n). Convert the map to a list of key/value pairs. -- -- 
--   toList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
--   toList empty == []
--
toList :: IntMap a -> [(Key, a)] -- | 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")]
--
fromList :: [(Key, a)] -> IntMap a -- | 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,"a")] == fromList [(3, "ab"), (5, "aba")]
--   fromListWith (++) [] == empty
--
fromListWith :: (a -> a -> a) -> [(Key, a)] -> IntMap a -- | O(n*min(n,W)). Build a map from a list of key/value pairs with -- a combining function. See also fromAscListWithKey'. -- -- 
--   fromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")]
--   fromListWith (++) [] == empty
--
fromListWithKey :: (Key -> a -> a -> a) -> [(Key, a)] -> IntMap a -- | O(n). Convert the map to a list of key/value pairs where the -- keys are in ascending order. -- -- 
--   toAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
--
toAscList :: IntMap a -> [(Key, a)] -- | 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")]
--
fromAscList :: [(Key, a)] -> IntMap a -- | 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")]
--
fromAscListWith :: (a -> a -> a) -> [(Key, a)] -> IntMap a -- | 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")]
--
fromAscListWithKey :: (Key -> a -> a -> a) -> [(Key, a)] -> IntMap 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")]
--
fromDistinctAscList :: [(Key, a)] -> IntMap 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
--
filter :: (a -> Bool) -> IntMap a -> IntMap a -- | O(n). Filter all keys/values that satisfy some predicate. -- -- 
--   filterWithKey (\k _ -> k > 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
--
filterWithKey :: (Key -> a -> Bool) -> IntMap a -> IntMap 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")])
--
partition :: (a -> Bool) -> IntMap a -> (IntMap a, IntMap 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")])
--
partitionWithKey :: (Key -> a -> Bool) -> IntMap a -> (IntMap a, IntMap 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"
--
mapMaybe :: (a -> Maybe b) -> IntMap a -> IntMap 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"
--
mapMaybeWithKey :: (Key -> a -> Maybe b) -> IntMap a -> IntMap b -- | 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")])
--
mapEither :: (a -> Either b c) -> IntMap a -> (IntMap b, IntMap 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")])
--
mapEitherWithKey :: (Key -> a -> Either b c) -> IntMap a -> (IntMap b, IntMap c) -- | O(log n). 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)
--
split :: Key -> IntMap a -> (IntMap a, IntMap a) -- | O(log n). 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)
--
splitLookup :: Key -> IntMap a -> (IntMap a, Maybe a, IntMap a) -- | O(n+m). Is this a submap? Defined as (isSubmapOf = -- isSubmapOfBy (==)). isSubmapOf :: Eq a => IntMap a -> IntMap a -> Bool -- | 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)])
--
isSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Bool -- | O(n+m). Is this a proper submap? (ie. a submap but not equal). -- Defined as (isProperSubmapOf = isProperSubmapOfBy -- (==)). isProperSubmapOf :: Eq a => IntMap a -> IntMap a -> 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)])
--
isProperSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Bool -- | O(log n). Retrieves the maximal key of the map, and the map -- stripped of that element, or Nothing if passed an empty map. maxView :: IntMap a -> Maybe (a, IntMap a) -- | O(log n). Retrieves the minimal key of the map, and the map -- stripped of that element, or Nothing if passed an empty map. minView :: IntMap a -> Maybe (a, IntMap a) -- | O(log n). The minimal key of the map. findMin :: IntMap a -> (Int, a) -- | O(log n). The maximal key of the map. findMax :: IntMap a -> (Int, a) -- | O(log n). Delete the minimal key. An error is thrown if the -- IntMap is already empty. Note, this is not the same behavior Map. deleteMin :: IntMap a -> IntMap a -- | O(log n). Delete the maximal key. An error is thrown if the -- IntMap is already empty. Note, this is not the same behavior Map. deleteMax :: IntMap a -> IntMap a -- | O(log n). Delete and find the minimal element. deleteFindMin :: IntMap a -> (a, IntMap a) -- | O(log n). Delete and find the maximal element. deleteFindMax :: IntMap a -> (a, IntMap 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"
--
updateMin :: (a -> a) -> IntMap a -> IntMap 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"
--
updateMax :: (a -> a) -> IntMap a -> IntMap 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"
--
updateMinWithKey :: (Key -> a -> a) -> IntMap a -> IntMap 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"
--
updateMaxWithKey :: (Key -> a -> a) -> IntMap a -> IntMap a -- | O(log n). 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
--
minViewWithKey :: IntMap a -> Maybe ((Key, a), IntMap a) -- | O(log n). 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
--
maxViewWithKey :: IntMap a -> Maybe ((Key, a), IntMap a) -- | O(n). Show the tree that implements the map. The tree is shown -- in a compressed, hanging format. showTree :: Show a => IntMap a -> String -- | 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. showTreeWith :: Show a => Bool -> Bool -> IntMap a -> String instance Typeable1 IntMap instance Read e => Read (IntMap e) instance Show a => Show (IntMap a) instance Functor IntMap instance Ord a => Ord (IntMap a) instance Eq a => Eq (IntMap a) instance Data a => Data (IntMap a) instance Traversable IntMap instance Foldable IntMap instance Monoid (IntMap a) -- | General purpose finite sequences. Apart from being finite and having -- strict operations, sequences also differ from lists in supporting a -- wider variety of operations efficiently. -- -- An amortized running time is given for each operation, with n -- referring to the length of the sequence and i being the -- integral index used by some operations. These bounds hold even in a -- persistent (shared) setting. -- -- The implementation uses 2-3 finger trees annotated with sizes, as -- described in section 4.2 of -- -- -- -- Note: Many of these operations have the same names as similar -- operations on lists in the Prelude. The ambiguity may be -- resolved using either qualification or the hiding clause. module Data.Sequence -- | General-purpose finite sequences. data Seq a -- | O(1). The empty sequence. empty :: Seq a -- | O(1). A singleton sequence. singleton :: a -> Seq a -- | O(1). Add an element to the left end of a sequence. Mnemonic: a -- triangle with the single element at the pointy end. (<|) :: a -> Seq a -> Seq a -- | O(1). Add an element to the right end of a sequence. Mnemonic: -- a triangle with the single element at the pointy end. (|>) :: Seq a -> a -> Seq a -- | O(log(min(n1,n2))). Concatenate two sequences. (><) :: Seq a -> Seq a -> Seq a -- | O(n). Create a sequence from a finite list of elements. There -- is a function toList in the opposite direction for all -- instances of the Foldable class, including Seq. fromList :: [a] -> Seq a -- | O(log n). replicate n x is a sequence consisting of -- n copies of x. replicate :: Int -> a -> Seq a -- | replicateA is an Applicative version of -- replicate, and makes O(log n) calls to <*> -- and pure. -- -- 
--   replicateA n x = sequenceA (replicate n x)
--
replicateA :: Applicative f => Int -> f a -> f (Seq a) -- | replicateM is a sequence counterpart of -- Control.Monad.replicateM. -- -- 
--   replicateM n x = sequence (replicate n x)
--
replicateM :: Monad m => Int -> m a -> m (Seq a) -- | O(n). Constructs a sequence by repeated application of a -- function to a seed value. -- -- 
--   iterateN n f x = fromList (Prelude.take n (Prelude.iterate f x))
--
iterateN :: Int -> (a -> a) -> a -> Seq a -- | Builds a sequence from a seed value. Takes time linear in the number -- of generated elements. WARNING: If the number of generated -- elements is infinite, this method will not terminate. unfoldr :: (b -> Maybe (a, b)) -> b -> Seq a -- | unfoldl f x is equivalent to reverse -- (unfoldr (swap . f) x). unfoldl :: (b -> Maybe (b, a)) -> b -> Seq a -- | O(1). Is this the empty sequence? null :: Seq a -> Bool -- | O(1). The number of elements in the sequence. length :: Seq a -> Int -- | View of the left end of a sequence. data ViewL a -- | empty sequence EmptyL :: ViewL a -- | leftmost element and the rest of the sequence (:<) :: a -> Seq a -> ViewL a -- | O(1). Analyse the left end of a sequence. viewl :: Seq a -> ViewL a -- | View of the right end of a sequence. data ViewR a -- | empty sequence EmptyR :: ViewR a -- | the sequence minus the rightmost element, and the rightmost element (:>) :: Seq a -> a -> ViewR a -- | O(1). Analyse the right end of a sequence. viewr :: Seq a -> ViewR a -- | scanl is similar to foldl, but returns a sequence of -- reduced values from the left: -- -- 
--   scanl f z (fromList [x1, x2, ...]) = fromList [z, z `f` x1, (z `f` x1) `f` x2, ...]
--
scanl :: (a -> b -> a) -> a -> Seq b -> Seq a -- | scanl1 is a variant of scanl that has no starting value -- argument: -- -- 
--   scanl1 f (fromList [x1, x2, ...]) = fromList [x1, x1 `f` x2, ...]
--
scanl1 :: (a -> a -> a) -> Seq a -> Seq a -- | scanr is the right-to-left dual of scanl. scanr :: (a -> b -> b) -> b -> Seq a -> Seq b -- | scanr1 is a variant of scanr that has no starting value -- argument. scanr1 :: (a -> a -> a) -> Seq a -> Seq a -- | O(n). Returns a sequence of all suffixes of this sequence, -- longest first. For example, -- -- 
--   tails (fromList "abc") = fromList [fromList "abc", fromList "bc", fromList "c", fromList ""]
--
-- -- Evaluating the ith suffix takes O(log(min(i, n-i))), but -- evaluating every suffix in the sequence takes O(n) due to -- sharing. tails :: Seq a -> Seq (Seq a) -- | O(n). Returns a sequence of all prefixes of this sequence, -- shortest first. For example, -- -- 
--   inits (fromList "abc") = fromList [fromList "", fromList "a", fromList "ab", fromList "abc"]
--
-- -- Evaluating the ith prefix takes O(log(min(i, n-i))), but -- evaluating every prefix in the sequence takes O(n) due to -- sharing. inits :: Seq a -> Seq (Seq a) -- | O(i) where i is the prefix length. takeWhileL, -- applied to a predicate p and a sequence xs, returns -- the longest prefix (possibly empty) of xs of elements that -- satisfy p. takeWhileL :: (a -> Bool) -> Seq a -> Seq a -- | O(i) where i is the suffix length. takeWhileR, -- applied to a predicate p and a sequence xs, returns -- the longest suffix (possibly empty) of xs of elements that -- satisfy p. -- -- takeWhileR p xs is equivalent to reverse -- (takeWhileL p (reverse xs)). takeWhileR :: (a -> Bool) -> Seq a -> Seq a -- | O(i) where i is the prefix length. dropWhileL -- p xs returns the suffix remaining after takeWhileL p -- xs. dropWhileL :: (a -> Bool) -> Seq a -> Seq a -- | O(i) where i is the suffix length. dropWhileR -- p xs returns the prefix remaining after takeWhileR p -- xs. -- -- dropWhileR p xs is equivalent to reverse -- (dropWhileL p (reverse xs)). dropWhileR :: (a -> Bool) -> Seq a -> Seq a -- | O(i) where i is the prefix length. spanl, applied -- to a predicate p and a sequence xs, returns a pair -- whose first element is the longest prefix (possibly empty) of -- xs of elements that satisfy p and the second element -- is the remainder of the sequence. spanl :: (a -> Bool) -> Seq a -> (Seq a, Seq a) -- | O(i) where i is the suffix length. spanr, applied -- to a predicate p and a sequence xs, returns a pair -- whose first element is the longest suffix (possibly -- empty) of xs of elements that satisfy p and the -- second element is the remainder of the sequence. spanr :: (a -> Bool) -> Seq a -> (Seq a, Seq a) -- | O(i) where i is the breakpoint index. breakl, -- applied to a predicate p and a sequence xs, returns -- a pair whose first element is the longest prefix (possibly empty) of -- xs of elements that do not satisfy p and the -- second element is the remainder of the sequence. -- -- breakl p is equivalent to spanl (not . -- p). breakl :: (a -> Bool) -> Seq a -> (Seq a, Seq a) -- | breakr p is equivalent to spanr (not . -- p). breakr :: (a -> Bool) -> Seq a -> (Seq a, Seq a) -- | O(n). The partition function takes a predicate -- p and a sequence xs and returns sequences of those -- elements which do and do not satisfy the predicate. partition :: (a -> Bool) -> Seq a -> (Seq a, Seq a) -- | O(n). The filter function takes a predicate p -- and a sequence xs and returns a sequence of those elements -- which satisfy the predicate. filter :: (a -> Bool) -> Seq a -> Seq a -- | O(n log n). sort sorts the specified Seq by the -- natural ordering of its elements. The sort is stable. If stability is -- not required, unstableSort can be considerably faster, and in -- particular uses less memory. sort :: Ord a => Seq a -> Seq a -- | O(n log n). sortBy sorts the specified Seq -- according to the specified comparator. The sort is stable. If -- stability is not required, unstableSortBy can be considerably -- faster, and in particular uses less memory. sortBy :: (a -> a -> Ordering) -> Seq a -> Seq a -- | O(n log n). unstableSort sorts the specified Seq -- by the natural ordering of its elements, but the sort is not stable. -- This algorithm is frequently faster and uses less memory than -- sort, and performs extremely well -- frequently twice as fast -- as sort -- when the sequence is already nearly sorted. unstableSort :: Ord a => Seq a -> Seq a -- | O(n log n). A generalization of unstableSort, -- unstableSortBy takes an arbitrary comparator and sorts the -- specified sequence. The sort is not stable. This algorithm is -- frequently faster and uses less memory than sortBy, and -- performs extremely well -- frequently twice as fast as sortBy -- -- when the sequence is already nearly sorted. unstableSortBy :: (a -> a -> Ordering) -> Seq a -> Seq a -- | O(log(min(i,n-i))). The element at the specified position, -- counting from 0. The argument should thus be a non-negative integer -- less than the size of the sequence. If the position is out of range, -- index fails with an error. index :: Seq a -> Int -> a -- | O(log(min(i,n-i))). Update the element at the specified -- position. If the position is out of range, the original sequence is -- returned. adjust :: (a -> a) -> Int -> Seq a -> Seq a -- | O(log(min(i,n-i))). Replace the element at the specified -- position. If the position is out of range, the original sequence is -- returned. update :: Int -> a -> Seq a -> Seq a -- | O(log(min(i,n-i))). The first i elements of a -- sequence. If i is negative, take i s yields -- the empty sequence. If the sequence contains fewer than i -- elements, the whole sequence is returned. take :: Int -> Seq a -> Seq a -- | O(log(min(i,n-i))). Elements of a sequence after the first -- i. If i is negative, drop i s yields -- the whole sequence. If the sequence contains fewer than i -- elements, the empty sequence is returned. drop :: Int -> Seq a -> Seq a -- | O(log(min(i,n-i))). Split a sequence at a given position. -- splitAt i s = (take i s, drop i s). splitAt :: Int -> Seq a -> (Seq a, Seq a) -- | elemIndexL finds the leftmost index of the specified element, -- if it is present, and otherwise Nothing. elemIndexL :: Eq a => a -> Seq a -> Maybe Int -- | elemIndicesL finds the indices of the specified element, from -- left to right (i.e. in ascending order). elemIndicesL :: Eq a => a -> Seq a -> [Int] -- | elemIndexR finds the rightmost index of the specified element, -- if it is present, and otherwise Nothing. elemIndexR :: Eq a => a -> Seq a -> Maybe Int -- | elemIndicesR finds the indices of the specified element, from -- right to left (i.e. in descending order). elemIndicesR :: Eq a => a -> Seq a -> [Int] -- | findIndexL p xs finds the index of the leftmost -- element that satisfies p, if any exist. findIndexL :: (a -> Bool) -> Seq a -> Maybe Int -- | findIndicesL p finds all indices of elements that -- satisfy p, in ascending order. findIndicesL :: (a -> Bool) -> Seq a -> [Int] -- | findIndexR p xs finds the index of the rightmost -- element that satisfies p, if any exist. findIndexR :: (a -> Bool) -> Seq a -> Maybe Int -- | findIndicesR p finds all indices of elements that -- satisfy p, in descending order. findIndicesR :: (a -> Bool) -> Seq a -> [Int] -- | foldlWithIndex is a version of foldl that also provides -- access to the index of each element. foldlWithIndex :: (b -> Int -> a -> b) -> b -> Seq a -> b -- | foldrWithIndex is a version of foldr that also provides -- access to the index of each element. foldrWithIndex :: (Int -> a -> b -> b) -> b -> Seq a -> b -- | A generalization of fmap, mapWithIndex takes a mapping -- function that also depends on the element's index, and applies it to -- every element in the sequence. mapWithIndex :: (Int -> a -> b) -> Seq a -> Seq b -- | O(n). The reverse of a sequence. reverse :: Seq a -> Seq a -- | O(min(n1,n2)). zip takes two sequences and returns a -- sequence of corresponding pairs. If one input is short, excess -- elements are discarded from the right end of the longer sequence. zip :: Seq a -> Seq b -> Seq (a, b) -- | O(min(n1,n2)). zipWith generalizes zip by zipping -- with the function given as the first argument, instead of a tupling -- function. For example, zipWith (+) is applied to two -- sequences to take the sequence of corresponding sums. zipWith :: (a -> b -> c) -> Seq a -> Seq b -> Seq c -- | O(min(n1,n2,n3)). zip3 takes three sequences and returns -- a sequence of triples, analogous to zip. zip3 :: Seq a -> Seq b -> Seq c -> Seq (a, b, c) -- | O(min(n1,n2,n3)). zipWith3 takes a function which -- combines three elements, as well as three sequences and returns a -- sequence of their point-wise combinations, analogous to -- zipWith. zipWith3 :: (a -> b -> c -> d) -> Seq a -> Seq b -> Seq c -> Seq d -- | O(min(n1,n2,n3,n4)). zip4 takes four sequences and -- returns a sequence of quadruples, analogous to zip. zip4 :: Seq a -> Seq b -> Seq c -> Seq d -> Seq (a, b, c, d) -- | O(min(n1,n2,n3,n4)). zipWith4 takes a function which -- combines four elements, as well as four sequences and returns a -- sequence of their point-wise combinations, analogous to -- zipWith. zipWith4 :: (a -> b -> c -> d -> e) -> Seq a -> Seq b -> Seq c -> Seq d -> Seq e instance Eq a => Eq (ViewR a) instance Ord a => Ord (ViewR a) instance Show a => Show (ViewR a) instance Read a => Read (ViewR a) instance Data a => Data (ViewR a) instance Eq a => Eq (ViewL a) instance Ord a => Ord (ViewL a) instance Show a => Show (ViewL a) instance Read a => Read (ViewL a) instance Data a => Data (ViewL a) instance Traversable ViewR instance Foldable ViewR instance Functor ViewR instance Typeable1 ViewR instance Traversable ViewL instance Foldable ViewL instance Functor ViewL instance Typeable1 ViewL instance Applicative (State s) instance Monad (State s) instance Functor (State s) instance Applicative Id instance Monad Id instance Functor Id instance Traversable Elem instance Foldable Elem instance Functor Elem instance Sized (Elem a) instance Sized (Node a) instance Traversable Node instance Functor Node instance Foldable Node instance Sized a => Sized (Digit a) instance Traversable Digit instance Functor Digit instance Foldable Digit instance Traversable FingerTree instance Functor FingerTree instance Foldable FingerTree instance Sized a => Sized (FingerTree a) instance Data a => Data (Seq a) instance Typeable1 Seq instance Monoid (Seq a) instance Read a => Read (Seq a) instance Show a => Show (Seq a) instance Ord a => Ord (Seq a) instance Eq a => Eq (Seq a) instance MonadPlus Seq instance Monad Seq instance Traversable Seq instance Foldable Seq instance Functor Seq -- | Multi-way trees (aka rose trees) and forests. module Data.Tree -- | Multi-way trees, also known as rose trees. data Tree a Node :: a -> Forest a -> Tree a -- | label value rootLabel :: Tree a -> a -- | zero or more child trees subForest :: Tree a -> Forest a type Forest a = [Tree a] -- | Neat 2-dimensional drawing of a tree. drawTree :: Tree String -> String -- | Neat 2-dimensional drawing of a forest. drawForest :: Forest String -> String -- | The elements of a tree in pre-order. flatten :: Tree a -> [a] -- | Lists of nodes at each level of the tree. levels :: Tree a -> [[a]] -- | Build a tree from a seed value unfoldTree :: (b -> (a, [b])) -> b -> Tree a -- | Build a forest from a list of seed values unfoldForest :: (b -> (a, [b])) -> [b] -> Forest a -- | Monadic tree builder, in depth-first order unfoldTreeM :: Monad m => (b -> m (a, [b])) -> b -> m (Tree a) -- | Monadic forest builder, in depth-first order unfoldForestM :: Monad m => (b -> m (a, [b])) -> [b] -> m (Forest a) -- | Monadic tree builder, in breadth-first order, using an algorithm -- adapted from Breadth-First Numbering: Lessons from a Small Exercise -- in Algorithm Design, by Chris Okasaki, ICFP'00. unfoldTreeM_BF :: Monad m => (b -> m (a, [b])) -> b -> m (Tree a) -- | Monadic forest builder, in breadth-first order, using an algorithm -- adapted from Breadth-First Numbering: Lessons from a Small Exercise -- in Algorithm Design, by Chris Okasaki, ICFP'00. unfoldForestM_BF :: Monad m => (b -> m (a, [b])) -> [b] -> m (Forest a) instance Eq a => Eq (Tree a) instance Read a => Read (Tree a) instance Show a => Show (Tree a) instance Data a => Data (Tree a) instance Foldable Tree instance Traversable Tree instance Monad Tree instance Applicative Tree instance Functor Tree instance Typeable1 Tree -- | A version of the graph algorithms described in: -- -- Lazy Depth-First Search and Linear Graph Algorithms in Haskell, -- by David King and John Launchbury. module Data.Graph -- | The strongly connected components of a directed graph, topologically -- sorted. stronglyConnComp :: Ord key => [(node, key, [key])] -> [SCC node] -- | The strongly connected components of a directed graph, topologically -- sorted. The function is the same as stronglyConnComp, except -- that all the information about each node retained. This interface is -- used when you expect to apply SCC to (some of) the result of -- SCC, so you don't want to lose the dependency information. stronglyConnCompR :: Ord key => [(node, key, [key])] -> [SCC (node, key, [key])] -- | Strongly connected component. data SCC vertex -- | A single vertex that is not in any cycle. AcyclicSCC :: vertex -> SCC vertex -- | A maximal set of mutually reachable vertices. CyclicSCC :: [vertex] -> SCC vertex -- | The vertices of a strongly connected component. flattenSCC :: SCC vertex -> [vertex] -- | The vertices of a list of strongly connected components. flattenSCCs :: [SCC a] -> [a] -- | Adjacency list representation of a graph, mapping each vertex to its -- list of successors. type Graph = Table [Vertex] -- | Table indexed by a contiguous set of vertices. type Table a = Array Vertex a -- | The bounds of a Table. type Bounds = (Vertex, Vertex) -- | An edge from the first vertex to the second. type Edge = (Vertex, Vertex) -- | Abstract representation of vertices. type Vertex = Int -- | Build a graph from a list of nodes uniquely identified by keys, with a -- list of keys of nodes this node should have edges to. The out-list may -- contain keys that don't correspond to nodes of the graph; they are -- ignored. graphFromEdges :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key]), key -> Maybe Vertex) -- | Identical to graphFromEdges, except that the return value does -- not include the function which maps keys to vertices. This version of -- graphFromEdges is for backwards compatibility. graphFromEdges' :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key])) -- | Build a graph from a list of edges. buildG :: Bounds -> [Edge] -> Graph -- | The graph obtained by reversing all edges. transposeG :: Graph -> Graph -- | All vertices of a graph. vertices :: Graph -> [Vertex] -- | All edges of a graph. edges :: Graph -> [Edge] -- | A table of the count of edges from each node. outdegree :: Graph -> Table Int -- | A table of the count of edges into each node. indegree :: Graph -> Table Int -- | A spanning forest of the part of the graph reachable from the listed -- vertices, obtained from a depth-first search of the graph starting at -- each of the listed vertices in order. dfs :: Graph -> [Vertex] -> Forest Vertex -- | A spanning forest of the graph, obtained from a depth-first search of -- the graph starting from each vertex in an unspecified order. dff :: Graph -> Forest Vertex -- | A topological sort of the graph. The order is partially specified by -- the condition that a vertex i precedes j whenever -- j is reachable from i but not vice versa. topSort :: Graph -> [Vertex] -- | The connected components of a graph. Two vertices are connected if -- there is a path between them, traversing edges in either direction. components :: Graph -> Forest Vertex -- | The strongly connected components of a graph. scc :: Graph -> Forest Vertex -- | The biconnected components of a graph. An undirected graph is -- biconnected if the deletion of any vertex leaves it connected. bcc :: Graph -> Forest [Vertex] -- | A list of vertices reachable from a given vertex. reachable :: Graph -> Vertex -> [Vertex] -- | Is the second vertex reachable from the first? path :: Graph -> Vertex -> Vertex -> Bool instance Monad (SetM s)