    {-# OPTIONS_GHC -cpp -XNoBangPatterns #-}
    -----------------------------------------------------------------------------
    -- |
    -- Module      :  Data.IntMap
    -- Copyright   :  (c) Daan Leijen 2002
    --                (c) Andriy Palamarchuk 2008
    -- License     :  BSD-style
    -- Maintainer  :  libraries@haskell.org
    -- Stability   :  provisional
    -- Portability :  portable
    --
    -- 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").
    --
    --    * Chris Okasaki and Andy Gill,  \"/Fast Mergeable Integer Maps/\",
    --      Workshop on ML, September 1998, pages 77-86,
    --      <http://citeseer.ist.psu.edu/okasaki98fast.html>
    --
    --    * D.R. Morrison, \"/PATRICIA -- Practical Algorithm To Retrieve
    --      Information Coded In Alphanumeric/\", Journal of the ACM, 15(4),
    --      October 1968, pages 514-534.
    --
    -- 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  (
                -- * Map type
                  IntMap, Key          -- instance Eq,Show

                -- * Operators
                , (!), (\\)

                -- * Query
                , null
                , size
                , member
                , notMember
    	    , lookup
                , findWithDefault

                -- * Construction
                , empty
                , singleton

                -- ** Insertion
                , insert
                , insertWith, insertWithKey, insertLookupWithKey

                -- ** Delete\/Update
                , delete
                , adjust
                , adjustWithKey
                , update
                , updateWithKey
                , updateLookupWithKey
                , alter

                -- * Combine

                -- ** Union
                , union
                , unionWith
                , unionWithKey
                , unions
                , unionsWith

                -- ** Difference
                , difference
                , differenceWith
                , differenceWithKey

                -- ** Intersection
                , intersection
                , intersectionWith
                , intersectionWithKey

                -- * Traversal
                -- ** Map
                , map
                , mapWithKey
                , mapAccum
                , mapAccumWithKey

                -- ** Fold
                , fold
                , foldWithKey

                -- * Conversion
                , elems
                , keys
    	    , keysSet
                , assocs

                -- ** Lists
                , toList
                , fromList
                , fromListWith
                , fromListWithKey

                -- ** Ordered lists
                , toAscList
                , fromAscList
                , fromAscListWith
                , fromAscListWithKey
                , fromDistinctAscList

                -- * Filter
                , filter
                , filterWithKey
                , partition
                , partitionWithKey

                , mapMaybe
                , mapMaybeWithKey
                , mapEither
                , mapEitherWithKey

                , split
                , splitLookup

                -- * Submap
                , isSubmapOf, isSubmapOfBy
                , isProperSubmapOf, isProperSubmapOfBy

                -- * Min\/Max

                , maxView
                , minView
                , findMin
                , findMax
                , deleteMin
                , deleteMax
                , deleteFindMin
                , deleteFindMax
                , updateMin
                , updateMax
                , updateMinWithKey
                , updateMaxWithKey
                , minViewWithKey
                , maxViewWithKey

                -- * Debugging
                , showTree
                , showTreeWith
                ) where

    import Prelude hiding (lookup,map,filter,foldr,foldl,null)
    import Data.Bits
    import qualified Data.IntSet as IntSet
    import Data.Monoid (Monoid(..))
    import Data.Maybe (fromMaybe)
    import Data.Typeable
    import Data.Foldable (Foldable(foldMap))
    import Control.Monad ( liftM )
    {-
    -- just for testing
    import qualified Prelude
    import Debug.QuickCheck
    import List (nub,sort)
    import qualified List
    -}

    #if __GLASGOW_HASKELL__
    import Text.Read
    import Data.Data (Data(..), mkNorepType)
    #endif

    #if __GLASGOW_HASKELL__ >= 503
    import GHC.Exts ( Word(..), Int(..), shiftRL# )
    #elif __GLASGOW_HASKELL__
    import Word
    import GlaExts ( Word(..), Int(..), shiftRL# )
    #else
    import Data.Word
    #endif

    infixl 9 \\{-This comment teaches CPP correct behaviour -}

    -- A "Nat" is a natural machine word (an unsigned Int)
    type Nat = Word

    natFromInt :: Key -> Nat
    natFromInt i = fromIntegral i

    intFromNat :: Nat -> Key
    intFromNat w = fromIntegral w

    shiftRL :: Nat -> Key -> Nat
    #if __GLASGOW_HASKELL__
    {--------------------------------------------------------------------
      GHC: use unboxing to get @shiftRL@ inlined.
    --------------------------------------------------------------------}
    shiftRL (W# x) (I# i)
      = W# (shiftRL# x i)
    #else
    shiftRL x i   = shiftR x i
    #endif

    {--------------------------------------------------------------------
      Operators
    --------------------------------------------------------------------}

    -- | /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
    m ! k    = find' k m

    -- | Same as 'difference'.
    (\\) :: IntMap a -> IntMap b -> IntMap a
    m1 \\ m2 = difference m1 m2

    {--------------------------------------------------------------------
      Types
    --------------------------------------------------------------------}
    -- | A map of integers to values @a@.
    data IntMap a = Nil
                  | Tip {-# UNPACK #-} !Key a
                  | Bin {-# UNPACK #-} !Prefix {-# UNPACK #-} !Mask !(IntMap a) !(IntMap a)

    type Prefix = Int
    type Mask   = Int
    type Key    = Int

    instance Monoid (IntMap a) where
        mempty  = empty
        mappend = union
        mconcat = unions

    instance Foldable IntMap where
        foldMap _ Nil = mempty
        foldMap f (Tip _k v) = f v
        foldMap f (Bin _ _ l r) = foldMap f l `mappend` foldMap f r

    #if __GLASGOW_HASKELL__

    {--------------------------------------------------------------------
      A Data instance
    --------------------------------------------------------------------}

    -- This instance preserves data abstraction at the cost of inefficiency.
    -- We omit reflection services for the sake of data abstraction.

    instance Data a => Data (IntMap a) where
      gfoldl f z im = z fromList `f` (toList im)
      toConstr _    = error "toConstr"
      gunfold _ _   = error "gunfold"
      dataTypeOf _  = mkNorepType "Data.IntMap.IntMap"
      dataCast1 f   = gcast1 f

    #endif

    {--------------------------------------------------------------------
      Query
    --------------------------------------------------------------------}
    -- | /O(1)/. Is the map empty?
    --
    -- > Data.IntMap.null (empty)           == True
    -- > Data.IntMap.null (singleton 1 'a') == False

    null :: IntMap a -> Bool
    null Nil = True
    null _   = False

    -- | /O(n)/. Number of elements in the map.
    --
    -- > size empty                                   == 0
    -- > size (singleton 1 'a')                       == 1
    -- > size (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3
    size :: IntMap a -> Int
    size t
      = case t of
          Bin _ _ l r -> size l + size r
          Tip _ _ -> 1
          Nil     -> 0

    -- | /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
    member k m
      = case lookup k m of
          Nothing -> False
          Just _  -> True

    -- | /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
    notMember k m = not $ member k m

    -- | /O(min(n,W))/. Lookup the value at a key in the map. See also 'Data.Map.lookup'.
    lookup :: Key -> IntMap a -> Maybe a
    lookup k t
      = let nk = natFromInt k  in seq nk (lookupN nk t)

    lookupN :: Nat -> IntMap a -> Maybe a
    lookupN k t
      = case t of
          Bin _ m l r
            | zeroN k (natFromInt m) -> lookupN k l
            | otherwise              -> lookupN k r
          Tip kx x
            | (k == natFromInt kx)  -> Just x
            | otherwise             -> Nothing
          Nil -> Nothing

    find' :: Key -> IntMap a -> a
    find' k m
      = case lookup k m of
          Nothing -> error ("IntMap.find: key " ++ show k ++ " is not an element of the map")
          Just x  -> x

    -- | /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
    findWithDefault def k m
      = case lookup k m of
          Nothing -> def
          Just x  -> x

    {--------------------------------------------------------------------
      Construction
    --------------------------------------------------------------------}
    -- | /O(1)/. The empty map.
    --
    -- > empty      == fromList []
    -- > size empty == 0

    empty :: IntMap a
    empty
      = Nil

    -- | /O(1)/. A map of one element.
    --
    -- > singleton 1 'a'        == fromList [(1, 'a')]
    -- > size (singleton 1 'a') == 1

    singleton :: Key -> a -> IntMap a
    singleton k x
      = Tip k x

    {--------------------------------------------------------------------
      Insert
    --------------------------------------------------------------------}
    -- | /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
    insert k x t
      = case t of
          Bin p m l r
            | nomatch k p m -> join k (Tip k x) p t
            | zero k m      -> Bin p m (insert k x l) r
            | otherwise     -> Bin p m l (insert k x r)
          Tip ky _
            | k==ky         -> Tip k x
            | otherwise     -> join k (Tip k x) ky t
          Nil -> Tip k x

    -- right-biased insertion, used by 'union'
    -- | /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
    insertWith f k x t
      = insertWithKey (\_ x' y' -> f x' y') k x t

    -- | /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
    insertWithKey f k x t
      = case t of
          Bin p m l r
            | nomatch k p m -> join k (Tip k x) p t
            | zero k m      -> Bin p m (insertWithKey f k x l) r
            | otherwise     -> Bin p m l (insertWithKey f k x r)
          Tip ky y
            | k==ky         -> Tip k (f k x y)
            | otherwise     -> join k (Tip k x) ky t
          Nil -> Tip k x

    -- | /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)
    insertLookupWithKey f k x t
      = case t of
          Bin p m l r
            | nomatch k p m -> (Nothing,join k (Tip k x) p t)
            | zero k m      -> let (found,l') = insertLookupWithKey f k x l in (found,Bin p m l' r)
            | otherwise     -> let (found,r') = insertLookupWithKey f k x r in (found,Bin p m l r')
          Tip ky y
            | k==ky         -> (Just y,Tip k (f k x y))
            | otherwise     -> (Nothing,join k (Tip k x) ky t)
          Nil -> (Nothing,Tip k x)

    {--------------------------------------------------------------------
      Deletion
      [delete] is the inlined version of [deleteWith (\k x -> Nothing)]
    --------------------------------------------------------------------}
    -- | /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
    delete k t
      = case t of
          Bin p m l r
            | nomatch k p m -> t
            | zero k m      -> bin p m (delete k l) r
            | otherwise     -> bin p m l (delete k r)
          Tip ky _
            | k==ky         -> Nil
            | otherwise     -> t
          Nil -> Nil

    -- | /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
    adjust f k m
      = adjustWithKey (\_ x -> f x) k m

    -- | /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
    adjustWithKey f k m
      = updateWithKey (\k' x -> Just (f k' x)) k m

    -- | /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
    update f k m
      = updateWithKey (\_ x -> f x) k m

    -- | /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
    updateWithKey f k t
      = case t of
          Bin p m l r
            | nomatch k p m -> t
            | zero k m      -> bin p m (updateWithKey f k l) r
            | otherwise     -> bin p m l (updateWithKey f k r)
          Tip ky y
            | k==ky         -> case (f k y) of
                                 Just y' -> Tip ky y'
                                 Nothing -> Nil
            | otherwise     -> t
          Nil -> Nil

    -- | /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)
    updateLookupWithKey f k t
      = case t of
          Bin p m l r
            | nomatch k p m -> (Nothing,t)
            | zero k m      -> let (found,l') = updateLookupWithKey f k l in (found,bin p m l' r)
            | otherwise     -> let (found,r') = updateLookupWithKey f k r in (found,bin p m l r')
          Tip ky y
            | k==ky         -> case (f k y) of
                                 Just y' -> (Just y,Tip ky y')
                                 Nothing -> (Just y,Nil)
            | otherwise     -> (Nothing,t)
          Nil -> (Nothing,Nil)

    -- | /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
    alter f k t
      = case t of
          Bin p m l r
            | nomatch k p m -> case f Nothing of
                                 Nothing -> t
                                 Just x -> join k (Tip k x) p t
            | zero k m      -> bin p m (alter f k l) r
            | otherwise     -> bin p m l (alter f k r)
          Tip ky y
            | k==ky         -> case f (Just y) of
                                 Just x -> Tip ky x
                                 Nothing -> Nil
            | otherwise     -> case f Nothing of
                                 Just x -> join k (Tip k x) ky t
                                 Nothing -> Tip ky y
          Nil               -> case f Nothing of
                                 Just x -> Tip k x
                                 Nothing -> Nil

    {--------------------------------------------------------------------
      Union
    --------------------------------------------------------------------}
    -- | 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
    unions xs
      = foldlStrict union empty xs

    -- | 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
    unionsWith f ts
      = foldlStrict (unionWith f) empty ts

    -- | /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
    union t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
      | shorter m1 m2  = union1
      | shorter m2 m1  = union2
      | p1 == p2       = Bin p1 m1 (union l1 l2) (union r1 r2)
      | otherwise      = join p1 t1 p2 t2
      where
        union1  | nomatch p2 p1 m1  = join p1 t1 p2 t2
                | zero p2 m1        = Bin p1 m1 (union l1 t2) r1
                | otherwise         = Bin p1 m1 l1 (union r1 t2)

        union2  | nomatch p1 p2 m2  = join p1 t1 p2 t2
                | zero p1 m2        = Bin p2 m2 (union t1 l2) r2
                | otherwise         = Bin p2 m2 l2 (union t1 r2)

    union (Tip k x) t = insert k x t
    union t (Tip k x) = insertWith (\_ y -> y) k x t  -- right bias
    union Nil t       = t
    union t Nil       = t

    -- | /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
    unionWith f m1 m2
      = unionWithKey (\_ x y -> f x y) m1 m2

    -- | /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
    unionWithKey f t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
      | shorter m1 m2  = union1
      | shorter m2 m1  = union2
      | p1 == p2       = Bin p1 m1 (unionWithKey f l1 l2) (unionWithKey f r1 r2)
      | otherwise      = join p1 t1 p2 t2
      where
        union1  | nomatch p2 p1 m1  = join p1 t1 p2 t2
                | zero p2 m1        = Bin p1 m1 (unionWithKey f l1 t2) r1
                | otherwise         = Bin p1 m1 l1 (unionWithKey f r1 t2)

        union2  | nomatch p1 p2 m2  = join p1 t1 p2 t2
                | zero p1 m2        = Bin p2 m2 (unionWithKey f t1 l2) r2
                | otherwise         = Bin p2 m2 l2 (unionWithKey f t1 r2)

    unionWithKey f (Tip k x) t = insertWithKey f k x t
    unionWithKey f t (Tip k x) = insertWithKey (\k' x' y' -> f k' y' x') k x t  -- right bias
    unionWithKey _ Nil t  = t
    unionWithKey _ t Nil  = t

    {--------------------------------------------------------------------
      Difference
    --------------------------------------------------------------------}
    -- | /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
    difference t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
      | shorter m1 m2  = difference1
      | shorter m2 m1  = difference2
      | p1 == p2       = bin p1 m1 (difference l1 l2) (difference r1 r2)
      | otherwise      = t1
      where
        difference1 | nomatch p2 p1 m1  = t1
                    | zero p2 m1        = bin p1 m1 (difference l1 t2) r1
                    | otherwise         = bin p1 m1 l1 (difference r1 t2)

        difference2 | nomatch p1 p2 m2  = t1
                    | zero p1 m2        = difference t1 l2
                    | otherwise         = difference t1 r2

    difference t1@(Tip k _) t2
      | member k t2  = Nil
      | otherwise    = t1

    difference Nil _       = Nil
    difference t (Tip k _) = delete k t
    difference t Nil       = t

    -- | /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
    differenceWith f m1 m2
      = differenceWithKey (\_ x y -> f x y) m1 m2

    -- | /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
    differenceWithKey f t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
      | shorter m1 m2  = difference1
      | shorter m2 m1  = difference2
      | p1 == p2       = bin p1 m1 (differenceWithKey f l1 l2) (differenceWithKey f r1 r2)
      | otherwise      = t1
      where
        difference1 | nomatch p2 p1 m1  = t1
                    | zero p2 m1        = bin p1 m1 (differenceWithKey f l1 t2) r1
                    | otherwise         = bin p1 m1 l1 (differenceWithKey f r1 t2)

        difference2 | nomatch p1 p2 m2  = t1
                    | zero p1 m2        = differenceWithKey f t1 l2
                    | otherwise         = differenceWithKey f t1 r2

    differenceWithKey f t1@(Tip k x) t2
      = case lookup k t2 of
          Just y  -> case f k x y of
                       Just y' -> Tip k y'
                       Nothing -> Nil
          Nothing -> t1

    differenceWithKey _ Nil _       = Nil
    differenceWithKey f t (Tip k y) = updateWithKey (\k' x -> f k' x y) k t
    differenceWithKey _ t Nil       = t

    {--------------------------------------------------------------------
      Intersection
    --------------------------------------------------------------------}
    -- | /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
    intersection t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
      | shorter m1 m2  = intersection1
      | shorter m2 m1  = intersection2
      | p1 == p2       = bin p1 m1 (intersection l1 l2) (intersection r1 r2)
      | otherwise      = Nil
      where
        intersection1 | nomatch p2 p1 m1  = Nil
                      | zero p2 m1        = intersection l1 t2
                      | otherwise         = intersection r1 t2

        intersection2 | nomatch p1 p2 m2  = Nil
                      | zero p1 m2        = intersection t1 l2
                      | otherwise         = intersection t1 r2

    intersection t1@(Tip k _) t2
      | member k t2  = t1
      | otherwise    = Nil
    intersection t (Tip k _)
      = case lookup k t of
          Just y  -> Tip k y
          Nothing -> Nil
    intersection Nil _ = Nil
    intersection _ Nil = Nil

    -- | /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 -> a) -> IntMap a -> IntMap b -> IntMap a
    intersectionWith f m1 m2
      = intersectionWithKey (\_ x y -> f x y) m1 m2

    -- | /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 -> a) -> IntMap a -> IntMap b -> IntMap a
    intersectionWithKey f t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
      | shorter m1 m2  = intersection1
      | shorter m2 m1  = intersection2
      | p1 == p2       = bin p1 m1 (intersectionWithKey f l1 l2) (intersectionWithKey f r1 r2)
      | otherwise      = Nil
      where
        intersection1 | nomatch p2 p1 m1  = Nil
                      | zero p2 m1        = intersectionWithKey f l1 t2
                      | otherwise         = intersectionWithKey f r1 t2

        intersection2 | nomatch p1 p2 m2  = Nil
                      | zero p1 m2        = intersectionWithKey f t1 l2
                      | otherwise         = intersectionWithKey f t1 r2

    intersectionWithKey f (Tip k x) t2
      = case lookup k t2 of
          Just y  -> Tip k (f k x y)
          Nothing -> Nil
    intersectionWithKey f t1 (Tip k y)
      = case lookup k t1 of
          Just x  -> Tip k (f k x y)
          Nothing -> Nil
    intersectionWithKey _ Nil _ = Nil
    intersectionWithKey _ _ Nil = Nil

    {--------------------------------------------------------------------
      Min\/Max
    --------------------------------------------------------------------}

    -- | /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
    updateMinWithKey f t
        = case t of
            Bin p m l r | m < 0 -> let t' = updateMinWithKeyUnsigned f r in Bin p m l t'
            Bin p m l r         -> let t' = updateMinWithKeyUnsigned f l in Bin p m t' r
            Tip k y -> Tip k (f k y)
            Nil -> error "maxView: empty map has no maximal element"

    updateMinWithKeyUnsigned :: (Key -> a -> a) -> IntMap a -> IntMap a
    updateMinWithKeyUnsigned f t
        = case t of
            Bin p m l r -> let t' = updateMinWithKeyUnsigned f l in Bin p m t' r
            Tip k y -> Tip k (f k y)
            Nil -> error "updateMinWithKeyUnsigned Nil"

    -- | /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
    updateMaxWithKey f t
        = case t of
            Bin p m l r | m < 0 -> let t' = updateMaxWithKeyUnsigned f l in Bin p m t' r
            Bin p m l r         -> let t' = updateMaxWithKeyUnsigned f r in Bin p m l t'
            Tip k y -> Tip k (f k y)
            Nil -> error "maxView: empty map has no maximal element"

    updateMaxWithKeyUnsigned :: (Key -> a -> a) -> IntMap a -> IntMap a
    updateMaxWithKeyUnsigned f t
        = case t of
            Bin p m l r -> let t' = updateMaxWithKeyUnsigned f r in Bin p m l t'
            Tip k y -> Tip k (f k y)
            Nil -> error "updateMaxWithKeyUnsigned Nil"

    -- | /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)
    maxViewWithKey t
        = case t of
            Bin p m l r | m < 0 -> let (result, t') = maxViewUnsigned l in Just (result, bin p m t' r)
            Bin p m l r         -> let (result, t') = maxViewUnsigned r in Just (result, bin p m l t')
            Tip k y -> Just ((k,y), Nil)
            Nil -> Nothing

    maxViewUnsigned :: IntMap a -> ((Key, a), IntMap a)
    maxViewUnsigned t
        = case t of
            Bin p m l r -> let (result,t') = maxViewUnsigned r in (result,bin p m l t')
            Tip k y -> ((k,y), Nil)
            Nil -> error "maxViewUnsigned Nil"

    -- | /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)
    minViewWithKey t
        = case t of
            Bin p m l r | m < 0 -> let (result, t') = minViewUnsigned r in Just (result, bin p m l t')
            Bin p m l r         -> let (result, t') = minViewUnsigned l in Just (result, bin p m t' r)
            Tip k y -> Just ((k,y),Nil)
            Nil -> Nothing

    minViewUnsigned :: IntMap a -> ((Key, a), IntMap a)
    minViewUnsigned t
        = case t of
            Bin p m l r -> let (result,t') = minViewUnsigned l in (result,bin p m t' r)
            Tip k y -> ((k,y),Nil)
            Nil -> error "minViewUnsigned Nil"

    -- | /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
    updateMax f = updateMaxWithKey (const f)

    -- | /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
    updateMin f = updateMinWithKey (const f)

    -- Similar to the Arrow instance.
    first :: (a -> c) -> (a, b) -> (c, b)
    first f (x,y) = (f x,y)

    -- | /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)
    maxView t = liftM (first snd) (maxViewWithKey t)

    -- | /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)
    minView t = liftM (first snd) (minViewWithKey t)

    -- | /O(log n)/. Delete and find the maximal element.
    deleteFindMax :: IntMap a -> (a, IntMap a)
    deleteFindMax = fromMaybe (error "deleteFindMax: empty map has no maximal element") . maxView

    -- | /O(log n)/. Delete and find the minimal element.
    deleteFindMin :: IntMap a -> (a, IntMap a)
    deleteFindMin = fromMaybe (error "deleteFindMin: empty map has no minimal element") . minView

    -- | /O(log n)/. The minimal key of the map.
    findMin :: IntMap a -> a
    findMin = maybe (error "findMin: empty map has no minimal element") fst . minView

    -- | /O(log n)/. The maximal key of the map.
    findMax :: IntMap a -> a
    findMax = maybe (error "findMax: empty map has no maximal element") fst . maxView

    -- | /O(log n)/. Delete the minimal key.
    deleteMin :: IntMap a -> IntMap a
    deleteMin = maybe (error "deleteMin: empty map has no minimal element") snd . minView

    -- | /O(log n)/. Delete the maximal key.
    deleteMax :: IntMap a -> IntMap a
    deleteMax = maybe (error "deleteMax: empty map has no maximal element") snd . maxView

    {--------------------------------------------------------------------
      Submap
    --------------------------------------------------------------------}
    -- | /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
    isProperSubmapOf m1 m2
      = isProperSubmapOfBy (==) m1 m2

    {- | /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
    isProperSubmapOfBy predicate t1 t2
      = case submapCmp predicate t1 t2 of
          LT -> True
          _  -> False

    submapCmp :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Ordering
    submapCmp predicate t1@(Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
      | shorter m1 m2  = GT
      | shorter m2 m1  = submapCmpLt
      | p1 == p2       = submapCmpEq
      | otherwise      = GT  -- disjoint
      where
        submapCmpLt | nomatch p1 p2 m2  = GT
                    | zero p1 m2        = submapCmp predicate t1 l2
                    | otherwise         = submapCmp predicate t1 r2
        submapCmpEq = case (submapCmp predicate l1 l2, submapCmp predicate r1 r2) of
                        (GT,_ ) -> GT
                        (_ ,GT) -> GT
                        (EQ,EQ) -> EQ
                        _       -> LT

    submapCmp _         (Bin _ _ _ _) _  = GT
    submapCmp predicate (Tip kx x) (Tip ky y)
      | (kx == ky) && predicate x y = EQ
      | otherwise                   = GT  -- disjoint
    submapCmp predicate (Tip k x) t
      = case lookup k t of
         Just y | predicate x y -> LT
         _                      -> GT -- disjoint
    submapCmp _    Nil Nil = EQ
    submapCmp _    Nil _   = LT

    -- | /O(n+m)/. Is this a submap?
    -- Defined as (@'isSubmapOf' = 'isSubmapOfBy' (==)@).
    isSubmapOf :: Eq a => IntMap a -> IntMap a -> Bool
    isSubmapOf m1 m2
      = isSubmapOfBy (==) m1 m2

    {- | /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
    isSubmapOfBy predicate t1@(Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
      | shorter m1 m2  = False
      | shorter m2 m1  = match p1 p2 m2 && (if zero p1 m2 then isSubmapOfBy predicate t1 l2
                                                          else isSubmapOfBy predicate t1 r2)
      | otherwise      = (p1==p2) && isSubmapOfBy predicate l1 l2 && isSubmapOfBy predicate r1 r2
    isSubmapOfBy _         (Bin _ _ _ _) _ = False
    isSubmapOfBy predicate (Tip k x) t     = case lookup k t of
                                             Just y  -> predicate x y
                                             Nothing -> False
    isSubmapOfBy _         Nil _           = True

    {--------------------------------------------------------------------
      Mapping
    --------------------------------------------------------------------}
    -- | /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
    map f m
      = mapWithKey (\_ x -> f x) m

    -- | /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
    mapWithKey f t
      = case t of
          Bin p m l r -> Bin p m (mapWithKey f l) (mapWithKey f r)
          Tip k x     -> Tip k (f k x)
          Nil         -> Nil

    -- | /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)
    mapAccum f a m
      = mapAccumWithKey (\a' _ x -> f a' x) a m

    -- | /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)
    mapAccumWithKey f a t
      = mapAccumL f a t

    -- | /O(n)/. The function @'mapAccumL'@ threads an accumulating
    -- argument through the map in ascending order of keys.
    mapAccumL :: (a -> Key -> b -> (a,c)) -> a -> IntMap b -> (a,IntMap c)
    mapAccumL f a t
      = case t of
          Bin p m l r -> let (a1,l') = mapAccumL f a l
                             (a2,r') = mapAccumL f a1 r
                         in (a2,Bin p m l' r')
          Tip k x     -> let (a',x') = f a k x in (a',Tip k x')
          Nil         -> (a,Nil)

    {-
    XXX unused code

    -- | /O(n)/. The function @'mapAccumR'@ threads an accumulating
    -- argument throught the map in descending order of keys.
    mapAccumR :: (a -> Key -> b -> (a,c)) -> a -> IntMap b -> (a,IntMap c)
    mapAccumR f a t
      = case t of
          Bin p m l r -> let (a1,r') = mapAccumR f a r
                             (a2,l') = mapAccumR f a1 l
                         in (a2,Bin p m l' r')
          Tip k x     -> let (a',x') = f a k x in (a',Tip k x')
          Nil         -> (a,Nil)
    -}

    {--------------------------------------------------------------------
      Filter
    --------------------------------------------------------------------}
    -- | /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
    filter p m
      = filterWithKey (\_ x -> p x) m

    -- | /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
    filterWithKey predicate t
      = case t of
          Bin p m l r
            -> bin p m (filterWithKey predicate l) (filterWithKey predicate r)
          Tip k x
            | predicate k x -> t
            | otherwise     -> Nil
          Nil -> Nil

    -- | /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)
    partition p m
      = partitionWithKey (\_ x -> p x) m

    -- | /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)
    partitionWithKey predicate t
      = case t of
          Bin p m l r
            -> let (l1,l2) = partitionWithKey predicate l
                   (r1,r2) = partitionWithKey predicate r
               in (bin p m l1 r1, bin p m l2 r2)
          Tip k x
            | predicate k x -> (t,Nil)
            | otherwise     -> (Nil,t)
          Nil -> (Nil,Nil)

    -- | /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
    mapMaybe f m
      = mapMaybeWithKey (\_ x -> f x) m

    -- | /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
    mapMaybeWithKey f (Bin p m l r)
      = bin p m (mapMaybeWithKey f l) (mapMaybeWithKey f r)
    mapMaybeWithKey f (Tip k x) = case f k x of
      Just y  -> Tip k y
      Nothing -> Nil
    mapMaybeWithKey _ Nil = Nil

    -- | /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)
    mapEither f m
      = mapEitherWithKey (\_ x -> f x) m

    -- | /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)
    mapEitherWithKey f (Bin p m l r)
      = (bin p m l1 r1, bin p m l2 r2)
      where
        (l1,l2) = mapEitherWithKey f l
        (r1,r2) = mapEitherWithKey f r
    mapEitherWithKey f (Tip k x) = case f k x of
      Left y  -> (Tip k y, Nil)
      Right z -> (Nil, Tip k z)
    mapEitherWithKey _ Nil = (Nil, Nil)

    -- | /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)
    split k t
      = case t of
          Bin _ m l r
              | m < 0 -> (if k >= 0 -- handle negative numbers.
                          then let (lt,gt) = split' k l in (union r lt, gt)
                          else let (lt,gt) = split' k r in (lt, union gt l))
              | otherwise   -> split' k t
          Tip ky _
            | k>ky      -> (t,Nil)
            | k<ky      -> (Nil,t)
            | otherwise -> (Nil,Nil)
          Nil -> (Nil,Nil)

    split' :: Key -> IntMap a -> (IntMap a,IntMap a)
    split' k t
      = case t of
          Bin p m l r
            | nomatch k p m -> if k>p then (t,Nil) else (Nil,t)
            | zero k m  -> let (lt,gt) = split k l in (lt,union gt r)
            | otherwise -> let (lt,gt) = split k r in (union l lt,gt)
          Tip ky _
            | k>ky      -> (t,Nil)
            | k<ky      -> (Nil,t)
            | otherwise -> (Nil,Nil)
          Nil -> (Nil,Nil)

    -- | /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)
    splitLookup k t
      = case t of
          Bin _ m l r
              | m < 0 -> (if k >= 0 -- handle negative numbers.
                          then let (lt,found,gt) = splitLookup' k l in (union r lt,found, gt)
                          else let (lt,found,gt) = splitLookup' k r in (lt,found, union gt l))
              | otherwise   -> splitLookup' k t
          Tip ky y
            | k>ky      -> (t,Nothing,Nil)
            | k<ky      -> (Nil,Nothing,t)
            | otherwise -> (Nil,Just y,Nil)
          Nil -> (Nil,Nothing,Nil)

    splitLookup' :: Key -> IntMap a -> (IntMap a,Maybe a,IntMap a)
    splitLookup' k t
      = case t of
          Bin p m l r
            | nomatch k p m -> if k>p then (t,Nothing,Nil) else (Nil,Nothing,t)
            | zero k m  -> let (lt,found,gt) = splitLookup k l in (lt,found,union gt r)
            | otherwise -> let (lt,found,gt) = splitLookup k r in (union l lt,found,gt)
          Tip ky y
            | k>ky      -> (t,Nothing,Nil)
            | k<ky      -> (Nil,Nothing,t)
            | otherwise -> (Nil,Just y,Nil)
          Nil -> (Nil,Nothing,Nil)

    {--------------------------------------------------------------------
      Fold
    --------------------------------------------------------------------}
    -- | /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
    fold f z t
      = foldWithKey (\_ x y -> f x y) z t

    -- | /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
    foldWithKey f z t
      = foldr f z t

    foldr :: (Key -> a -> b -> b) -> b -> IntMap a -> b
    foldr f z t
      = case t of
          Bin 0 m l r | m < 0 -> foldr' f (foldr' f z l) r  -- put negative numbers before.
          Bin _ _ _ _ -> foldr' f z t
          Tip k x     -> f k x z
          Nil         -> z

    foldr' :: (Key -> a -> b -> b) -> b -> IntMap a -> b
    foldr' f z t
      = case t of
          Bin _ _ l r -> foldr' f (foldr' f z r) l
          Tip k x     -> f k x z
          Nil         -> z

    {--------------------------------------------------------------------
      List variations
    --------------------------------------------------------------------}
    -- | /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]
    elems m
      = foldWithKey (\_ x xs -> x:xs) [] m

    -- | /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]
    keys m
      = foldWithKey (\k _ ks -> k:ks) [] m

    -- | /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.IntSet
    keysSet m = IntSet.fromDistinctAscList (keys m)

    -- | /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)]
    assocs m
      = toList m

    {--------------------------------------------------------------------
      Lists
    --------------------------------------------------------------------}
    -- | /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)]
    toList t
      = foldWithKey (\k x xs -> (k,x):xs) [] t

    -- | /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)]
    toAscList t
      = -- NOTE: the following algorithm only works for big-endian trees
        let (pos,neg) = span (\(k,_) -> k >=0) (foldr (\k x xs -> (k,x):xs) [] t) in neg ++ pos

    -- | /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
    fromList xs
      = foldlStrict ins empty xs
      where
        ins t (k,x)  = insert k x t

    -- | /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
    fromListWith f xs
      = fromListWithKey (\_ x y -> f x y) xs

    -- | /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
    fromListWithKey f xs
      = foldlStrict ins empty xs
      where
        ins t (k,x) = insertWithKey f k x t

    -- | /O(n*min(n,W))/. 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
    fromAscList xs
      = fromList xs

    -- | /O(n*min(n,W))/. Build a map from a list of key\/value pairs where
    -- the keys are in ascending order, with a combining function on equal keys.
    --
    -- > fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]

    fromAscListWith :: (a -> a -> a) -> [(Key,a)] -> IntMap a
    fromAscListWith f xs
      = fromListWith f xs

    -- | /O(n*min(n,W))/. Build a map from a list of key\/value pairs where
    -- the keys are in ascending order, with a combining function on equal keys.
    --
    -- > fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]

    fromAscListWithKey :: (Key -> a -> a -> a) -> [(Key,a)] -> IntMap a
    fromAscListWithKey f xs
      = fromListWithKey f xs

    -- | /O(n*min(n,W))/. Build a map from a list of key\/value pairs where
    -- the keys are in ascending order and all distinct.
    --
    -- > fromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]

    fromDistinctAscList :: [(Key,a)] -> IntMap a
    fromDistinctAscList xs
      = fromList xs

    {--------------------------------------------------------------------
      Eq
    --------------------------------------------------------------------}
    instance Eq a => Eq (IntMap a) where
      t1 == t2  = equal t1 t2
      t1 /= t2  = nequal t1 t2

    equal :: Eq a => IntMap a -> IntMap a -> Bool
    equal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
      = (m1 == m2) && (p1 == p2) && (equal l1 l2) && (equal r1 r2)
    equal (Tip kx x) (Tip ky y)
      = (kx == ky) && (x==y)
    equal Nil Nil = True
    equal _   _   = False

    nequal :: Eq a => IntMap a -> IntMap a -> Bool
    nequal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
      = (m1 /= m2) || (p1 /= p2) || (nequal l1 l2) || (nequal r1 r2)
    nequal (Tip kx x) (Tip ky y)
      = (kx /= ky) || (x/=y)
    nequal Nil Nil = False
    nequal _   _   = True

    {--------------------------------------------------------------------
      Ord
    --------------------------------------------------------------------}

    instance Ord a => Ord (IntMap a) where
        compare m1 m2 = compare (toList m1) (toList m2)

    {--------------------------------------------------------------------
      Functor
    --------------------------------------------------------------------}

    instance Functor IntMap where
        fmap = map

    {--------------------------------------------------------------------
      Show
    --------------------------------------------------------------------}

    instance Show a => Show (IntMap a) where
      showsPrec d m   = showParen (d > 10) $
        showString "fromList " . shows (toList m)

    {-
    XXX unused code

    showMap :: (Show a) => [(Key,a)] -> ShowS
    showMap []
      = showString "{}"
    showMap (x:xs)
      = showChar '{' . showElem x . showTail xs
      where
        showTail []     = showChar '}'
        showTail (x':xs') = showChar ',' . showElem x' . showTail xs'

        showElem (k,v)  = shows k . showString ":=" . shows v
    -}

    {--------------------------------------------------------------------
      Read
    --------------------------------------------------------------------}
    instance (Read e) => Read (IntMap e) where
    #ifdef __GLASGOW_HASKELL__
      readPrec = parens $ prec 10 $ do
        Ident "fromList" <- lexP
        xs <- readPrec
        return (fromList xs)

      readListPrec = readListPrecDefault
    #else
      readsPrec p = readParen (p > 10) $ \ r -> do
        ("fromList",s) <- lex r
        (xs,t) <- reads s
        return (fromList xs,t)
    #endif

    {--------------------------------------------------------------------
      Typeable
    --------------------------------------------------------------------}

    #include "Typeable.h"
    INSTANCE_TYPEABLE1(IntMap,intMapTc,"IntMap")

    {--------------------------------------------------------------------
      Debugging
    --------------------------------------------------------------------}
    -- | /O(n)/. Show the tree that implements the map. The tree is shown
    -- in a compressed, hanging format.
    showTree :: Show a => IntMap a -> String
    showTree s
      = showTreeWith True False s

    {- | /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
    showTreeWith hang wide t
      | hang      = (showsTreeHang wide [] t) ""
      | otherwise = (showsTree wide [] [] t) ""

    showsTree :: Show a => Bool -> [String] -> [String] -> IntMap a -> ShowS
    showsTree wide lbars rbars t
      = case t of
          Bin p m l r
              -> showsTree wide (withBar rbars) (withEmpty rbars) r .
                 showWide wide rbars .
                 showsBars lbars . showString (showBin p m) . showString "\n" .
                 showWide wide lbars .
                 showsTree wide (withEmpty lbars) (withBar lbars) l
          Tip k x
              -> showsBars lbars . showString " " . shows k . showString ":=" . shows x . showString "\n"
          Nil -> showsBars lbars . showString "|\n"

    showsTreeHang :: Show a => Bool -> [String] -> IntMap a -> ShowS
    showsTreeHang wide bars t
      = case t of
          Bin p m l r
              -> showsBars bars . showString (showBin p m) . showString "\n" .
                 showWide wide bars .
                 showsTreeHang wide (withBar bars) l .
                 showWide wide bars .
                 showsTreeHang wide (withEmpty bars) r
          Tip k x
              -> showsBars bars . showString " " . shows k . showString ":=" . shows x . showString "\n"
          Nil -> showsBars bars . showString "|\n"

    showBin :: Prefix -> Mask -> String
    showBin _ _
      = "*" -- ++ show (p,m)

    showWide :: Bool -> [String] -> String -> String
    showWide wide bars
      | wide      = showString (concat (reverse bars)) . showString "|\n"
      | otherwise = id

    showsBars :: [String] -> ShowS
    showsBars bars
      = case bars of
          [] -> id
          _  -> showString (concat (reverse (tail bars))) . showString node

    node :: String
    node           = "+--"

    withBar, withEmpty :: [String] -> [String]
    withBar bars   = "|  ":bars
    withEmpty bars = "   ":bars

    {--------------------------------------------------------------------
      Helpers
    --------------------------------------------------------------------}
    {--------------------------------------------------------------------
      Join
    --------------------------------------------------------------------}
    join :: Prefix -> IntMap a -> Prefix -> IntMap a -> IntMap a
    join p1 t1 p2 t2
      | zero p1 m = Bin p m t1 t2
      | otherwise = Bin p m t2 t1
      where
        m = branchMask p1 p2
        p = mask p1 m

    {--------------------------------------------------------------------
      @bin@ assures that we never have empty trees within a tree.
    --------------------------------------------------------------------}
    bin :: Prefix -> Mask -> IntMap a -> IntMap a -> IntMap a
    bin _ _ l Nil = l
    bin _ _ Nil r = r
    bin p m l r   = Bin p m l r

    {--------------------------------------------------------------------
      Endian independent bit twiddling
    --------------------------------------------------------------------}
    zero :: Key -> Mask -> Bool
    zero i m
      = (natFromInt i) .&. (natFromInt m) == 0

    nomatch,match :: Key -> Prefix -> Mask -> Bool
    nomatch i p m
      = (mask i m) /= p

    match i p m
      = (mask i m) == p

    mask :: Key -> Mask -> Prefix
    mask i m
      = maskW (natFromInt i) (natFromInt m)

    zeroN :: Nat -> Nat -> Bool
    zeroN i m = (i .&. m) == 0

    {--------------------------------------------------------------------
      Big endian operations
    --------------------------------------------------------------------}
    maskW :: Nat -> Nat -> Prefix
    maskW i m
      = intFromNat (i .&. (complement (m-1) `xor` m))

    shorter :: Mask -> Mask -> Bool
    shorter m1 m2
      = (natFromInt m1) > (natFromInt m2)

    branchMask :: Prefix -> Prefix -> Mask
    branchMask p1 p2
      = intFromNat (highestBitMask (natFromInt p1 `xor` natFromInt p2))

    {----------------------------------------------------------------------
      Finding the highest bit (mask) in a word [x] can be done efficiently in
      three ways:
      * convert to a floating point value and the mantissa tells us the
        [log2(x)] that corresponds with the highest bit position. The mantissa
        is retrieved either via the standard C function [frexp] or by some bit
        twiddling on IEEE compatible numbers (float). Note that one needs to
        use at least [double] precision for an accurate mantissa of 32 bit
        numbers.
      * use bit twiddling, a logarithmic sequence of bitwise or's and shifts (bit).
      * use processor specific assembler instruction (asm).

      The most portable way would be [bit], but is it efficient enough?
      I have measured the cycle counts of the different methods on an AMD
      Athlon-XP 1800 (~ Pentium III 1.8Ghz) using the RDTSC instruction:

      highestBitMask: method  cycles
                      --------------
                       frexp   200
                       float    33
                       bit      11
                       asm      12

      highestBit:     method  cycles
                      --------------
                       frexp   195
                       float    33
                       bit      11
                       asm      11

      Wow, the bit twiddling is on today's RISC like machines even faster
      than a single CISC instruction (BSR)!
    ----------------------------------------------------------------------}

    {----------------------------------------------------------------------
      [highestBitMask] returns a word where only the highest bit is set.
      It is found by first setting all bits in lower positions than the
      highest bit and than taking an exclusive or with the original value.
      Allthough the function may look expensive, GHC compiles this into
      excellent C code that subsequently compiled into highly efficient
      machine code. The algorithm is derived from Jorg Arndt's FXT library.
    ----------------------------------------------------------------------}
    highestBitMask :: Nat -> Nat
    highestBitMask x0
      = case (x0 .|. shiftRL x0 1) of
         x1 -> case (x1 .|. shiftRL x1 2) of
          x2 -> case (x2 .|. shiftRL x2 4) of
           x3 -> case (x3 .|. shiftRL x3 8) of
            x4 -> case (x4 .|. shiftRL x4 16) of
             x5 -> case (x5 .|. shiftRL x5 32) of   -- for 64 bit platforms
              x6 -> (x6 `xor` (shiftRL x6 1))

    {--------------------------------------------------------------------
      Utilities
    --------------------------------------------------------------------}
    foldlStrict :: (a -> b -> a) -> a -> [b] -> a
    foldlStrict f z xs
      = case xs of
          []     -> z
          (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)

    {-
    {--------------------------------------------------------------------
      Testing
    --------------------------------------------------------------------}
    testTree :: [Int] -> IntMap Int
    testTree xs   = fromList [(x,x*x*30696 `mod` 65521) | x <- xs]
    test1 = testTree [1..20]
    test2 = testTree [30,29..10]
    test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]

    {--------------------------------------------------------------------
      QuickCheck
    --------------------------------------------------------------------}
    qcheck prop
      = check config prop
      where
        config = Config
          { configMaxTest = 500
          , configMaxFail = 5000
          , configSize    = \n -> (div n 2 + 3)
          , configEvery   = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
          }

    {--------------------------------------------------------------------
      Arbitrary, reasonably balanced trees
    --------------------------------------------------------------------}
    instance Arbitrary a => Arbitrary (IntMap a) where
      arbitrary = do{ ks <- arbitrary
                    ; xs <- mapM (\k -> do{ x <- arbitrary; return (k,x)}) ks
                    ; return (fromList xs)
                    }

    {--------------------------------------------------------------------
      Single, Insert, Delete
    --------------------------------------------------------------------}
    prop_Single :: Key -> Int -> Bool
    prop_Single k x
      = (insert k x empty == singleton k x)

    prop_InsertDelete :: Key -> Int -> IntMap Int -> Property
    prop_InsertDelete k x t
      = not (member k t) ==> delete k (insert k x t) == t

    prop_UpdateDelete :: Key -> IntMap Int -> Bool
    prop_UpdateDelete k t
      = update (const Nothing) k t == delete k t

    {--------------------------------------------------------------------
      Union
    --------------------------------------------------------------------}
    prop_UnionInsert :: Key -> Int -> IntMap Int -> Bool
    prop_UnionInsert k x t
      = union (singleton k x) t == insert k x t

    prop_UnionAssoc :: IntMap Int -> IntMap Int -> IntMap Int -> Bool
    prop_UnionAssoc t1 t2 t3
      = union t1 (union t2 t3) == union (union t1 t2) t3

    prop_UnionComm :: IntMap Int -> IntMap Int -> Bool
    prop_UnionComm t1 t2
      = (union t1 t2 == unionWith (\x y -> y) t2 t1)

    prop_Diff :: [(Key,Int)] -> [(Key,Int)] -> Bool
    prop_Diff xs ys
      =  List.sort (keys (difference (fromListWith (+) xs) (fromListWith (+) ys)))
        == List.sort ((List.\\) (nub (Prelude.map fst xs))  (nub (Prelude.map fst ys)))

    prop_Int :: [(Key,Int)] -> [(Key,Int)] -> Bool
    prop_Int xs ys
      =  List.sort (keys (intersection (fromListWith (+) xs) (fromListWith (+) ys)))
        == List.sort (nub ((List.intersect) (Prelude.map fst xs)  (Prelude.map fst ys)))

    {--------------------------------------------------------------------
      Lists
    --------------------------------------------------------------------}
    prop_Ordered
      = forAll (choose (5,100)) $ \n ->
        let xs = [(x,()) | x <- [0..n::Int]]
        in fromAscList xs == fromList xs

    prop_List :: [Key] -> Bool
    prop_List xs
      = (sort (nub xs) == [x | (x,()) <- toAscList (fromList [(x,()) | x <- xs])])

    {--------------------------------------------------------------------
      updateMin / updateMax
    --------------------------------------------------------------------}
    prop_UpdateMinMax :: [Key] -> Bool
    prop_UpdateMinMax xs =
      let m = fromList [(x,0)|x<-xs]
          minKey = fst . head . Prelude.filter ((==1).snd) . assocs . updateMin succ $ m
          maxKey = fst . head . Prelude.filter ((==1).snd) . assocs . updateMax succ $ m
      in  all (>=minKey) xs && all (<=maxKey) xs

    -}