content: {-# LANGUAGE CPP #-}
  #if __GLASGOW_HASKELL__
  {-# LANGUAGE DeriveDataTypeable, StandaloneDeriving #-}
  #endif
  #if !defined(TESTING) && __GLASGOW_HASKELL__ >= 703
  {-# LANGUAGE Trustworthy #-}
  #endif
  -----------------------------------------------------------------------------
  -- |
  -- Module      :  Data.Map.Base
  -- 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 keys to values (dictionaries).
  --
  -- Since many function names (but not the type name) clash with
  -- "Prelude" names, this module is usually imported @qualified@, e.g.
  --
  -- >  import Data.Map (Map)
  -- >  import qualified Data.Map as Map
  --
  -- The implementation of 'Map' is based on /size balanced/ binary trees (or
  -- trees of /bounded balance/) as described by:
  --
  --    * Stephen Adams, \"/Efficient sets: a balancing act/\",
  --     Journal of Functional Programming 3(4):553-562, October 1993,
  --     <http://www.swiss.ai.mit.edu/~adams/BB/>.
  --
  --    * J. Nievergelt and E.M. Reingold,
  --      \"/Binary search trees of bounded balance/\",
  --      SIAM journal of computing 2(1), March 1973.
  --
  -- Note that the implementation is /left-biased/ -- the elements of a
  -- first argument are always preferred to the second, for example in
  -- 'union' or 'insert'.
  --
  -- Operation comments contain the operation time complexity in
  -- the Big-O notation <http://en.wikipedia.org/wiki/Big_O_notation>.
  -----------------------------------------------------------------------------
  
  -- [Note: Using INLINABLE]
  -- ~~~~~~~~~~~~~~~~~~~~~~~
  -- It is crucial to the performance that the functions specialize on the Ord
  -- type when possible. GHC 7.0 and higher does this by itself when it sees th
  -- unfolding of a function -- that is why all public functions are marked
  -- INLINABLE (that exposes the unfolding).
  
  
  -- [Note: Using INLINE]
  -- ~~~~~~~~~~~~~~~~~~~~
  -- For other compilers and GHC pre 7.0, we mark some of the functions INLINE.
  -- We mark the functions that just navigate down the tree (lookup, insert,
  -- delete and similar). That navigation code gets inlined and thus specialized
  -- when possible. There is a price to pay -- code growth. The code INLINED is
  -- therefore only the tree navigation, all the real work (rebalancing) is not
  -- INLINED by using a NOINLINE.
  --
  -- All methods marked INLINE have to be nonrecursive -- a 'go' function doing
  -- the real work is provided.
  
  
  -- [Note: Type of local 'go' function]
  -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  -- If the local 'go' function uses an Ord class, it sometimes heap-allocates
  -- the Ord dictionary when the 'go' function does not have explicit type.
  -- In that case we give 'go' explicit type. But this slightly decrease
  -- performance, as the resulting 'go' function can float out to top level.
  
  
  -- [Note: Local 'go' functions and capturing]
  -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  -- As opposed to IntMap, when 'go' function captures an argument, increased
  -- heap-allocation can occur: sometimes in a polymorphic function, the 'go'
  -- floats out of its enclosing function and then it heap-allocates the
  -- dictionary and the argument. Maybe it floats out too late and strictness
  -- analyzer cannot see that these could be passed on stack.
  --
  -- For example, change 'member' so that its local 'go' function is not passing
  -- argument k and then look at the resulting code for hedgeInt.
  
  
  -- [Note: Order of constructors]
  -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  -- The order of constructors of Map matters when considering performance.
  -- Currently in GHC 7.0, when type has 2 constructors, a forward conditional
  -- jump is made when successfully matching second constructor. Successful match
  -- of first constructor results in the forward jump not taken.
  -- On GHC 7.0, reordering constructors from Tip | Bin to Bin | Tip
  -- improves the benchmark by up to 10% on x86.
  
  module Data.Map.Base (
              -- * Map type
                Map(..)          -- instance Eq,Show,Read
  
              -- * Operators
              , (!), (\\)
  
              -- * Query
              , null
              , size
              , member
              , notMember
              , lookup
              , findWithDefault
              , lookupLT
              , lookupGT
              , lookupLE
              , lookupGE
  
              -- * 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
  
              -- ** Universal combining function
              , mergeWithKey
  
              -- * Traversal
              -- ** Map
              , map
              , mapWithKey
              , traverseWithKey
              , mapAccum
              , mapAccumWithKey
              , mapAccumRWithKey
              , mapKeys
              , mapKeysWith
              , mapKeysMonotonic
  
              -- * Folds
              , foldr
              , foldl
              , foldrWithKey
              , foldlWithKey
              -- ** Strict folds
              , foldr'
              , foldl'
              , foldrWithKey'
              , foldlWithKey'
  
              -- * Conversion
              , elems
              , keys
              , assocs
              , keysSet
              , fromSet
  
              -- ** Lists
              , toList
              , fromList
              , fromListWith
              , fromListWithKey
  
              -- ** Ordered lists
              , toAscList
              , toDescList
              , fromAscList
              , fromAscListWith
              , fromAscListWithKey
              , fromDistinctAscList
  
              -- * Filter
              , filter
              , filterWithKey
              , partition
              , partitionWithKey
  
              , mapMaybe
              , mapMaybeWithKey
              , mapEither
              , mapEitherWithKey
  
              , split
              , splitLookup
  
              -- * Submap
              , isSubmapOf, isSubmapOfBy
              , isProperSubmapOf, isProperSubmapOfBy
  
              -- * Indexed
              , lookupIndex
              , findIndex
              , elemAt
              , updateAt
              , deleteAt
  
              -- * Min\/Max
              , findMin
              , findMax
              , deleteMin
              , deleteMax
              , deleteFindMin
              , deleteFindMax
              , updateMin
              , updateMax
              , updateMinWithKey
              , updateMaxWithKey
              , minView
              , maxView
              , minViewWithKey
              , maxViewWithKey
  
              -- * Debugging
              , showTree
              , showTreeWith
              , valid
  
              -- Used by the strict version
              , bin
              , balance
              , balanced
              , balanceL
              , balanceR
              , delta
              , join
              , merge
              , glue
              , trim
              , trimLookupLo
              , foldlStrict
              , MaybeS(..)
              , filterGt
              , filterLt
              ) where
  
  import Prelude hiding (lookup,map,filter,foldr,foldl,null)
  import qualified Data.Set.Base as Set
  import Data.StrictPair
  import Data.Monoid (Monoid(..))
  import Control.Applicative (Applicative(..), (<$>))
  import Data.Traversable (Traversable(traverse))
  import qualified Data.Foldable as Foldable
  -- import Data.Typeable
  import Control.DeepSeq (NFData(rnf))
  
  #if __GLASGOW_HASKELL__
  import GHC.Exts ( build )
  import Text.Read
  import Data.Data
  #endif
  
  -- Use macros to define strictness of functions.
  -- STRICT_x_OF_y denotes an y-ary function strict in the x-th parameter.
  -- We do not use BangPatterns, because they are not in any standard and we
  -- want the compilers to be compiled by as many compilers as possible.
  #define STRICT_1_OF_2(fn) fn arg _ | arg `seq` False = undefined
  #define STRICT_1_OF_3(fn) fn arg _ _ | arg `seq` False = undefined
  #define STRICT_2_OF_3(fn) fn _ arg _ | arg `seq` False = undefined
  #define STRICT_1_OF_4(fn) fn arg _ _ _ | arg `seq` False = undefined
  #define STRICT_2_OF_4(fn) fn _ arg _ _ | arg `seq` False = undefined
  
  {--------------------------------------------------------------------
    Operators
  --------------------------------------------------------------------}
  infixl 9 !,\\ --
  
  -- | /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
  m ! k = find k m
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE (!) #-}
  #endif
  
  -- | Same as 'difference'.
  (\\) :: Ord k => Map k a -> Map k b -> Map k a
  m1 \\ m2 = difference m1 m2
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE (\\) #-}
  #endif
  
  {--------------------------------------------------------------------
    Size balanced trees.
  --------------------------------------------------------------------}
  -- | A Map from keys @k@ to values @a@.
  
  -- See Note: Order of constructors
  data Map k a  = Bin {-# UNPACK #-} !Size !k a !(Map k a) !(Map k a)
                | Tip
  
  type Size     = Int
  
  instance (Ord k) => Monoid (Map k v) where
      mempty  = empty
      mappend = union
      mconcat = unions
  
  #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 k, Data a, Ord k) => Data (Map k a) where
    gfoldl f z m   = z fromList `f` toList m
    toConstr _     = error "toConstr"
    gunfold _ _    = error "gunfold"
    dataTypeOf _   = mkNoRepType "Data.Map.Map"
    dataCast2 f    = gcast2 f
  #endif
  
  {--------------------------------------------------------------------
    Query
  --------------------------------------------------------------------}
  -- | /O(1)/. Is the map empty?
  --
  -- > Data.Map.null (empty)           == True
  -- > Data.Map.null (singleton 1 'a') == False
  
  null :: Map k a -> Bool
  null Tip      = True
  null (Bin {}) = False
  {-# INLINE null #-}
  
  -- | /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
  size Tip              = 0
  size (Bin sz _ _ _ _) = sz
  {-# INLINE size #-}
  
  
  -- | /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
  lookup = go
    where
      STRICT_1_OF_2(go)
      go _ Tip = Nothing
      go k (Bin _ kx x l r) = case compare k kx of
        LT -> go k l
        GT -> go k r
        EQ -> Just x
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE lookup #-}
  #else
  {-# INLINE lookup #-}
  #endif
  
  -- | /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
  member = go
    where
      STRICT_1_OF_2(go)
      go _ Tip = False
      go k (Bin _ kx _ l r) = case compare k kx of
        LT -> go k l
        GT -> go k r
        EQ -> True
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE member #-}
  #else
  {-# INLINE member #-}
  #endif
  
  -- | /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
  notMember k m = not $ member k m
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE notMember #-}
  #else
  {-# INLINE notMember #-}
  #endif
  
  -- | /O(log n)/. Find the value at a key.
  -- Calls 'error' when the element can not be found.
  find :: Ord k => k -> Map k a -> a
  find = go
    where
      STRICT_1_OF_2(go)
      go _ Tip = error "Map.!: given key is not an element in the map"
      go k (Bin _ kx x l r) = case compare k kx of
        LT -> go k l
        GT -> go k r
        EQ -> x
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE find #-}
  #else
  {-# INLINE find #-}
  #endif
  
  -- | /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
  findWithDefault = go
    where
      STRICT_2_OF_3(go)
      go def _ Tip = def
      go def k (Bin _ kx x l r) = case compare k kx of
        LT -> go def k l
        GT -> go def k r
        EQ -> x
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE findWithDefault #-}
  #else
  {-# INLINE findWithDefault #-}
  #endif
  
  -- | /O(log n)/. Find largest key smaller than the given one and return the
  -- corresponding (key, value) pair.
  --
  -- > lookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing
  -- > lookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
  lookupLT :: Ord k => k -> Map k v -> Maybe (k, v)
  lookupLT = goNothing
    where
      STRICT_1_OF_2(goNothing)
      goNothing _ Tip = Nothing
      goNothing k (Bin _ kx x l r) | k <= kx = goNothing k l
                                   | otherwise = goJust k kx x r
  
      STRICT_1_OF_4(goJust)
      goJust _ kx' x' Tip = Just (kx', x')
      goJust k kx' x' (Bin _ kx x l r) | k <= kx = goJust k kx' x' l
                                       | otherwise = goJust k kx x r
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE lookupLT #-}
  #else
  {-# INLINE lookupLT #-}
  #endif
  
  -- | /O(log n)/. Find smallest key greater than the given one and return the
  -- corresponding (key, value) pair.
  --
  -- > lookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
  -- > lookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing
  lookupGT :: Ord k => k -> Map k v -> Maybe (k, v)
  lookupGT = goNothing
    where
      STRICT_1_OF_2(goNothing)
      goNothing _ Tip = Nothing
      goNothing k (Bin _ kx x l r) | k < kx = goJust k kx x l
                                   | otherwise = goNothing k r
  
      STRICT_1_OF_4(goJust)
      goJust _ kx' x' Tip = Just (kx', x')
      goJust k kx' x' (Bin _ kx x l r) | k < kx = goJust k kx x l
                                       | otherwise = goJust k kx' x' r
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE lookupGT #-}
  #else
  {-# INLINE lookupGT #-}
  #endif
  
  -- | /O(log n)/. Find largest key smaller or equal to the given one and return
  -- the corresponding (key, value) pair.
  --
  -- > lookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing
  -- > lookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
  -- > lookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
  lookupLE :: Ord k => k -> Map k v -> Maybe (k, v)
  lookupLE = goNothing
    where
      STRICT_1_OF_2(goNothing)
      goNothing _ Tip = Nothing
      goNothing k (Bin _ kx x l r) = case compare k kx of LT -> goNothing k l
                                                          EQ -> Just (kx, x)
                                                          GT -> goJust k kx x r
  
      STRICT_1_OF_4(goJust)
      goJust _ kx' x' Tip = Just (kx', x')
      goJust k kx' x' (Bin _ kx x l r) = case compare k kx of LT -> goJust k kx' x' l
                                                              EQ -> Just (kx, x)
                                                              GT -> goJust k kx x r
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE lookupLE #-}
  #else
  {-# INLINE lookupLE #-}
  #endif
  
  -- | /O(log n)/. Find smallest key greater or equal to the given one and return
  -- the corresponding (key, value) pair.
  --
  -- > lookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
  -- > lookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
  -- > lookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing
  lookupGE :: Ord k => k -> Map k v -> Maybe (k, v)
  lookupGE = goNothing
    where
      STRICT_1_OF_2(goNothing)
      goNothing _ Tip = Nothing
      goNothing k (Bin _ kx x l r) = case compare k kx of LT -> goJust k kx x l
                                                          EQ -> Just (kx, x)
                                                          GT -> goNothing k r
  
      STRICT_1_OF_4(goJust)
      goJust _ kx' x' Tip = Just (kx', x')
      goJust k kx' x' (Bin _ kx x l r) = case compare k kx of LT -> goJust k kx x l
                                                              EQ -> Just (kx, x)
                                                              GT -> goJust k kx' x' r
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE lookupGE #-}
  #else
  {-# INLINE lookupGE #-}
  #endif
  
  {--------------------------------------------------------------------
    Construction
  --------------------------------------------------------------------}
  -- | /O(1)/. The empty map.
  --
  -- > empty      == fromList []
  -- > size empty == 0
  
  empty :: Map k a
  empty = Tip
  {-# INLINE empty #-}
  
  -- | /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
  singleton k x = Bin 1 k x Tip Tip
  {-# INLINE singleton #-}
  
  {--------------------------------------------------------------------
    Insertion
  --------------------------------------------------------------------}
  -- | /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'
  
  -- See Note: Type of local 'go' function
  insert :: Ord k => k -> a -> Map k a -> Map k a
  insert = go
    where
      go :: Ord k => k -> a -> Map k a -> Map k a
      STRICT_1_OF_3(go)
      go kx x Tip = singleton kx x
      go kx x (Bin sz ky y l r) =
          case compare kx ky of
              LT -> balanceL ky y (go kx x l) r
              GT -> balanceR ky y l (go kx x r)
              EQ -> Bin sz kx x l r
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE insert #-}
  #else
  {-# INLINE insert #-}
  #endif
  
  -- Insert a new key and value in the map if it is not already present.
  -- Used by `union`.
  
  -- See Note: Type of local 'go' function
  insertR :: Ord k => k -> a -> Map k a -> Map k a
  insertR = go
    where
      go :: Ord k => k -> a -> Map k a -> Map k a
      STRICT_1_OF_3(go)
      go kx x Tip = singleton kx x
      go kx x t@(Bin _ ky y l r) =
          case compare kx ky of
              LT -> balanceL ky y (go kx x l) r
              GT -> balanceR ky y l (go kx x r)
              EQ -> t
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE insertR #-}
  #else
  {-# INLINE insertR #-}
  #endif
  
  -- | /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
  insertWith f = insertWithKey (\_ x' y' -> f x' y')
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE insertWith #-}
  #else
  {-# INLINE insertWith #-}
  #endif
  
  -- | /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"
  
  -- See Note: Type of local 'go' function
  insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
  insertWithKey = go
    where
      go :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
      STRICT_2_OF_4(go)
      go _ kx x Tip = singleton kx x
      go f kx x (Bin sy ky y l r) =
          case compare kx ky of
              LT -> balanceL ky y (go f kx x l) r
              GT -> balanceR ky y l (go f kx x r)
              EQ -> Bin sy kx (f kx x y) l r
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE insertWithKey #-}
  #else
  {-# INLINE insertWithKey #-}
  #endif
  
  -- | /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")])
  
  -- See Note: Type of local 'go' function
  insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a
                      -> (Maybe a, Map k a)
  insertLookupWithKey = go
    where
      go :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)
      STRICT_2_OF_4(go)
      go _ kx x Tip = (Nothing, singleton kx x)
      go f kx x (Bin sy ky y l r) =
          case compare kx ky of
              LT -> let (found, l') = go f kx x l
                    in (found, balanceL ky y l' r)
              GT -> let (found, r') = go f kx x r
                    in (found, balanceR ky y l r')
              EQ -> (Just y, Bin sy kx (f kx x y) l r)
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE insertLookupWithKey #-}
  #else
  {-# INLINE insertLookupWithKey #-}
  #endif
  
  {--------------------------------------------------------------------
    Deletion
  --------------------------------------------------------------------}
  -- | /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
  
  -- See Note: Type of local 'go' function
  delete :: Ord k => k -> Map k a -> Map k a
  delete = go
    where
      go :: Ord k => k -> Map k a -> Map k a
      STRICT_1_OF_2(go)
      go _ Tip = Tip
      go k (Bin _ kx x l r) =
          case compare k kx of
              LT -> balanceR kx x (go k l) r
              GT -> balanceL kx x l (go k r)
              EQ -> glue l r
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE delete #-}
  #else
  {-# INLINE delete #-}
  #endif
  
  -- | /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
  adjust f = adjustWithKey (\_ x -> f x)
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE adjust #-}
  #else
  {-# INLINE adjust #-}
  #endif
  
  -- | /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
  adjustWithKey f = updateWithKey (\k' x' -> Just (f k' x'))
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE adjustWithKey #-}
  #else
  {-# INLINE adjustWithKey #-}
  #endif
  
  -- | /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
  update f = updateWithKey (\_ x -> f x)
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE update #-}
  #else
  {-# INLINE update #-}
  #endif
  
  -- | /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"
  
  -- See Note: Type of local 'go' function
  updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a
  updateWithKey = go
    where
      go :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a
      STRICT_2_OF_3(go)
      go _ _ Tip = Tip
      go f k(Bin sx kx x l r) =
          case compare k kx of
             LT -> balanceR kx x (go f k l) r
             GT -> balanceL kx x l (go f k r)
             EQ -> case f kx x of
                     Just x' -> Bin sx kx x' l r
                     Nothing -> glue l r
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE updateWithKey #-}
  #else
  {-# INLINE updateWithKey #-}
  #endif
  
  -- | /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")
  
  -- See Note: Type of local 'go' function
  updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a,Map k a)
  updateLookupWithKey = go
   where
     go :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a,Map k a)
     STRICT_2_OF_3(go)
     go _ _ Tip = (Nothing,Tip)
     go f k (Bin sx kx x l r) =
            case compare k kx of
                 LT -> let (found,l') = go f k l in (found,balanceR kx x l' r)
                 GT -> let (found,r') = go f k r in (found,balanceL kx x l r')
                 EQ -> case f kx x of
                         Just x' -> (Just x',Bin sx kx x' l r)
                         Nothing -> (Just x,glue l r)
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE updateLookupWithKey #-}
  #else
  {-# INLINE updateLookupWithKey #-}
  #endif
  
  -- | /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")]
  
  -- See Note: Type of local 'go' function
  alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a
  alter = go
    where
      go :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a
      STRICT_2_OF_3(go)
      go f k Tip = case f Nothing of
                 Nothing -> Tip
                 Just x  -> singleton k x
  
      go f k (Bin sx kx x l r) = case compare k kx of
                 LT -> balance kx x (go f k l) r
                 GT -> balance kx x l (go f k r)
                 EQ -> case f (Just x) of
                         Just x' -> Bin sx kx x' l r
                         Nothing -> glue l r
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE alter #-}
  #else
  {-# INLINE alter #-}
  #endif
  
  {--------------------------------------------------------------------
    Indexing
  --------------------------------------------------------------------}
  -- | /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
  
  -- See Note: Type of local 'go' function
  findIndex :: Ord k => k -> Map k a -> Int
  findIndex = go 0
    where
      go :: Ord k => Int -> k -> Map k a -> Int
      STRICT_1_OF_3(go)
      STRICT_2_OF_3(go)
      go _   _ Tip  = error "Map.findIndex: element is not in the map"
      go idx k (Bin _ kx _ l r) = case compare k kx of
        LT -> go idx k l
        GT -> go (idx + size l + 1) k r
        EQ -> idx + size l
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE findIndex #-}
  #endif
  
  -- | /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
  
  -- See Note: Type of local 'go' function
  lookupIndex :: Ord k => k -> Map k a -> Maybe Int
  lookupIndex = go 0
    where
      go :: Ord k => Int -> k -> Map k a -> Maybe Int
      STRICT_1_OF_3(go)
      STRICT_2_OF_3(go)
      go _   _ Tip  = Nothing
      go idx k (Bin _ kx _ l r) = case compare k kx of
        LT -> go idx k l
        GT -> go (idx + size l + 1) k r
        EQ -> Just $! idx + size l
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE lookupIndex #-}
  #endif
  
  -- | /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)
  STRICT_1_OF_2(elemAt)
  elemAt _ Tip = error "Map.elemAt: index out of range"
  elemAt i (Bin _ kx x l r)
    = case compare i sizeL of
        LT -> elemAt i l
        GT -> elemAt (i-sizeL-1) r
        EQ -> (kx,x)
    where
      sizeL = size l
  
  -- | /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
  updateAt f i t = i `seq`
    case t of
      Tip -> error "Map.updateAt: index out of range"
      Bin sx kx x l r -> case compare i sizeL of
        LT -> balanceR kx x (updateAt f i l) r
        GT -> balanceL kx x l (updateAt f (i-sizeL-1) r)
        EQ -> case f kx x of
                Just x' -> Bin sx kx x' l r
                Nothing -> glue l r
        where
          sizeL = size l
  
  -- | /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
  deleteAt i t = i `seq`
    case t of
      Tip -> error "Map.deleteAt: index out of range"
      Bin _ kx x l r -> case compare i sizeL of
        LT -> balanceR kx x (deleteAt i l) r
        GT -> balanceL kx x l (deleteAt (i-sizeL-1) r)
        EQ -> glue l r
        where
          sizeL = size l
  
  
  {--------------------------------------------------------------------
    Minimal, Maximal
  --------------------------------------------------------------------}
  -- | /O(log n)/. The minimal key of the map. Calls 'error' if 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)
  findMin (Bin _ kx x Tip _)  = (kx,x)
  findMin (Bin _ _  _ l _)    = findMin l
  findMin Tip                 = error "Map.findMin: empty map has no minimal element"
  
  -- | /O(log n)/. The maximal key of the map. Calls 'error' if 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)
  findMax (Bin _ kx x _ Tip)  = (kx,x)
  findMax (Bin _ _  _ _ r)    = findMax r
  findMax Tip                 = error "Map.findMax: empty map has no maximal element"
  
  -- | /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
  deleteMin (Bin _ _  _ Tip r)  = r
  deleteMin (Bin _ kx x l r)    = balanceR kx x (deleteMin l) r
  deleteMin Tip                 = Tip
  
  -- | /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
  deleteMax (Bin _ _  _ l Tip)  = l
  deleteMax (Bin _ kx x l r)    = balanceL kx x l (deleteMax r)
  deleteMax Tip                 = Tip
  
  -- | /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
  updateMin f m
    = updateMinWithKey (\_ x -> f x) m
  
  -- | /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
  updateMax f m
    = updateMaxWithKey (\_ x -> f x) m
  
  
  -- | /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
  updateMinWithKey _ Tip                 = Tip
  updateMinWithKey f (Bin sx kx x Tip r) = case f kx x of
                                             Nothing -> r
                                             Just x' -> Bin sx kx x' Tip r
  updateMinWithKey f (Bin _ kx x l r)    = balanceR kx x (updateMinWithKey f l) r
  
  -- | /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
  updateMaxWithKey _ Tip                 = Tip
  updateMaxWithKey f (Bin sx kx x l Tip) = case f kx x of
                                             Nothing -> l
                                             Just x' -> Bin sx kx x' l Tip
  updateMaxWithKey f (Bin _ kx x l r)    = balanceL kx x l (updateMaxWithKey f r)
  
  -- | /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)
  minViewWithKey Tip = Nothing
  minViewWithKey x   = Just (deleteFindMin x)
  
  -- | /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)
  maxViewWithKey Tip = Nothing
  maxViewWithKey x   = Just (deleteFindMax x)
  
  -- | /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)
  minView Tip = Nothing
  minView x   = Just (first snd $ deleteFindMin x)
  
  -- | /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)
  maxView Tip = Nothing
  maxView x   = Just (first snd $ deleteFindMax x)
  
  -- Update the 1st component of a tuple (special case of Control.Arrow.first)
  first :: (a -> b) -> (a,c) -> (b,c)
  first f (x,y) = (f x, y)
  
  {--------------------------------------------------------------------
    Union.
  --------------------------------------------------------------------}
  -- | The union of a list of maps:
  --   (@'unions' == 'Prelude.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
  unions ts
    = foldlStrict union empty ts
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE unions #-}
  #endif
  
  -- | The union of a list of maps, with a combining operation:
  --   (@'unionsWith' f == 'Prelude.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
  unionsWith f ts
    = foldlStrict (unionWith f) empty ts
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE unionsWith #-}
  #endif
  
  -- | /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
  union Tip t2  = t2
  union t1 Tip  = t1
  union t1 t2 = hedgeUnion NothingS NothingS t1 t2
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE union #-}
  #endif
  
  -- left-biased hedge union
  hedgeUnion :: Ord a => MaybeS a -> MaybeS a -> Map a b -> Map a b -> Map a b
  hedgeUnion _   _   t1  Tip = t1
  hedgeUnion blo bhi Tip (Bin _ kx x l r) = join kx x (filterGt blo l) (filterLt bhi r)
  hedgeUnion _   _   t1  (Bin _ kx x Tip Tip) = insertR kx x t1  -- According to benchmarks, this special case increases
                                                                -- performance up to 30%. It does not help in difference or intersection.
  hedgeUnion blo bhi (Bin _ kx x l r) t2 = join kx x (hedgeUnion blo bmi l (trim blo bmi t2))
                                                     (hedgeUnion bmi bhi r (trim bmi bhi t2))
    where bmi = JustS kx
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE hedgeUnion #-}
  #endif
  
  {--------------------------------------------------------------------
    Union with a combining function
  --------------------------------------------------------------------}
  -- | /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
  unionWith f m1 m2
    = unionWithKey (\_ x y -> f x y) m1 m2
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE unionWith #-}
  #endif
  
  -- | /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
  unionWithKey f t1 t2 = mergeWithKey (\k x1 x2 -> Just $ f k x1 x2) id id t1 t2
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE unionWithKey #-}
  #endif
  
  {--------------------------------------------------------------------
    Difference
  --------------------------------------------------------------------}
  -- | /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
  difference Tip _   = Tip
  difference t1 Tip  = t1
  difference t1 t2   = hedgeDiff NothingS NothingS t1 t2
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE difference #-}
  #endif
  
  hedgeDiff :: Ord a => MaybeS a -> MaybeS a -> Map a b -> Map a c -> Map a b
  hedgeDiff _   _   Tip              _ = Tip
  hedgeDiff blo bhi (Bin _ kx x l r) Tip = join kx x (filterGt blo l) (filterLt bhi r)
  hedgeDiff blo bhi t (Bin _ kx _ l r) = merge (hedgeDiff blo bmi (trim blo bmi t) l)
                                               (hedgeDiff bmi bhi (trim bmi bhi t) r)
    where bmi = JustS kx
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE hedgeDiff #-}
  #endif
  
  -- | /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
  differenceWith f m1 m2
    = differenceWithKey (\_ x y -> f x y) m1 m2
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE differenceWith #-}
  #endif
  
  -- | /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
  differenceWithKey f t1 t2 = mergeWithKey f id (const Tip) t1 t2
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE differenceWithKey #-}
  #endif
  
  
  {--------------------------------------------------------------------
    Intersection
  --------------------------------------------------------------------}
  -- | /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
  intersection Tip _ = Tip
  intersection _ Tip = Tip
  intersection t1 t2 = hedgeInt NothingS NothingS t1 t2
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE intersection #-}
  #endif
  
  hedgeInt :: Ord k => MaybeS k -> MaybeS k -> Map k a -> Map k b -> Map k a
  hedgeInt _ _ _   Tip = Tip
  hedgeInt _ _ Tip _   = Tip
  hedgeInt blo bhi (Bin _ kx x l r) t2 = let l' = hedgeInt blo bmi l (trim blo bmi t2)
                                             r' = hedgeInt bmi bhi r (trim bmi bhi t2)
                                         in if kx `member` t2 then join kx x l' r' else merge l' r'
    where bmi = JustS kx
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE hedgeInt #-}
  #endif
  
  -- | /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
  intersectionWith f m1 m2
    = intersectionWithKey (\_ x y -> f x y) m1 m2
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE intersectionWith #-}
  #endif
  
  -- | /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
  intersectionWithKey f t1 t2 = mergeWithKey (\k x1 x2 -> Just $ f k x1 x2) (const Tip) (const Tip) t1 t2
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE intersectionWithKey #-}
  #endif
  
  
  {--------------------------------------------------------------------
    MergeWithKey
  --------------------------------------------------------------------}
  
  -- | /O(n+m)/. A high-performance universal combining function. This function
  -- is used to define 'unionWith', 'unionWithKey', 'differenceWith',
  -- 'differenceWithKey', 'intersectionWith', 'intersectionWithKey' and can be
  -- used to define other custom combine functions.
  --
  -- Please make sure you know what is going on when using 'mergeWithKey',
  -- otherwise you can be surprised by unexpected code growth or even
  -- corruption of the data structure.
  --
  -- When 'mergeWithKey' is given three arguments, it is inlined to the call
  -- site. You should therefore use 'mergeWithKey' only to define your custom
  -- combining functions. For example, you could define 'unionWithKey',
  -- 'differenceWithKey' and 'intersectionWithKey' as
  --
  -- > myUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) id id m1 m2
  -- > myDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2
  -- > myIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) (const empty) (const empty) m1 m2
  --
  -- When calling @'mergeWithKey' combine only1 only2@, a function combining two
  -- 'IntMap's is created, such that
  --
  -- * if a key is present in both maps, it is passed with both corresponding
  --   values to the @combine@ function. Depending on the result, the key is either
  --   present in the result with specified value, or is left out;
  --
  -- * a nonempty subtree present only in the first map is passed to @only1@ and
  --   the output is added to the result;
  --
  -- * a nonempty subtree present only in the second map is passed to @only2@ and
  --   the output is added to the result.
  --
  -- The @only1@ and @only2@ methods /must return a map with a subset (possibly empty) of the keys of the given map/.
  -- The values can be modified arbitrarily. Most common variants of @only1@ and
  -- @only2@ are 'id' and @'const' 'empty'@, but for example @'map' f@ or
  -- @'filterWithKey' f@ could be used for any @f@.
  
  mergeWithKey :: Ord k => (k -> a -> b -> Maybe c) -> (Map k a -> Map k c) -> (Map k b -> Map k c)
               -> Map k a -> Map k b -> Map k c
  mergeWithKey f g1 g2 = go
    where
      go Tip t2 = g2 t2
      go t1 Tip = g1 t1
      go t1 t2 = hedgeMerge NothingS NothingS t1 t2
  
      hedgeMerge _   _   t1  Tip = g1 t1
      hedgeMerge blo bhi Tip (Bin _ kx x l r) = g2 $ join kx x (filterGt blo l) (filterLt bhi r)
      hedgeMerge blo bhi (Bin _ kx x l r) t2 = let l' = hedgeMerge blo bmi l (trim blo bmi t2)
                                                   (found, trim_t2) = trimLookupLo kx bhi t2
                                                   r' = hedgeMerge bmi bhi r trim_t2
                                               in case found of
                                                    Nothing -> case g1 (singleton kx x) of
                                                                 Tip -> merge l' r'
                                                                 (Bin _ _ x' Tip Tip) -> join kx x' l' r'
                                                                 _ -> error "mergeWithKey: Given function only1 does not fulfil required conditions (see documentation)"
                                                    Just x2 -> case f kx x x2 of
                                                                 Nothing -> merge l' r'
                                                                 Just x' -> join kx x' l' r'
        where bmi = JustS kx
  {-# INLINE mergeWithKey #-}
  
  {--------------------------------------------------------------------
    Submap
  --------------------------------------------------------------------}
  -- | /O(n+m)/.
  -- This function is defined as (@'isSubmapOf' = 'isSubmapOfBy' (==)@).
  --
  isSubmapOf :: (Ord k,Eq a) => Map k a -> Map k a -> Bool
  isSubmapOf m1 m2 = isSubmapOfBy (==) m1 m2
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE isSubmapOf #-}
  #endif
  
  {- | /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
  isSubmapOfBy f t1 t2
    = (size t1 <= size t2) && (submap' f t1 t2)
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE isSubmapOfBy #-}
  #endif
  
  submap' :: Ord a => (b -> c -> Bool) -> Map a b -> Map a c -> Bool
  submap' _ Tip _ = True
  submap' _ _ Tip = False
  submap' f (Bin _ kx x l r) t
    = case found of
        Nothing -> False
        Just y  -> f x y && submap' f l lt && submap' f r gt
    where
      (lt,found,gt) = splitLookup kx t
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE submap' #-}
  #endif
  
  -- | /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
  isProperSubmapOf m1 m2
    = isProperSubmapOfBy (==) m1 m2
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE isProperSubmapOf #-}
  #endif
  
  {- | /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
  isProperSubmapOfBy f t1 t2
    = (size t1 < size t2) && (submap' f t1 t2)
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE isProperSubmapOfBy #-}
  #endif
  
  {--------------------------------------------------------------------
    Filter and partition
  --------------------------------------------------------------------}
  -- | /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 :: (a -> Bool) -> Map k a -> Map k a
  filter p m
    = filterWithKey (\_ x -> p x) m
  
  -- | /O(n)/. Filter all keys\/values that satisfy the predicate.
  --
  -- > filterWithKey (\k _ -> k > 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
  
  filterWithKey :: (k -> a -> Bool) -> Map k a -> Map k a
  filterWithKey _ Tip = Tip
  filterWithKey p (Bin _ kx x l r)
    | p kx x    = join kx x (filterWithKey p l) (filterWithKey p r)
    | otherwise = merge (filterWithKey p l) (filterWithKey p r)
  
  -- | /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 :: (a -> Bool) -> Map k a -> (Map k a,Map k a)
  partition p m
    = partitionWithKey (\_ x -> p x) m
  
  -- | /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 :: (k -> a -> Bool) -> Map k a -> (Map k a,Map k a)
  partitionWithKey _ Tip = (Tip,Tip)
  partitionWithKey p (Bin _ kx x l r)
    | p kx x    = (join kx x l1 r1,merge l2 r2)
    | otherwise = (merge l1 r1,join kx x l2 r2)
    where
      (l1,l2) = partitionWithKey p l
      (r1,r2) = partitionWithKey p r
  
  -- | /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) -> Map k a -> Map k b
  mapMaybe f = mapMaybeWithKey (\_ x -> f x)
  
  -- | /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 :: (k -> a -> Maybe b) -> Map k a -> Map k b
  mapMaybeWithKey _ Tip = Tip
  mapMaybeWithKey f (Bin _ kx x l r) = case f kx x of
    Just y  -> join kx y (mapMaybeWithKey f l) (mapMaybeWithKey f r)
    Nothing -> merge (mapMaybeWithKey f l) (mapMaybeWithKey f r)
  
  -- | /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) -> Map k a -> (Map k b, Map k 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 :: (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c)
  mapEitherWithKey _ Tip = (Tip, Tip)
  mapEitherWithKey f (Bin _ kx x l r) = case f kx x of
    Left y  -> (join kx y l1 r1, merge l2 r2)
    Right z -> (merge l1 r1, join kx z l2 r2)
   where
      (l1,l2) = mapEitherWithKey f l
      (r1,r2) = mapEitherWithKey f r
  
  {--------------------------------------------------------------------
    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) -> Map k a -> Map k b
  map _ Tip = Tip
  map f (Bin sx kx x l r) = Bin sx kx (f x) (map f l) (map f r)
  
  -- | /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
  mapWithKey _ Tip = Tip
  mapWithKey f (Bin sx kx x l r) = Bin sx kx (f kx x) (mapWithKey f l) (mapWithKey f r)
  
  -- | /O(n)/.
  -- @'traverseWithKey' f s == 'fromList' <$> 'traverse' (\(k, v) -> (,) k <$> f k v) ('toList' m)@
  -- That is, behaves exactly like a regular 'traverse' except that the traversing
  -- function also has access to the key associated with a value.
  --
  -- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
  -- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing
  {-# INLINE traverseWithKey #-}
  traverseWithKey :: Applicative t => (k -> a -> t b) -> Map k a -> t (Map k b)
  traverseWithKey f = go
    where
      go Tip = pure Tip
      go (Bin s k v l r)
        = flip (Bin s k) <$> go l <*> f k v <*> go r
  
  -- | /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)
  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 -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k 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 -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
  mapAccumL _ a Tip               = (a,Tip)
  mapAccumL f a (Bin sx kx x l r) =
    let (a1,l') = mapAccumL f a l
        (a2,x') = f a1 kx x
        (a3,r') = mapAccumL f a2 r
    in (a3,Bin sx kx x' l' r')
  
  -- | /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)
  mapAccumRWithKey _ a Tip = (a,Tip)
  mapAccumRWithKey f a (Bin sx kx x l r) =
    let (a1,r') = mapAccumRWithKey f a r
        (a2,x') = f a1 kx x
        (a3,l') = mapAccumRWithKey f a2 l
    in (a3,Bin sx kx x' l' r')
  
  -- | /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 greatest of the
  -- original keys is retained.
  --
  -- > mapKeys (+ 1) (fromList [(5,"a"), (3,"b")])                        == fromList [(4, "b"), (6, "a")]
  -- > mapKeys (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
  -- > mapKeys (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"
  
  mapKeys :: Ord k2 => (k1->k2) -> Map k1 a -> Map k2 a
  mapKeys f = fromList . foldrWithKey (\k x xs -> (f k, x) : xs) []
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE mapKeys #-}
  #endif
  
  -- | /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
  mapKeysWith c f = fromListWith c . foldrWithKey (\k x xs -> (f k, x) : xs) []
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE mapKeysWith #-}
  #endif
  
  
  -- | /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
  mapKeysMonotonic _ Tip = Tip
  mapKeysMonotonic f (Bin sz k x l r) =
      Bin sz (f k) x (mapKeysMonotonic f l) (mapKeysMonotonic f r)
  
  {--------------------------------------------------------------------
    Folds
  --------------------------------------------------------------------}
  
  -- | /O(n)/. Fold the values in the map using the given right-associative
  -- binary operator, such that @'foldr' f z == 'Prelude.foldr' f z . 'elems'@.
  --
  -- For example,
  --
  -- > elems map = foldr (:) [] map
  --
  -- > let f a len = len + (length a)
  -- > foldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
  foldr :: (a -> b -> b) -> b -> Map k a -> b
  foldr f z = go z
    where
      go z' Tip             = z'
      go z' (Bin _ _ x l r) = go (f x (go z' r)) l
  {-# INLINE foldr #-}
  
  -- | /O(n)/. A strict version of 'foldr'. Each application of the operator is
  -- evaluated before using the result in the next application. This
  -- function is strict in the starting value.
  foldr' :: (a -> b -> b) -> b -> Map k a -> b
  foldr' f z = go z
    where
      STRICT_1_OF_2(go)
      go z' Tip             = z'
      go z' (Bin _ _ x l r) = go (f x (go z' r)) l
  {-# INLINE foldr' #-}
  
  -- | /O(n)/. Fold the values in the map using the given left-associative
  -- binary operator, such that @'foldl' f z == 'Prelude.foldl' f z . 'elems'@.
  --
  -- For example,
  --
  -- > elems = reverse . foldl (flip (:)) []
  --
  -- > let f len a = len + (length a)
  -- > foldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
  foldl :: (a -> b -> a) -> a -> Map k b -> a
  foldl f z = go z
    where
      go z' Tip             = z'
      go z' (Bin _ _ x l r) = go (f (go z' l) x) r
  {-# INLINE foldl #-}
  
  -- | /O(n)/. A strict version of 'foldl'. Each application of the operator is
  -- evaluated before using the result in the next application. This
  -- function is strict in the starting value.
  foldl' :: (a -> b -> a) -> a -> Map k b -> a
  foldl' f z = go z
    where
      STRICT_1_OF_2(go)
      go z' Tip             = z'
      go z' (Bin _ _ x l r) = go (f (go z' l) x) r
  {-# INLINE foldl' #-}
  
  -- | /O(n)/. Fold the keys and values in the map using the given right-associative
  -- binary operator, such that
  -- @'foldrWithKey' f z == 'Prelude.foldr' ('uncurry' f) z . 'toAscList'@.
  --
  -- For example,
  --
  -- > keys map = foldrWithKey (\k x ks -> k:ks) [] map
  --
  -- > let f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
  -- > foldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)"
  foldrWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b
  foldrWithKey f z = go z
    where
      go z' Tip             = z'
      go z' (Bin _ kx x l r) = go (f kx x (go z' r)) l
  {-# INLINE foldrWithKey #-}
  
  -- | /O(n)/. A strict version of 'foldrWithKey'. Each application of the operator is
  -- evaluated before using the result in the next application. This
  -- function is strict in the starting value.
  foldrWithKey' :: (k -> a -> b -> b) -> b -> Map k a -> b
  foldrWithKey' f z = go z
    where
      STRICT_1_OF_2(go)
      go z' Tip              = z'
      go z' (Bin _ kx x l r) = go (f kx x (go z' r)) l
  {-# INLINE foldrWithKey' #-}
  
  -- | /O(n)/. Fold the keys and values in the map using the given left-associative
  -- binary operator, such that
  -- @'foldlWithKey' f z == 'Prelude.foldl' (\\z' (kx, x) -> f z' kx x) z . 'toAscList'@.
  --
  -- For example,
  --
  -- > keys = reverse . foldlWithKey (\ks k x -> k:ks) []
  --
  -- > let f result k a = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
  -- > foldlWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (3:b)(5:a)"
  foldlWithKey :: (a -> k -> b -> a) -> a -> Map k b -> a
  foldlWithKey f z = go z
    where
      go z' Tip              = z'
      go z' (Bin _ kx x l r) = go (f (go z' l) kx x) r
  {-# INLINE foldlWithKey #-}
  
  -- | /O(n)/. A strict version of 'foldlWithKey'. Each application of the operator is
  -- evaluated before using the result in the next application. This
  -- function is strict in the starting value.
  foldlWithKey' :: (a -> k -> b -> a) -> a -> Map k b -> a
  foldlWithKey' f z = go z
    where
      STRICT_1_OF_2(go)
      go z' Tip              = z'
      go z' (Bin _ kx x l r) = go (f (go z' l) kx x) r
  {-# INLINE foldlWithKey' #-}
  
  {--------------------------------------------------------------------
    List variations
  --------------------------------------------------------------------}
  -- | /O(n)/.
  -- Return all elements of the map in the ascending order of their keys.
  -- Subject to list fusion.
  --
  -- > elems (fromList [(5,"a"), (3,"b")]) == ["b","a"]
  -- > elems empty == []
  
  elems :: Map k a -> [a]
  elems = foldr (:) []
  
  -- | /O(n)/. Return all keys of the map in ascending order. Subject to list
  -- fusion.
  --
  -- > keys (fromList [(5,"a"), (3,"b")]) == [3,5]
  -- > keys empty == []
  
  keys  :: Map k a -> [k]
  keys = foldrWithKey (\k _ ks -> k : ks) []
  
  -- | /O(n)/. An alias for 'toAscList'. Return all key\/value pairs in the map
  -- in ascending key order. Subject to list fusion.
  --
  -- > assocs (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
  -- > assocs empty == []
  
  assocs :: Map k a -> [(k,a)]
  assocs m
    = toAscList m
  
  -- | /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.Set k
  keysSet Tip = Set.Tip
  keysSet (Bin sz kx _ l r) = Set.Bin sz kx (keysSet l) (keysSet r)
  
  -- | /O(n)/. Build a map from a set of keys and a function which for each key
  -- computes its value.
  --
  -- > fromSet (\k -> replicate k 'a') (Data.Set.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")]
  -- > fromSet undefined Data.Set.empty == empty
  fromSet :: (k -> a) -> Set.Set k -> Map k a
  fromSet _ Set.Tip = Tip
  fromSet f (Set.Bin sz x l r) = Bin sz x (f x) (fromSet f l) (fromSet f r)
  
  {--------------------------------------------------------------------
    Lists
    use [foldlStrict] to reduce demand on the control-stack
  --------------------------------------------------------------------}
  -- | /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
  fromList xs
    = foldlStrict ins empty xs
    where
      ins t (k,x) = insert k x t
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE fromList #-}
  #endif
  
  -- | /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
  fromListWith f xs
    = fromListWithKey (\_ x y -> f x y) xs
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE fromListWith #-}
  #endif
  
  -- | /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
  fromListWithKey f xs
    = foldlStrict ins empty xs
    where
      ins t (k,x) = insertWithKey f k x t
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE fromListWithKey #-}
  #endif
  
  -- | /O(n)/. Convert the map to a list of key\/value pairs. Subject to list fusion.
  --
  -- > toList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
  -- > toList empty == []
  
  toList :: Map k a -> [(k,a)]
  toList = toAscList
  
  -- | /O(n)/. Convert the map to a list of key\/value pairs where the keys are
  -- in ascending order. Subject to list fusion.
  --
  -- > toAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
  
  toAscList :: Map k a -> [(k,a)]
  toAscList = foldrWithKey (\k x xs -> (k,x):xs) []
  
  -- | /O(n)/. Convert the map to a list of key\/value pairs where the keys
  -- are in descending order. Subject to list fusion.
  --
  -- > toDescList (fromList [(5,"a"), (3,"b")]) == [(5,"a"), (3,"b")]
  
  toDescList :: Map k a -> [(k,a)]
  toDescList = foldlWithKey (\xs k x -> (k,x):xs) []
  
  -- List fusion for the list generating functions.
  #if __GLASGOW_HASKELL__
  -- The foldrFB and foldlFB are fold{r,l}WithKey equivalents, used for list fusion.
  -- They are important to convert unfused methods back, see mapFB in prelude.
  foldrFB :: (k -> a -> b -> b) -> b -> Map k a -> b
  foldrFB = foldrWithKey
  {-# INLINE[0] foldrFB #-}
  foldlFB :: (a -> k -> b -> a) -> a -> Map k b -> a
  foldlFB = foldlWithKey
  {-# INLINE[0] foldlFB #-}
  
  -- Inline assocs and toList, so that we need to fuse only toAscList.
  {-# INLINE assocs #-}
  {-# INLINE toList #-}
  
  -- The fusion is enabled up to phase 2 included. If it does not succeed,
  -- convert in phase 1 the expanded elems,keys,to{Asc,Desc}List calls back to
  -- elems,keys,to{Asc,Desc}List.  In phase 0, we inline fold{lr}FB (which were
  -- used in a list fusion, otherwise it would go away in phase 1), and let compiler
  -- do whatever it wants with elems,keys,to{Asc,Desc}List -- it was forbidden to
  -- inline it before phase 0, otherwise the fusion rules would not fire at all.
  {-# NOINLINE[0] elems #-}
  {-# NOINLINE[0] keys #-}
  {-# NOINLINE[0] toAscList #-}
  {-# NOINLINE[0] toDescList #-}
  {-# RULES "Map.elems" [~1] forall m . elems m = build (\c n -> foldrFB (\_ x xs -> c x xs) n m) #-}
  {-# RULES "Map.elemsBack" [1] foldrFB (\_ x xs -> x : xs) [] = elems #-}
  {-# RULES "Map.keys" [~1] forall m . keys m = build (\c n -> foldrFB (\k _ xs -> c k xs) n m) #-}
  {-# RULES "Map.keysBack" [1] foldrFB (\k _ xs -> k : xs) [] = keys #-}
  {-# RULES "Map.toAscList" [~1] forall m . toAscList m = build (\c n -> foldrFB (\k x xs -> c (k,x) xs) n m) #-}
  {-# RULES "Map.toAscListBack" [1] foldrFB (\k x xs -> (k, x) : xs) [] = toAscList #-}
  {-# RULES "Map.toDescList" [~1] forall m . toDescList m = build (\c n -> foldlFB (\xs k x -> c (k,x) xs) n m) #-}
  {-# RULES "Map.toDescListBack" [1] foldlFB (\xs k x -> (k, x) : xs) [] = toDescList #-}
  #endif
  
  {--------------------------------------------------------------------
    Building trees from ascending/descending lists can be done in linear time.
  
    Note that if [xs] is ascending that:
      fromAscList xs       == fromList xs
      fromAscListWith f xs == fromListWith f xs
  --------------------------------------------------------------------}
  -- | /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
  fromAscList xs
    = fromAscListWithKey (\_ x _ -> x) xs
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE fromAscList #-}
  #endif
  
  -- | /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
  fromAscListWith f xs
    = fromAscListWithKey (\_ x y -> f x y) xs
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE fromAscListWith #-}
  #endif
  
  -- | /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
  fromAscListWithKey f xs
    = fromDistinctAscList (combineEq f xs)
    where
    -- [combineEq f xs] combines equal elements with function [f] in an ordered list [xs]
    combineEq _ xs'
      = case xs' of
          []     -> []
          [x]    -> [x]
          (x:xx) -> combineEq' x xx
  
    combineEq' z [] = [z]
    combineEq' z@(kz,zz) (x@(kx,xx):xs')
      | kx==kz    = let yy = f kx xx zz in combineEq' (kx,yy) xs'
      | otherwise = z:combineEq' x xs'
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE fromAscListWithKey #-}
  #endif
  
  
  -- | /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
  fromDistinctAscList xs
    = create const (length xs) xs
    where
      -- 1) use continuations so that we use heap space instead of stack space.
      -- 2) special case for n==5 to create bushier trees.
      create c 0 xs' = c Tip xs'
      create c 5 xs' = case xs' of
                         ((k1,x1):(k2,x2):(k3,x3):(k4,x4):(k5,x5):xx)
                              -> c (bin k4 x4 (bin k2 x2 (singleton k1 x1) (singleton k3 x3)) (singleton k5 x5)) xx
                         _ -> error "fromDistinctAscList create"
      create c n xs' = seq nr $ create (createR nr c) nl xs'
        where nl = n `div` 2
              nr = n - nl - 1
  
      createR n c l ((k,x):ys) = create (createB l k x c) n ys
      createR _ _ _ []         = error "fromDistinctAscList createR []"
      createB l k x c r zs     = c (bin k x l r) zs
  
  
  {--------------------------------------------------------------------
    Utility functions that return sub-ranges of the original
    tree. Some functions take a `Maybe value` as an argument to
    allow comparisons against infinite values. These are called `blow`
    (Nothing is -\infty) and `bhigh` (here Nothing is +\infty).
    We use MaybeS value, which is a Maybe strict in the Just case.
  
    [trim blow bhigh t]   A tree that is either empty or where [x > blow]
                          and [x < bhigh] for the value [x] of the root.
    [filterGt blow t]     A tree where for all values [k]. [k > blow]
    [filterLt bhigh t]    A tree where for all values [k]. [k < bhigh]
  
    [split k t]           Returns two trees [l] and [r] where all keys
                          in [l] are <[k] and all keys in [r] are >[k].
    [splitLookup k t]     Just like [split] but also returns whether [k]
                          was found in the tree.
  --------------------------------------------------------------------}
  
  data MaybeS a = NothingS | JustS !a
  
  {--------------------------------------------------------------------
    [trim blo bhi t] trims away all subtrees that surely contain no
    values between the range [blo] to [bhi]. The returned tree is either
    empty or the key of the root is between @blo@ and @bhi@.
  --------------------------------------------------------------------}
  trim :: Ord k => MaybeS k -> MaybeS k -> Map k a -> Map k a
  trim NothingS   NothingS   t = t
  trim (JustS lk) NothingS   t = greater lk t where greater lo (Bin _ k _ _ r) | k <= lo = greater lo r
                                                    greater _  t' = t'
  trim NothingS   (JustS hk) t = lesser hk t  where lesser  hi (Bin _ k _ l _) | k >= hi = lesser  hi l
                                                    lesser  _  t' = t'
  trim (JustS lk) (JustS hk) t = middle lk hk t  where middle lo hi (Bin _ k _ _ r) | k <= lo = middle lo hi r
                                                       middle lo hi (Bin _ k _ l _) | k >= hi = middle lo hi l
                                                       middle _  _  t' = t'
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE trim #-}
  #endif
  
  -- Helper function for 'mergeWithKey'. The @'trimLookupLo' lk hk t@ performs both
  -- @'trim' (JustS lk) hk t@ and @'lookup' lk t@.
  
  -- See Note: Type of local 'go' function
  trimLookupLo :: Ord k => k -> MaybeS k -> Map k a -> (Maybe a, Map k a)
  trimLookupLo lk NothingS t = greater lk t
    where greater :: Ord k => k -> Map k a -> (Maybe a, Map k a)
          greater lo t'@(Bin _ kx x l r) = case compare lo kx of LT -> (lookup lo l, {-`strictPair`-} t')
                                                                 EQ -> (Just x, r)
                                                                 GT -> greater lo r
          greater _ Tip = (Nothing, Tip)
  trimLookupLo lk (JustS hk) t = middle lk hk t
    where middle :: Ord k => k -> k -> Map k a -> (Maybe a, Map k a)
          middle lo hi t'@(Bin _ kx x l r) = case compare lo kx of LT | kx < hi -> (lookup lo l, {- `strictPair` -} t')
                                                                      | otherwise -> middle lo hi l
                                                                   EQ -> (Just x, {-`strictPair`-} lesser hi r)
                                                                   GT -> middle lo hi r
          middle _ _ Tip = (Nothing, Tip)
  
          lesser :: Ord k => k -> Map k a -> Map k a
          lesser hi (Bin _ k _ l _) | k >= hi = lesser hi l
          lesser _ t' = t'
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE trimLookupLo #-}
  #endif
  
  
  {--------------------------------------------------------------------
    [filterGt b t] filter all keys >[b] from tree [t]
    [filterLt b t] filter all keys <[b] from tree [t]
  --------------------------------------------------------------------}
  filterGt :: Ord k => MaybeS k -> Map k v -> Map k v
  filterGt NothingS t = t
  filterGt (JustS b) t = filter' b t
    where filter' _   Tip = Tip
          filter' b' (Bin _ kx x l r) =
            case compare b' kx of LT -> join kx x (filter' b' l) r
                                  EQ -> r
                                  GT -> filter' b' r
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE filterGt #-}
  #endif
  
  filterLt :: Ord k => MaybeS k -> Map k v -> Map k v
  filterLt NothingS t = t
  filterLt (JustS b) t = filter' b t
    where filter' _   Tip = Tip
          filter' b' (Bin _ kx x l r) =
            case compare kx b' of LT -> join kx x l (filter' b' r)
                                  EQ -> l
                                  GT -> filter' b' l
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE filterLt #-}
  #endif
  
  {--------------------------------------------------------------------
    Split
  --------------------------------------------------------------------}
  -- | /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)
  split k t = k `seq`
    case t of
      Tip            -> (Tip, Tip)
      Bin _ kx x l r -> case compare k kx of
        LT -> let (lt,gt) = split k l in (lt,join kx x gt r)
        GT -> let (lt,gt) = split k r in (join kx x l lt,gt)
        EQ -> (l,r)
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE split #-}
  #endif
  
  -- | /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)
  splitLookup k t = k `seq`
    case t of
      Tip            -> (Tip,Nothing,Tip)
      Bin _ kx x l r -> case compare k kx of
        LT -> let (lt,z,gt) = splitLookup k l in (lt,z,join kx x gt r)
        GT -> let (lt,z,gt) = splitLookup k r in (join kx x l lt,z,gt)
        EQ -> (l,Just x,r)
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE splitLookup #-}
  #endif
  
  {--------------------------------------------------------------------
    Utility functions that maintain the balance properties of the tree.
    All constructors assume that all values in [l] < [k] and all values
    in [r] > [k], and that [l] and [r] are valid trees.
  
    In order of sophistication:
      [Bin sz k x l r]  The type constructor.
      [bin k x l r]     Maintains the correct size, assumes that both [l]
                        and [r] are balanced with respect to each other.
      [balance k x l r] Restores the balance and size.
                        Assumes that the original tree was balanced and
                        that [l] or [r] has changed by at most one element.
      [join k x l r]    Restores balance and size.
  
    Furthermore, we can construct a new tree from two trees. Both operations
    assume that all values in [l] < all values in [r] and that [l] and [r]
    are valid:
      [glue l r]        Glues [l] and [r] together. Assumes that [l] and
                        [r] are already balanced with respect to each other.
      [merge l r]       Merges two trees and restores balance.
  
    Note: in contrast to Adam's paper, we use (<=) comparisons instead
    of (<) comparisons in [join], [merge] and [balance].
    Quickcheck (on [difference]) showed that this was necessary in order
    to maintain the invariants. It is quite unsatisfactory that I haven't
    been able to find out why this is actually the case! Fortunately, it
    doesn't hurt to be a bit more conservative.
  --------------------------------------------------------------------}
  
  {--------------------------------------------------------------------
    Join
  --------------------------------------------------------------------}
  join :: k -> a -> Map k a -> Map k a -> Map k a
  join kx x Tip r  = insertMin kx x r
  join kx x l Tip  = insertMax kx x l
  join kx x l@(Bin sizeL ky y ly ry) r@(Bin sizeR kz z lz rz)
    | delta*sizeL < sizeR  = balanceL kz z (join kx x l lz) rz
    | delta*sizeR < sizeL  = balanceR ky y ly (join kx x ry r)
    | otherwise            = bin kx x l r
  
  
  -- insertMin and insertMax don't perform potentially expensive comparisons.
  insertMax,insertMin :: k -> a -> Map k a -> Map k a
  insertMax kx x t
    = case t of
        Tip -> singleton kx x
        Bin _ ky y l r
            -> balanceR ky y l (insertMax kx x r)
  
  insertMin kx x t
    = case t of
        Tip -> singleton kx x
        Bin _ ky y l r
            -> balanceL ky y (insertMin kx x l) r
  
  {--------------------------------------------------------------------
    [merge l r]: merges two trees.
  --------------------------------------------------------------------}
  merge :: Map k a -> Map k a -> Map k a
  merge Tip r   = r
  merge l Tip   = l
  merge l@(Bin sizeL kx x lx rx) r@(Bin sizeR ky y ly ry)
    | delta*sizeL < sizeR = balanceL ky y (merge l ly) ry
    | delta*sizeR < sizeL = balanceR kx x lx (merge rx r)
    | otherwise           = glue l r
  
  {--------------------------------------------------------------------
    [glue l r]: glues two trees together.
    Assumes that [l] and [r] are already balanced with respect to each other.
  --------------------------------------------------------------------}
  glue :: Map k a -> Map k a -> Map k a
  glue Tip r = r
  glue l Tip = l
  glue l r
    | size l > size r = let ((km,m),l') = deleteFindMax l in balanceR km m l' r
    | otherwise       = let ((km,m),r') = deleteFindMin r in balanceL km m l r'
  
  
  -- | /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)
  deleteFindMin t
    = case t of
        Bin _ k x Tip r -> ((k,x),r)
        Bin _ k x l r   -> let (km,l') = deleteFindMin l in (km,balanceR k x l' r)
        Tip             -> (error "Map.deleteFindMin: can not return the minimal element of an empty map", Tip)
  
  -- | /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)
  deleteFindMax t
    = case t of
        Bin _ k x l Tip -> ((k,x),l)
        Bin _ k x l r   -> let (km,r') = deleteFindMax r in (km,balanceL k x l r')
        Tip             -> (error "Map.deleteFindMax: can not return the maximal element of an empty map", Tip)
  
  
  {--------------------------------------------------------------------
    [balance l x r] balances two trees with value x.
    The sizes of the trees should balance after decreasing the
    size of one of them. (a rotation).
  
    [delta] is the maximal relative difference between the sizes of
            two trees, it corresponds with the [w] in Adams' paper.
    [ratio] is the ratio between an outer and inner sibling of the
            heavier subtree in an unbalanced setting. It determines
            whether a double or single rotation should be performed
            to restore balance. It is corresponds with the inverse
            of $\alpha$ in Adam's article.
  
    Note that according to the Adam's paper:
    - [delta] should be larger than 4.646 with a [ratio] of 2.
    - [delta] should be larger than 3.745 with a [ratio] of 1.534.
  
    But the Adam's paper is erroneous:
    - It can be proved that for delta=2 and delta>=5 there does
      not exist any ratio that would work.
    - Delta=4.5 and ratio=2 does not work.
  
    That leaves two reasonable variants, delta=3 and delta=4,
    both with ratio=2.
  
    - A lower [delta] leads to a more 'perfectly' balanced tree.
    - A higher [delta] performs less rebalancing.
  
    In the benchmarks, delta=3 is faster on insert operations,
    and delta=4 has slightly better deletes. As the insert speedup
    is larger, we currently use delta=3.
  
  --------------------------------------------------------------------}
  delta,ratio :: Int
  delta = 3
  ratio = 2
  
  -- The balance function is equivalent to the following:
  --
  --   balance :: k -> a -> Map k a -> Map k a -> Map k a
  --   balance k x l r
  --     | sizeL + sizeR <= 1    = Bin sizeX k x l r
  --     | sizeR > delta*sizeL   = rotateL k x l r
  --     | sizeL > delta*sizeR   = rotateR k x l r
  --     | otherwise             = Bin sizeX k x l r
  --     where
  --       sizeL = size l
  --       sizeR = size r
  --       sizeX = sizeL + sizeR + 1
  --
  --   rotateL :: a -> b -> Map a b -> Map a b -> Map a b
  --   rotateL k x l r@(Bin _ _ _ ly ry) | size ly < ratio*size ry = singleL k x l r
  --                                     | otherwise               = doubleL k x l r
  --
  --   rotateR :: a -> b -> Map a b -> Map a b -> Map a b
  --   rotateR k x l@(Bin _ _ _ ly ry) r | size ry < ratio*size ly = singleR k x l r
  --                                     | otherwise               = doubleR k x l r
  --
  --   singleL, singleR :: a -> b -> Map a b -> Map a b -> Map a b
  --   singleL k1 x1 t1 (Bin _ k2 x2 t2 t3)  = bin k2 x2 (bin k1 x1 t1 t2) t3
  --   singleR k1 x1 (Bin _ k2 x2 t1 t2) t3  = bin k2 x2 t1 (bin k1 x1 t2 t3)
  --
  --   doubleL, doubleR :: a -> b -> Map a b -> Map a b -> Map a b
  --   doubleL k1 x1 t1 (Bin _ k2 x2 (Bin _ k3 x3 t2 t3) t4) = bin k3 x3 (bin k1 x1 t1 t2) (bin k2 x2 t3 t4)
  --   doubleR k1 x1 (Bin _ k2 x2 t1 (Bin _ k3 x3 t2 t3)) t4 = bin k3 x3 (bin k2 x2 t1 t2) (bin k1 x1 t3 t4)
  --
  -- It is only written in such a way that every node is pattern-matched only once.
  
  balance :: k -> a -> Map k a -> Map k a -> Map k a
  balance k x l r = case l of
    Tip -> case r of
             Tip -> Bin 1 k x Tip Tip
             (Bin _ _ _ Tip Tip) -> Bin 2 k x Tip r
             (Bin _ rk rx Tip rr@(Bin _ _ _ _ _)) -> Bin 3 rk rx (Bin 1 k x Tip Tip) rr
             (Bin _ rk rx (Bin _ rlk rlx _ _) Tip) -> Bin 3 rlk rlx (Bin 1 k x Tip Tip) (Bin 1 rk rx Tip Tip)
             (Bin rs rk rx rl@(Bin rls rlk rlx rll rlr) rr@(Bin rrs _ _ _ _))
               | rls < ratio*rrs -> Bin (1+rs) rk rx (Bin (1+rls) k x Tip rl) rr
               | otherwise -> Bin (1+rs) rlk rlx (Bin (1+size rll) k x Tip rll) (Bin (1+rrs+size rlr) rk rx rlr rr)
  
    (Bin ls lk lx ll lr) -> case r of
             Tip -> case (ll, lr) of
                      (Tip, Tip) -> Bin 2 k x l Tip
                      (Tip, (Bin _ lrk lrx _ _)) -> Bin 3 lrk lrx (Bin 1 lk lx Tip Tip) (Bin 1 k x Tip Tip)
                      ((Bin _ _ _ _ _), Tip) -> Bin 3 lk lx ll (Bin 1 k x Tip Tip)
                      ((Bin lls _ _ _ _), (Bin lrs lrk lrx lrl lrr))
                        | lrs < ratio*lls -> Bin (1+ls) lk lx ll (Bin (1+lrs) k x lr Tip)
                        | otherwise -> Bin (1+ls) lrk lrx (Bin (1+lls+size lrl) lk lx ll lrl) (Bin (1+size lrr) k x lrr Tip)
             (Bin rs rk rx rl rr)
                | rs > delta*ls  -> case (rl, rr) of
                     (Bin rls rlk rlx rll rlr, Bin rrs _ _ _ _)
                       | rls < ratio*rrs -> Bin (1+ls+rs) rk rx (Bin (1+ls+rls) k x l rl) rr
                       | otherwise -> Bin (1+ls+rs) rlk rlx (Bin (1+ls+size rll) k x l rll) (Bin (1+rrs+size rlr) rk rx rlr rr)
                     (_, _) -> error "Failure in Data.Map.balance"
                | ls > delta*rs  -> case (ll, lr) of
                     (Bin lls _ _ _ _, Bin lrs lrk lrx lrl lrr)
                       | lrs < ratio*lls -> Bin (1+ls+rs) lk lx ll (Bin (1+rs+lrs) k x lr r)
                       | otherwise -> Bin (1+ls+rs) lrk lrx (Bin (1+lls+size lrl) lk lx ll lrl) (Bin (1+rs+size lrr) k x lrr r)
                     (_, _) -> error "Failure in Data.Map.balance"
                | otherwise -> Bin (1+ls+rs) k x l r
  {-# NOINLINE balance #-}
  
  -- Functions balanceL and balanceR are specialised versions of balance.
  -- balanceL only checks whether the left subtree is too big,
  -- balanceR only checks whether the right subtree is too big.
  
  -- balanceL is called when left subtree might have been inserted to or when
  -- right subtree might have been deleted from.
  balanceL :: k -> a -> Map k a -> Map k a -> Map k a
  balanceL k x l r = case r of
    Tip -> case l of
             Tip -> Bin 1 k x Tip Tip
             (Bin _ _ _ Tip Tip) -> Bin 2 k x l Tip
             (Bin _ lk lx Tip (Bin _ lrk lrx _ _)) -> Bin 3 lrk lrx (Bin 1 lk lx Tip Tip) (Bin 1 k x Tip Tip)
             (Bin _ lk lx ll@(Bin _ _ _ _ _) Tip) -> Bin 3 lk lx ll (Bin 1 k x Tip Tip)
             (Bin ls lk lx ll@(Bin lls _ _ _ _) lr@(Bin lrs lrk lrx lrl lrr))
               | lrs < ratio*lls -> Bin (1+ls) lk lx ll (Bin (1+lrs) k x lr Tip)
               | otherwise -> Bin (1+ls) lrk lrx (Bin (1+lls+size lrl) lk lx ll lrl) (Bin (1+size lrr) k x lrr Tip)
  
    (Bin rs _ _ _ _) -> case l of
             Tip -> Bin (1+rs) k x Tip r
  
             (Bin ls lk lx ll lr)
                | ls > delta*rs  -> case (ll, lr) of
                     (Bin lls _ _ _ _, Bin lrs lrk lrx lrl lrr)
                       | lrs < ratio*lls -> Bin (1+ls+rs) lk lx ll (Bin (1+rs+lrs) k x lr r)
                       | otherwise -> Bin (1+ls+rs) lrk lrx (Bin (1+lls+size lrl) lk lx ll lrl) (Bin (1+rs+size lrr) k x lrr r)
                     (_, _) -> error "Failure in Data.Map.balanceL"
                | otherwise -> Bin (1+ls+rs) k x l r
  {-# NOINLINE balanceL #-}
  
  -- balanceR is called when right subtree might have been inserted to or when
  -- left subtree might have been deleted from.
  balanceR :: k -> a -> Map k a -> Map k a -> Map k a
  balanceR k x l r = case l of
    Tip -> case r of
             Tip -> Bin 1 k x Tip Tip
             (Bin _ _ _ Tip Tip) -> Bin 2 k x Tip r
             (Bin _ rk rx Tip rr@(Bin _ _ _ _ _)) -> Bin 3 rk rx (Bin 1 k x Tip Tip) rr
             (Bin _ rk rx (Bin _ rlk rlx _ _) Tip) -> Bin 3 rlk rlx (Bin 1 k x Tip Tip) (Bin 1 rk rx Tip Tip)
             (Bin rs rk rx rl@(Bin rls rlk rlx rll rlr) rr@(Bin rrs _ _ _ _))
               | rls < ratio*rrs -> Bin (1+rs) rk rx (Bin (1+rls) k x Tip rl) rr
               | otherwise -> Bin (1+rs) rlk rlx (Bin (1+size rll) k x Tip rll) (Bin (1+rrs+size rlr) rk rx rlr rr)
  
    (Bin ls _ _ _ _) -> case r of
             Tip -> Bin (1+ls) k x l Tip
  
             (Bin rs rk rx rl rr)
                | rs > delta*ls  -> case (rl, rr) of
                     (Bin rls rlk rlx rll rlr, Bin rrs _ _ _ _)
                       | rls < ratio*rrs -> Bin (1+ls+rs) rk rx (Bin (1+ls+rls) k x l rl) rr
                       | otherwise -> Bin (1+ls+rs) rlk rlx (Bin (1+ls+size rll) k x l rll) (Bin (1+rrs+size rlr) rk rx rlr rr)
                     (_, _) -> error "Failure in Data.Map.balanceR"
                | otherwise -> Bin (1+ls+rs) k x l r
  {-# NOINLINE balanceR #-}
  
  
  {--------------------------------------------------------------------
    The bin constructor maintains the size of the tree
  --------------------------------------------------------------------}
  bin :: k -> a -> Map k a -> Map k a -> Map k a
  bin k x l r
    = Bin (size l + size r + 1) k x l r
  {-# INLINE bin #-}
  
  
  {--------------------------------------------------------------------
    Eq converts the tree to a list. In a lazy setting, this
    actually seems one of the faster methods to compare two trees
    and it is certainly the simplest :-)
  --------------------------------------------------------------------}
  instance (Eq k,Eq a) => Eq (Map k a) where
    t1 == t2  = (size t1 == size t2) && (toAscList t1 == toAscList t2)
  
  {--------------------------------------------------------------------
    Ord
  --------------------------------------------------------------------}
  
  instance (Ord k, Ord v) => Ord (Map k v) where
      compare m1 m2 = compare (toAscList m1) (toAscList m2)
  
  {--------------------------------------------------------------------
    Functor
  --------------------------------------------------------------------}
  instance Functor (Map k) where
    fmap f m  = map f m
  
  instance Traversable (Map k) where
    traverse f = traverseWithKey (\_ -> f)
  
  instance Foldable.Foldable (Map k) where
    fold Tip = mempty
    fold (Bin _ _ v l r) = Foldable.fold l `mappend` v `mappend` Foldable.fold r
    foldr = foldr
    foldl = foldl
    foldMap _ Tip = mempty
    foldMap f (Bin _ _ v l r) = Foldable.foldMap f l `mappend` f v `mappend` Foldable.foldMap f r
  
  instance (NFData k, NFData a) => NFData (Map k a) where
      rnf Tip = ()
      rnf (Bin _ kx x l r) = rnf kx `seq` rnf x `seq` rnf l `seq` rnf r
  
  {--------------------------------------------------------------------
    Read
  --------------------------------------------------------------------}
  instance (Ord k, Read k, Read e) => Read (Map k 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
  
  {--------------------------------------------------------------------
    Show
  --------------------------------------------------------------------}
  instance (Show k, Show a) => Show (Map k a) where
    showsPrec d m  = showParen (d > 10) $
      showString "fromList " . shows (toList m)
  
  -- | /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
  showTree m
    = showTreeWith showElem True False m
    where
      showElem k x  = show k ++ ":=" ++ show x
  
  
  {- | /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
  showTreeWith showelem hang wide t
    | hang      = (showsTreeHang showelem wide [] t) ""
    | otherwise = (showsTree showelem wide [] [] t) ""
  
  showsTree :: (k -> a -> String) -> Bool -> [String] -> [String] -> Map k a -> ShowS
  showsTree showelem wide lbars rbars t
    = case t of
        Tip -> showsBars lbars . showString "|\n"
        Bin _ kx x Tip Tip
            -> showsBars lbars . showString (showelem kx x) . showString "\n"
        Bin _ kx x l r
            -> showsTree showelem wide (withBar rbars) (withEmpty rbars) r .
               showWide wide rbars .
               showsBars lbars . showString (showelem kx x) . showString "\n" .
               showWide wide lbars .
               showsTree showelem wide (withEmpty lbars) (withBar lbars) l
  
  showsTreeHang :: (k -> a -> String) -> Bool -> [String] -> Map k a -> ShowS
  showsTreeHang showelem wide bars t
    = case t of
        Tip -> showsBars bars . showString "|\n"
        Bin _ kx x Tip Tip
            -> showsBars bars . showString (showelem kx x) . showString "\n"
        Bin _ kx x l r
            -> showsBars bars . showString (showelem kx x) . showString "\n" .
               showWide wide bars .
               showsTreeHang showelem wide (withBar bars) l .
               showWide wide bars .
               showsTreeHang showelem wide (withEmpty bars) r
  
  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
  
  {--------------------------------------------------------------------
    Typeable
  --------------------------------------------------------------------}
  
  #include "Typeable.h"
  INSTANCE_TYPEABLE2(Map,mapTc,"Map")
  
  {--------------------------------------------------------------------
    Assertions
  --------------------------------------------------------------------}
  -- | /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
  valid t
    = balanced t && ordered t && validsize t
  
  ordered :: Ord a => Map a b -> Bool
  ordered t
    = bounded (const True) (const True) t
    where
      bounded lo hi t'
        = case t' of
            Tip              -> True
            Bin _ kx _ l r  -> (lo kx) && (hi kx) && bounded lo (<kx) l && bounded (>kx) hi r
  
  -- | Exported only for "Debug.QuickCheck"
  balanced :: Map k a -> Bool
  balanced t
    = case t of
        Tip            -> True
        Bin _ _ _ l r  -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) &&
                          balanced l && balanced r
  
  validsize :: Map a b -> Bool
  validsize t
    = (realsize t == Just (size t))
    where
      realsize t'
        = case t' of
            Tip            -> Just 0
            Bin sz _ _ l r -> case (realsize l,realsize r) of
                              (Just n,Just m)  | n+m+1 == sz  -> Just sz
                              _                               -> Nothing
  
  {--------------------------------------------------------------------
    Utilities
  --------------------------------------------------------------------}
  foldlStrict :: (a -> b -> a) -> a -> [b] -> a
  foldlStrict f = go
    where
      go z []     = z
      go z (x:xs) = let z' = f z x in z' `seq` go z' xs
  {-# INLINE foldlStrict #-}
  