content: {-# LANGUAGE CPP #-}
  #if __GLASGOW_HASKELL__
  {-# LANGUAGE MagicHash, DeriveDataTypeable, StandaloneDeriving #-}
  #endif
  #if !defined(TESTING) && __GLASGOW_HASKELL__ >= 703
  {-# LANGUAGE Trustworthy #-}
  #endif
  -----------------------------------------------------------------------------
  -- |
  -- Module      :  Data.IntMap.Base
  -- Copyright   :  (c) Daan Leijen 2002
  --                (c) Andriy Palamarchuk 2008
  -- License     :  BSD-style
  -- Maintainer  :  libraries@haskell.org
  -- Stability   :  provisional
  -- Portability :  portable
  --
  -- This defines the data structures and core (hidden) manipulations
  -- on representations.
  -----------------------------------------------------------------------------
  
  -- [Note: INLINE bit fiddling]
  -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~
  -- It is essential that the bit fiddling functions like mask, zero, branchMask
  -- etc are inlined. If they do not, the memory allocation skyrockets. The GHC
  -- usually gets it right, but it is disastrous if it does not. Therefore we
  -- explicitly mark these functions INLINE.
  
  
  -- [Note: Local 'go' functions and capturing]
  -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  -- Care must be taken when using 'go' function which captures an argument.
  -- Sometimes (for example when the argument is passed to a data constructor,
  -- as in insert), GHC heap-allocates more than necessary. Therefore C-- code
  -- must be checked for increased allocation when creating and modifying such
  -- functions.
  
  
  -- [Note: Order of constructors]
  -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  -- The order of constructors of IntMap matters when considering performance.
  -- Currently in GHC 7.0, when type has 3 constructors, they are matched from
  -- the first to the last -- the best performance is achieved when the
  -- constructors are ordered by frequency.
  -- On GHC 7.0, reordering constructors from Nil | Tip | Bin to Bin | Tip | Nil
  -- improves the benchmark by circa 10%.
  
  module Data.IntMap.Base (
              -- * Map type
                IntMap(..), Key          -- instance Eq,Show
  
              -- * 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
              , 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
  
              -- * Min\/Max
              , findMin
              , findMax
              , deleteMin
              , deleteMax
              , deleteFindMin
              , deleteFindMax
              , updateMin
              , updateMax
              , updateMinWithKey
              , updateMaxWithKey
              , minView
              , maxView
              , minViewWithKey
              , maxViewWithKey
  
              -- * Debugging
              , showTree
              , showTreeWith
  
              -- * Internal types
              , Mask, Prefix, Nat
  
              -- * Utility
              , natFromInt
              , intFromNat
              , shiftRL
              , shiftLL
              , join
              , bin
              , zero
              , nomatch
              , match
              , mask
              , maskW
              , shorter
              , branchMask
              , highestBitMask
              , foldlStrict
              ) where
  
  import Data.Bits
  
  import Prelude hiding (lookup,map,filter,foldr,foldl,null)
  import qualified Data.IntSet.Base as IntSet
  import Data.Monoid (Monoid(..))
  import Data.Maybe (fromMaybe)
  import Data.Typeable
  import qualified Data.Foldable as Foldable
  import Data.Traversable (Traversable(traverse))
  import Control.Applicative (Applicative(pure,(<*>)),(<$>))
  import Control.Monad ( liftM )
  import Control.DeepSeq (NFData(rnf))
  
  #if __GLASGOW_HASKELL__
  import Text.Read
  import Data.Data (Data(..), mkNoRepType)
  #endif
  
  #if __GLASGOW_HASKELL__
  import GHC.Exts ( Word(..), Int(..), build )
  import GHC.Prim ( uncheckedShiftL#, uncheckedShiftRL# )
  #else
  import Data.Word
  #endif
  
  -- On GHC, include MachDeps.h to get WORD_SIZE_IN_BITS macro.
  #if defined(__GLASGOW_HASKELL__)
  #include "MachDeps.h"
  #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
  
  -- A "Nat" is a natural machine word (an unsigned Int)
  type Nat = Word
  
  natFromInt :: Key -> Nat
  natFromInt = fromIntegral
  {-# INLINE natFromInt #-}
  
  intFromNat :: Nat -> Key
  intFromNat = fromIntegral
  {-# INLINE intFromNat #-}
  
  -- Right and left logical shifts.
  shiftRL, shiftLL :: Nat -> Key -> Nat
  #if __GLASGOW_HASKELL__
  {--------------------------------------------------------------------
    GHC: use unboxing to get @shiftRL@ inlined.
  --------------------------------------------------------------------}
  shiftRL (W# x) (I# i) = W# (uncheckedShiftRL# x i)
  shiftLL (W# x) (I# i) = W# (uncheckedShiftL#  x i)
  #else
  shiftRL x i   = shiftR x i
  shiftLL x i   = shiftL x i
  #endif
  {-# INLINE shiftRL #-}
  {-# INLINE shiftLL #-}
  
  {--------------------------------------------------------------------
    Types
  --------------------------------------------------------------------}
  
  
  -- | A map of integers to values @a@.
  
  -- See Note: Order of constructors
  data IntMap a = Bin {-# UNPACK #-} !Prefix {-# UNPACK #-} !Mask !(IntMap a) !(IntMap a)
                | Tip {-# UNPACK #-} !Key a
                | Nil
  
  type Prefix = Int
  type Mask   = Int
  type Key    = Int
  
  {--------------------------------------------------------------------
    Operators
  --------------------------------------------------------------------}
  
  -- | /O(min(n,W))/. Find the value at a key.
  -- Calls 'error' when the element can not be found.
  --
  -- > fromList [(5,'a'), (3,'b')] ! 1    Error: element not in the map
  -- > fromList [(5,'a'), (3,'b')] ! 5 == 'a'
  
  (!) :: IntMap a -> Key -> a
  m ! k = find k m
  
  -- | Same as 'difference'.
  (\\) :: IntMap a -> IntMap b -> IntMap a
  m1 \\ m2 = difference m1 m2
  
  infixl 9 \\{-This comment teaches CPP correct behaviour -}
  
  {--------------------------------------------------------------------
    Types
  --------------------------------------------------------------------}
  
  instance Monoid (IntMap a) where
      mempty  = empty
      mappend = union
      mconcat = unions
  
  instance Foldable.Foldable IntMap where
    fold Nil = mempty
    fold (Tip _ v) = v
    fold (Bin _ _ l r) = Foldable.fold l `mappend` Foldable.fold r
    foldr = foldr
    foldl = foldl
    foldMap _ Nil = mempty
    foldMap f (Tip _k v) = f v
    foldMap f (Bin _ _ l r) = Foldable.foldMap f l `mappend` Foldable.foldMap f r
  
  instance Traversable IntMap where
      traverse f = traverseWithKey (\_ -> f)
  
  instance NFData a => NFData (IntMap a) where
      rnf Nil = ()
      rnf (Tip _ v) = rnf v
      rnf (Bin _ _ l r) = rnf l `seq` rnf r
  
  #if __GLASGOW_HASKELL__
  
  {--------------------------------------------------------------------
    A Data instance
  --------------------------------------------------------------------}
  
  -- This instance preserves data abstraction at the cost of inefficiency.
  -- We omit reflection services for the sake of data abstraction.
  
  instance Data a => Data (IntMap a) where
    gfoldl f z im = z fromList `f` (toList im)
    toConstr _    = error "toConstr"
    gunfold _ _   = error "gunfold"
    dataTypeOf _  = mkNoRepType "Data.IntMap.IntMap"
    dataCast1 f   = gcast1 f
  
  #endif
  
  {--------------------------------------------------------------------
    Query
  --------------------------------------------------------------------}
  -- | /O(1)/. Is the map empty?
  --
  -- > Data.IntMap.null (empty)           == True
  -- > Data.IntMap.null (singleton 1 'a') == False
  
  null :: IntMap a -> Bool
  null Nil = True
  null _   = False
  {-# INLINE null #-}
  
  -- | /O(n)/. Number of elements in the map.
  --
  -- > size empty                                   == 0
  -- > size (singleton 1 'a')                       == 1
  -- > size (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3
  size :: IntMap a -> Int
  size t
    = case t of
        Bin _ _ l r -> size l + size r
        Tip _ _ -> 1
        Nil     -> 0
  
  -- | /O(min(n,W))/. Is the key a member of the map?
  --
  -- > member 5 (fromList [(5,'a'), (3,'b')]) == True
  -- > member 1 (fromList [(5,'a'), (3,'b')]) == False
  
  -- See Note: Local 'go' functions and capturing]
  member :: Key -> IntMap a -> Bool
  member k = k `seq` go
    where
      go (Bin p m l r) | nomatch k p m = False
                       | zero k m  = go l
                       | otherwise = go r
      go (Tip kx _) = k == kx
      go Nil = False
  
  -- | /O(min(n,W))/. Is the key not a member of the map?
  --
  -- > notMember 5 (fromList [(5,'a'), (3,'b')]) == False
  -- > notMember 1 (fromList [(5,'a'), (3,'b')]) == True
  
  notMember :: Key -> IntMap a -> Bool
  notMember k m = not $ member k m
  
  -- | /O(min(n,W))/. Lookup the value at a key in the map. See also 'Data.Map.lookup'.
  
  -- See Note: Local 'go' functions and capturing]
  lookup :: Key -> IntMap a -> Maybe a
  lookup k = k `seq` go
    where
      go (Bin p m l r) | nomatch k p m = Nothing
                       | zero k m  = go l
                       | otherwise = go r
      go (Tip kx x) | k == kx   = Just x
                    | otherwise = Nothing
      go Nil = Nothing
  
  
  -- See Note: Local 'go' functions and capturing]
  find :: Key -> IntMap a -> a
  find k = k `seq` go
    where
      go (Bin p m l r) | nomatch k p m = not_found
                       | zero k m  = go l
                       | otherwise = go r
      go (Tip kx x) | k == kx   = x
                    | otherwise = not_found
      go Nil = not_found
  
      not_found = error ("IntMap.!: key " ++ show k ++ " is not an element of the map")
  
  -- | /O(min(n,W))/. The expression @('findWithDefault' def k map)@
  -- returns the value at key @k@ or returns @def@ when the key is not an
  -- element of the map.
  --
  -- > findWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x'
  -- > findWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a'
  
  -- See Note: Local 'go' functions and capturing]
  findWithDefault :: a -> Key -> IntMap a -> a
  findWithDefault def k = k `seq` go
    where
      go (Bin p m l r) | nomatch k p m = def
                       | zero k m  = go l
                       | otherwise = go r
      go (Tip kx x) | k == kx   = x
                    | otherwise = def
      go Nil = def
  
  -- | /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')
  
  -- See Note: Local 'go' functions and capturing.
  lookupLT :: Key -> IntMap a -> Maybe (Key, a)
  lookupLT k t = k `seq` case t of
      Bin _ m l r | m < 0 -> if k >= 0 then go r l else go Nil r
      _ -> go Nil t
    where
      go def (Bin p m l r) | nomatch k p m = if k < p then unsafeFindMax def else unsafeFindMax r
                           | zero k m  = go def l
                           | otherwise = go l r
      go def (Tip ky y) | k <= ky   = unsafeFindMax def
                        | otherwise = Just (ky, y)
      go def Nil = unsafeFindMax def
  
  -- | /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
  
  -- See Note: Local 'go' functions and capturing.
  lookupGT :: Key -> IntMap a -> Maybe (Key, a)
  lookupGT k t = k `seq` case t of
      Bin _ m l r | m < 0 -> if k >= 0 then go Nil l else go l r
      _ -> go Nil t
    where
      go def (Bin p m l r) | nomatch k p m = if k < p then unsafeFindMin l else unsafeFindMin def
                           | zero k m  = go r l
                           | otherwise = go def r
      go def (Tip ky y) | k >= ky   = unsafeFindMin def
                        | otherwise = Just (ky, y)
      go def Nil = unsafeFindMin def
  
  -- | /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')
  
  -- See Note: Local 'go' functions and capturing.
  lookupLE :: Key -> IntMap a -> Maybe (Key, a)
  lookupLE k t = k `seq` case t of
      Bin _ m l r | m < 0 -> if k >= 0 then go r l else go Nil r
      _ -> go Nil t
    where
      go def (Bin p m l r) | nomatch k p m = if k < p then unsafeFindMax def else unsafeFindMax r
                           | zero k m  = go def l
                           | otherwise = go l r
      go def (Tip ky y) | k < ky    = unsafeFindMax def
                        | otherwise = Just (ky, y)
      go def Nil = unsafeFindMax def
  
  -- | /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
  
  -- See Note: Local 'go' functions and capturing.
  lookupGE :: Key -> IntMap a -> Maybe (Key, a)
  lookupGE k t = k `seq` case t of
      Bin _ m l r | m < 0 -> if k >= 0 then go Nil l else go l r
      _ -> go Nil t
    where
      go def (Bin p m l r) | nomatch k p m = if k < p then unsafeFindMin l else unsafeFindMin def
                           | zero k m  = go r l
                           | otherwise = go def r
      go def (Tip ky y) | k > ky    = unsafeFindMin def
                        | otherwise = Just (ky, y)
      go def Nil = unsafeFindMin def
  
  
  -- Helper function for lookupGE and lookupGT. It assumes that if a Bin node is
  -- given, it has m > 0.
  unsafeFindMin :: IntMap a -> Maybe (Key, a)
  unsafeFindMin Nil = Nothing
  unsafeFindMin (Tip ky y) = Just (ky, y)
  unsafeFindMin (Bin _ _ l _) = unsafeFindMin l
  
  -- Helper function for lookupLE and lookupLT. It assumes that if a Bin node is
  -- given, it has m > 0.
  unsafeFindMax :: IntMap a -> Maybe (Key, a)
  unsafeFindMax Nil = Nothing
  unsafeFindMax (Tip ky y) = Just (ky, y)
  unsafeFindMax (Bin _ _ _ r) = unsafeFindMax r
  
  {--------------------------------------------------------------------
    Construction
  --------------------------------------------------------------------}
  -- | /O(1)/. The empty map.
  --
  -- > empty      == fromList []
  -- > size empty == 0
  
  empty :: IntMap a
  empty
    = Nil
  {-# INLINE empty #-}
  
  -- | /O(1)/. A map of one element.
  --
  -- > singleton 1 'a'        == fromList [(1, 'a')]
  -- > size (singleton 1 'a') == 1
  
  singleton :: Key -> a -> IntMap a
  singleton k x
    = Tip k x
  {-# INLINE singleton #-}
  
  {--------------------------------------------------------------------
    Insert
  --------------------------------------------------------------------}
  -- | /O(min(n,W))/. Insert a new key\/value pair in the map.
  -- If the key is already present in the map, the associated value is
  -- replaced with the supplied value, i.e. 'insert' is equivalent to
  -- @'insertWith' 'const'@.
  --
  -- > insert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')]
  -- > insert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')]
  -- > insert 5 'x' empty                         == singleton 5 'x'
  
  insert :: Key -> a -> IntMap a -> IntMap a
  insert k x t = k `seq`
    case t of
      Bin p m l r
        | nomatch k p m -> join k (Tip k x) p t
        | zero k m      -> Bin p m (insert k x l) r
        | otherwise     -> Bin p m l (insert k x r)
      Tip ky _
        | k==ky         -> Tip k x
        | otherwise     -> join k (Tip k x) ky t
      Nil -> Tip k x
  
  -- right-biased insertion, used by 'union'
  -- | /O(min(n,W))/. Insert with a combining function.
  -- @'insertWith' f key value mp@
  -- will insert the pair (key, value) into @mp@ if key does
  -- not exist in the map. If the key does exist, the function will
  -- insert @f new_value old_value@.
  --
  -- > insertWith (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")]
  -- > insertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
  -- > insertWith (++) 5 "xxx" empty                         == singleton 5 "xxx"
  
  insertWith :: (a -> a -> a) -> Key -> a -> IntMap a -> IntMap a
  insertWith f k x t
    = insertWithKey (\_ x' y' -> f x' y') k x t
  
  -- | /O(min(n,W))/. Insert with a combining function.
  -- @'insertWithKey' f key value mp@
  -- will insert the pair (key, value) into @mp@ if key does
  -- not exist in the map. If the key does exist, the function will
  -- insert @f key new_value old_value@.
  --
  -- > let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
  -- > insertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")]
  -- > insertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
  -- > insertWithKey f 5 "xxx" empty                         == singleton 5 "xxx"
  
  insertWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> IntMap a
  insertWithKey f k x t = k `seq`
    case t of
      Bin p m l r
        | nomatch k p m -> join k (Tip k x) p t
        | zero k m      -> Bin p m (insertWithKey f k x l) r
        | otherwise     -> Bin p m l (insertWithKey f k x r)
      Tip ky y
        | k==ky         -> Tip k (f k x y)
        | otherwise     -> join k (Tip k x) ky t
      Nil -> Tip k x
  
  -- | /O(min(n,W))/. The expression (@'insertLookupWithKey' f k x map@)
  -- is a pair where the first element is equal to (@'lookup' k map@)
  -- and the second element equal to (@'insertWithKey' f k x map@).
  --
  -- > let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
  -- > insertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")])
  -- > insertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "xxx")])
  -- > insertLookupWithKey f 5 "xxx" empty                         == (Nothing,  singleton 5 "xxx")
  --
  -- This is how to define @insertLookup@ using @insertLookupWithKey@:
  --
  -- > let insertLookup kx x t = insertLookupWithKey (\_ a _ -> a) kx x t
  -- > insertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")])
  -- > insertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "x")])
  
  insertLookupWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> (Maybe a, IntMap a)
  insertLookupWithKey f k x t = k `seq`
    case t of
      Bin p m l r
        | nomatch k p m -> (Nothing,join k (Tip k x) p t)
        | zero k m      -> let (found,l') = insertLookupWithKey f k x l in (found,Bin p m l' r)
        | otherwise     -> let (found,r') = insertLookupWithKey f k x r in (found,Bin p m l r')
      Tip ky y
        | k==ky         -> (Just y,Tip k (f k x y))
        | otherwise     -> (Nothing,join k (Tip k x) ky t)
      Nil -> (Nothing,Tip k x)
  
  
  {--------------------------------------------------------------------
    Deletion
  --------------------------------------------------------------------}
  -- | /O(min(n,W))/. Delete a key and its value from the map. When the key is not
  -- a member of the map, the original map is returned.
  --
  -- > delete 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
  -- > delete 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
  -- > delete 5 empty                         == empty
  
  delete :: Key -> IntMap a -> IntMap a
  delete k t = k `seq`
    case t of
      Bin p m l r
        | nomatch k p m -> t
        | zero k m      -> bin p m (delete k l) r
        | otherwise     -> bin p m l (delete k r)
      Tip ky _
        | k==ky         -> Nil
        | otherwise     -> t
      Nil -> Nil
  
  -- | /O(min(n,W))/. Adjust a value at a specific key. When the key is not
  -- a member of the map, the original map is returned.
  --
  -- > adjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
  -- > adjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
  -- > adjust ("new " ++) 7 empty                         == empty
  
  adjust ::  (a -> a) -> Key -> IntMap a -> IntMap a
  adjust f k m
    = adjustWithKey (\_ x -> f x) k m
  
  -- | /O(min(n,W))/. Adjust a value at a specific key. When the key is not
  -- a member of the map, the original map is returned.
  --
  -- > let f key x = (show key) ++ ":new " ++ x
  -- > adjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
  -- > adjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
  -- > adjustWithKey f 7 empty                         == empty
  
  adjustWithKey ::  (Key -> a -> a) -> Key -> IntMap a -> IntMap a
  adjustWithKey f
    = updateWithKey (\k' x -> Just (f k' x))
  
  -- | /O(min(n,W))/. The expression (@'update' f k map@) updates the value @x@
  -- at @k@ (if it is in the map). If (@f x@) is 'Nothing', the element is
  -- deleted. If it is (@'Just' y@), the key @k@ is bound to the new value @y@.
  --
  -- > let f x = if x == "a" then Just "new a" else Nothing
  -- > update f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
  -- > update f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
  -- > update f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
  
  update ::  (a -> Maybe a) -> Key -> IntMap a -> IntMap a
  update f
    = updateWithKey (\_ x -> f x)
  
  -- | /O(min(n,W))/. The expression (@'update' f k map@) updates the value @x@
  -- at @k@ (if it is in the map). If (@f k x@) is 'Nothing', the element is
  -- deleted. If it is (@'Just' y@), the key @k@ is bound to the new value @y@.
  --
  -- > let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
  -- > updateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
  -- > updateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
  -- > updateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
  
  updateWithKey ::  (Key -> a -> Maybe a) -> Key -> IntMap a -> IntMap a
  updateWithKey f k t = k `seq`
    case t of
      Bin p m l r
        | nomatch k p m -> t
        | zero k m      -> bin p m (updateWithKey f k l) r
        | otherwise     -> bin p m l (updateWithKey f k r)
      Tip ky y
        | k==ky         -> case (f k y) of
                             Just y' -> Tip ky y'
                             Nothing -> Nil
        | otherwise     -> t
      Nil -> Nil
  
  -- | /O(min(n,W))/. Lookup and update.
  -- The function returns original value, if it is updated.
  -- This is different behavior than 'Data.Map.updateLookupWithKey'.
  -- Returns the original key value if the map entry is deleted.
  --
  -- > let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
  -- > updateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:new a")])
  -- > updateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a")])
  -- > updateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a")
  
  updateLookupWithKey ::  (Key -> a -> Maybe a) -> Key -> IntMap a -> (Maybe a,IntMap a)
  updateLookupWithKey f k t = k `seq`
    case t of
      Bin p m l r
        | nomatch k p m -> (Nothing,t)
        | zero k m      -> let (found,l') = updateLookupWithKey f k l in (found,bin p m l' r)
        | otherwise     -> let (found,r') = updateLookupWithKey f k r in (found,bin p m l r')
      Tip ky y
        | k==ky         -> case (f k y) of
                             Just y' -> (Just y,Tip ky y')
                             Nothing -> (Just y,Nil)
        | otherwise     -> (Nothing,t)
      Nil -> (Nothing,Nil)
  
  
  
  -- | /O(min(n,W))/. The expression (@'alter' f k map@) alters the value @x@ at @k@, or absence thereof.
  -- 'alter' can be used to insert, delete, or update a value in an 'IntMap'.
  -- In short : @'lookup' k ('alter' f k m) = f ('lookup' k m)@.
  alter :: (Maybe a -> Maybe a) -> Key -> IntMap a -> IntMap a
  alter f k t = k `seq`
    case t of
      Bin p m l r
        | nomatch k p m -> case f Nothing of
                             Nothing -> t
                             Just x -> join k (Tip k x) p t
        | zero k m      -> bin p m (alter f k l) r
        | otherwise     -> bin p m l (alter f k r)
      Tip ky y
        | k==ky         -> case f (Just y) of
                             Just x -> Tip ky x
                             Nothing -> Nil
        | otherwise     -> case f Nothing of
                             Just x -> join k (Tip k x) ky t
                             Nothing -> Tip ky y
      Nil               -> case f Nothing of
                             Just x -> Tip k x
                             Nothing -> Nil
  
  
  {--------------------------------------------------------------------
    Union
  --------------------------------------------------------------------}
  -- | The union of a list of maps.
  --
  -- > unions [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
  -- >     == fromList [(3, "b"), (5, "a"), (7, "C")]
  -- > unions [(fromList [(5, "A3"), (3, "B3")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "a"), (3, "b")])]
  -- >     == fromList [(3, "B3"), (5, "A3"), (7, "C")]
  
  unions :: [IntMap a] -> IntMap a
  unions xs
    = foldlStrict union empty xs
  
  -- | The union of a list of maps, with a combining operation.
  --
  -- > unionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
  -- >     == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")]
  
  unionsWith :: (a->a->a) -> [IntMap a] -> IntMap a
  unionsWith f ts
    = foldlStrict (unionWith f) empty ts
  
  -- | /O(n+m)/. The (left-biased) union of two maps.
  -- It prefers the first map when duplicate keys are encountered,
  -- i.e. (@'union' == 'unionWith' 'const'@).
  --
  -- > union (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")]
  
  union :: IntMap a -> IntMap a -> IntMap a
  union m1 m2
    = mergeWithKey' Bin const id id m1 m2
  
  -- | /O(n+m)/. The union with a combining function.
  --
  -- > unionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")]
  
  unionWith :: (a -> a -> a) -> IntMap a -> IntMap a -> IntMap a
  unionWith f m1 m2
    = unionWithKey (\_ x y -> f x y) m1 m2
  
  -- | /O(n+m)/. The union with a combining function.
  --
  -- > let f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value
  -- > unionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")]
  
  unionWithKey :: (Key -> a -> a -> a) -> IntMap a -> IntMap a -> IntMap a
  unionWithKey f m1 m2
    = mergeWithKey' Bin (\(Tip k1 x1) (Tip _k2 x2) -> Tip k1 (f k1 x1 x2)) id id m1 m2
  
  {--------------------------------------------------------------------
    Difference
  --------------------------------------------------------------------}
  -- | /O(n+m)/. Difference between two maps (based on keys).
  --
  -- > difference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b"
  
  difference :: IntMap a -> IntMap b -> IntMap a
  difference m1 m2
    = mergeWithKey (\_ _ _ -> Nothing) id (const Nil) m1 m2
  
  -- | /O(n+m)/. Difference with a combining function.
  --
  -- > let f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing
  -- > differenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")])
  -- >     == singleton 3 "b:B"
  
  differenceWith :: (a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap a
  differenceWith f m1 m2
    = differenceWithKey (\_ x y -> f x y) m1 m2
  
  -- | /O(n+m)/. Difference with a combining function. When two equal keys are
  -- encountered, the combining function is applied to the key and both values.
  -- If it returns 'Nothing', the element is discarded (proper set difference).
  -- If it returns (@'Just' y@), the element is updated with a new value @y@.
  --
  -- > let f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing
  -- > differenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")])
  -- >     == singleton 3 "3:b|B"
  
  differenceWithKey :: (Key -> a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap a
  differenceWithKey f m1 m2
    = mergeWithKey f id (const Nil) m1 m2
  
  
  {--------------------------------------------------------------------
    Intersection
  --------------------------------------------------------------------}
  -- | /O(n+m)/. The (left-biased) intersection of two maps (based on keys).
  --
  -- > intersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a"
  
  intersection :: IntMap a -> IntMap b -> IntMap a
  intersection m1 m2
    = mergeWithKey' bin const (const Nil) (const Nil) m1 m2
  
  -- | /O(n+m)/. The intersection with a combining function.
  --
  -- > intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"
  
  intersectionWith :: (a -> b -> c) -> IntMap a -> IntMap b -> IntMap c
  intersectionWith f m1 m2
    = intersectionWithKey (\_ x y -> f x y) m1 m2
  
  -- | /O(n+m)/. The intersection with a combining function.
  --
  -- > let f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar
  -- > intersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A"
  
  intersectionWithKey :: (Key -> a -> b -> c) -> IntMap a -> IntMap b -> IntMap c
  intersectionWithKey f m1 m2
    = mergeWithKey' bin (\(Tip k1 x1) (Tip _k2 x2) -> Tip k1 (f k1 x1 x2)) (const Nil) (const Nil) m1 m2
  
  {--------------------------------------------------------------------
    MergeWithKey
  --------------------------------------------------------------------}
  
  -- | /O(n+m)/. A high-performance universal combining function. Using
  -- 'mergeWithKey', all combining functions can be defined without any loss of
  -- efficiency (with exception of 'union', 'difference' and 'intersection',
  -- where sharing of some nodes is lost with 'mergeWithKey').
  --
  -- Please make sure you know what is going on when using 'mergeWithKey',
  -- otherwise you can be surprised by unexpected code growth or even
  -- corruption of the data structure.
  --
  -- When 'mergeWithKey' is given three arguments, it is inlined to the call
  -- site. You should therefore use 'mergeWithKey' only to define your custom
  -- combining functions. For example, you could define 'unionWithKey',
  -- 'differenceWithKey' and 'intersectionWithKey' as
  --
  -- > myUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) id id m1 m2
  -- > myDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2
  -- > myIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) (const empty) (const empty) m1 m2
  --
  -- When calling @'mergeWithKey' combine only1 only2@, a function combining two
  -- '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 :: (Key -> a -> b -> Maybe c) -> (IntMap a -> IntMap c) -> (IntMap b -> IntMap c)
               -> IntMap a -> IntMap b -> IntMap c
  mergeWithKey f g1 g2 = mergeWithKey' bin combine g1 g2
    where -- We use the lambda form to avoid non-exhaustive pattern matches warning.
          combine = \(Tip k1 x1) (Tip _k2 x2) -> case f k1 x1 x2 of Nothing -> Nil
                                                                    Just x -> Tip k1 x
          {-# INLINE combine #-}
  {-# INLINE mergeWithKey #-}
  
  -- Slightly more general version of mergeWithKey. It differs in the following:
  --
  -- * the combining function operates on maps instead of keys and values. The
  --   reason is to enable sharing in union, difference and intersection.
  --
  -- * mergeWithKey' is given an equivalent of bin. The reason is that in union*,
  --   Bin constructor can be used, because we know both subtrees are nonempty.
  
  mergeWithKey' :: (Prefix -> Mask -> IntMap c -> IntMap c -> IntMap c)
                -> (IntMap a -> IntMap b -> IntMap c) -> (IntMap a -> IntMap c) -> (IntMap b -> IntMap c)
                -> IntMap a -> IntMap b -> IntMap c
  mergeWithKey' bin' f g1 g2 = go
    where
      go t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
        | shorter m1 m2  = merge1
        | shorter m2 m1  = merge2
        | p1 == p2       = bin' p1 m1 (go l1 l2) (go r1 r2)
        | otherwise      = maybe_join p1 (g1 t1) p2 (g2 t2)
        where
          merge1 | nomatch p2 p1 m1  = maybe_join p1 (g1 t1) p2 (g2 t2)
                 | zero p2 m1        = bin' p1 m1 (go l1 t2) (g1 r1)
                 | otherwise         = bin' p1 m1 (g1 l1) (go r1 t2)
          merge2 | nomatch p1 p2 m2  = maybe_join p1 (g1 t1) p2 (g2 t2)
                 | zero p1 m2        = bin' p2 m2 (go t1 l2) (g2 r2)
                 | otherwise         = bin' p2 m2 (g2 l2) (go t1 r2)
  
      go t1'@(Bin _ _ _ _) t2'@(Tip k2' _) = merge t2' k2' t1'
        where merge t2 k2 t1@(Bin p1 m1 l1 r1) | nomatch k2 p1 m1 = maybe_join p1 (g1 t1) k2 (g2 t2)
                                               | zero k2 m1 = bin' p1 m1 (merge t2 k2 l1) (g1 r1)
                                               | otherwise  = bin' p1 m1 (g1 l1) (merge t2 k2 r1)
              merge t2 k2 t1@(Tip k1 _) | k1 == k2 = f t1 t2
                                        | otherwise = maybe_join k1 (g1 t1) k2 (g2 t2)
              merge t2 _  Nil = g2 t2
  
      go t1@(Bin _ _ _ _) Nil = g1 t1
  
      go t1'@(Tip k1' _) t2' = merge t1' k1' t2'
        where merge t1 k1 t2@(Bin p2 m2 l2 r2) | nomatch k1 p2 m2 = maybe_join k1 (g1 t1) p2 (g2 t2)
                                               | zero k1 m2 = bin' p2 m2 (merge t1 k1 l2) (g2 r2)
                                               | otherwise  = bin' p2 m2 (g2 l2) (merge t1 k1 r2)
              merge t1 k1 t2@(Tip k2 _) | k1 == k2 = f t1 t2
                                        | otherwise = maybe_join k1 (g1 t1) k2 (g2 t2)
              merge t1 _  Nil = g1 t1
  
      go Nil t2 = g2 t2
  
      maybe_join _ Nil _ t2 = t2
      maybe_join _ t1 _ Nil = t1
      maybe_join p1 t1 p2 t2 = join p1 t1 p2 t2
      {-# INLINE maybe_join #-}
  {-# INLINE mergeWithKey' #-}
  
  {--------------------------------------------------------------------
    Min\/Max
  --------------------------------------------------------------------}
  
  -- | /O(min(n,W))/. Update the value at the minimal key.
  --
  -- > updateMinWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")]
  -- > updateMinWithKey (\ _ _ -> Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
  
  updateMinWithKey :: (Key -> a -> Maybe a) -> IntMap a -> IntMap a
  updateMinWithKey f t =
    case t of Bin p m l r | m < 0 -> bin p m l (go f r)
              _ -> go f t
    where
      go f' (Bin p m l r) = bin p m (go f' l) r
      go f' (Tip k y) = case f' k y of
                          Just y' -> Tip k y'
                          Nothing -> Nil
      go _ Nil = error "updateMinWithKey Nil"
  
  -- | /O(min(n,W))/. Update the value at the maximal key.
  --
  -- > updateMaxWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")]
  -- > updateMaxWithKey (\ _ _ -> Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
  
  updateMaxWithKey :: (Key -> a -> Maybe a) -> IntMap a -> IntMap a
  updateMaxWithKey f t =
    case t of Bin p m l r | m < 0 -> bin p m (go f l) r
              _ -> go f t
    where
      go f' (Bin p m l r) = bin p m l (go f' r)
      go f' (Tip k y) = case f' k y of
                          Just y' -> Tip k y'
                          Nothing -> Nil
      go _ Nil = error "updateMaxWithKey Nil"
  
  -- | /O(min(n,W))/. Retrieves the maximal (key,value) pair of the map, and
  -- the map stripped of that element, or 'Nothing' if passed an empty map.
  --
  -- > maxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b")
  -- > maxViewWithKey empty == Nothing
  
  maxViewWithKey :: IntMap a -> Maybe ((Key, a), IntMap a)
  maxViewWithKey t =
    case t of Nil -> Nothing
              Bin p m l r | m < 0 -> case go l of (result, l') -> Just (result, bin p m l' r)
              _ -> Just (go t)
    where
      go (Bin p m l r) = case go r of (result, r') -> (result, bin p m l r')
      go (Tip k y) = ((k, y), Nil)
      go Nil = error "maxViewWithKey Nil"
  
  -- | /O(min(n,W))/. Retrieves the minimal (key,value) pair of the map, and
  -- the map stripped of that element, or 'Nothing' if passed an empty map.
  --
  -- > minViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((3,"b"), singleton 5 "a")
  -- > minViewWithKey empty == Nothing
  
  minViewWithKey :: IntMap a -> Maybe ((Key, a), IntMap a)
  minViewWithKey t =
    case t of Nil -> Nothing
              Bin p m l r | m < 0 -> case go r of (result, r') -> Just (result, bin p m l r')
              _ -> Just (go t)
    where
      go (Bin p m l r) = case go l of (result, l') -> (result, bin p m l' r)
      go (Tip k y) = ((k, y), Nil)
      go Nil = error "minViewWithKey Nil"
  
  -- | /O(min(n,W))/. Update the value at the maximal key.
  --
  -- > updateMax (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")]
  -- > updateMax (\ _ -> Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
  
  updateMax :: (a -> Maybe a) -> IntMap a -> IntMap a
  updateMax f = updateMaxWithKey (const f)
  
  -- | /O(min(n,W))/. Update the value at the minimal key.
  --
  -- > updateMin (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")]
  -- > updateMin (\ _ -> Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
  
  updateMin :: (a -> Maybe a) -> IntMap a -> IntMap a
  updateMin f = updateMinWithKey (const f)
  
  -- Similar to the Arrow instance.
  first :: (a -> c) -> (a, b) -> (c, b)
  first f (x,y) = (f x,y)
  
  -- | /O(min(n,W))/. Retrieves the maximal key of the map, and the map
  -- stripped of that element, or 'Nothing' if passed an empty map.
  maxView :: IntMap a -> Maybe (a, IntMap a)
  maxView t = liftM (first snd) (maxViewWithKey t)
  
  -- | /O(min(n,W))/. Retrieves the minimal key of the map, and the map
  -- stripped of that element, or 'Nothing' if passed an empty map.
  minView :: IntMap a -> Maybe (a, IntMap a)
  minView t = liftM (first snd) (minViewWithKey t)
  
  -- | /O(min(n,W))/. Delete and find the maximal element.
  deleteFindMax :: IntMap a -> ((Key, a), IntMap a)
  deleteFindMax = fromMaybe (error "deleteFindMax: empty map has no maximal element") . maxViewWithKey
  
  -- | /O(min(n,W))/. Delete and find the minimal element.
  deleteFindMin :: IntMap a -> ((Key, a), IntMap a)
  deleteFindMin = fromMaybe (error "deleteFindMin: empty map has no minimal element") . minViewWithKey
  
  -- | /O(min(n,W))/. The minimal key of the map.
  findMin :: IntMap a -> (Key, a)
  findMin Nil = error $ "findMin: empty map has no minimal element"
  findMin (Tip k v) = (k,v)
  findMin (Bin _ m l r)
    |   m < 0   = go r
    | otherwise = go l
      where go (Tip k v)      = (k,v)
            go (Bin _ _ l' _) = go l'
            go Nil            = error "findMax Nil"
  
  -- | /O(min(n,W))/. The maximal key of the map.
  findMax :: IntMap a -> (Key, a)
  findMax Nil = error $ "findMax: empty map has no maximal element"
  findMax (Tip k v) = (k,v)
  findMax (Bin _ m l r)
    |   m < 0   = go l
    | otherwise = go r
      where go (Tip k v)      = (k,v)
            go (Bin _ _ _ r') = go r'
            go Nil            = error "findMax Nil"
  
  -- | /O(min(n,W))/. Delete the minimal key. An error is thrown if the IntMap is already empty.
  -- Note, this is not the same behavior Map.
  deleteMin :: IntMap a -> IntMap a
  deleteMin = maybe Nil snd . minView
  
  -- | /O(min(n,W))/. Delete the maximal key. An error is thrown if the IntMap is already empty.
  -- Note, this is not the same behavior Map.
  deleteMax :: IntMap a -> IntMap a
  deleteMax = maybe Nil snd . maxView
  
  
  {--------------------------------------------------------------------
    Submap
  --------------------------------------------------------------------}
  -- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).
  -- Defined as (@'isProperSubmapOf' = 'isProperSubmapOfBy' (==)@).
  isProperSubmapOf :: Eq a => IntMap a -> IntMap a -> Bool
  isProperSubmapOf m1 m2
    = isProperSubmapOfBy (==) m1 m2
  
  {- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).
   The expression (@'isProperSubmapOfBy' f m1 m2@) returns 'True' when
   @m1@ and @m2@ are not equal,
   all keys in @m1@ are in @m2@, and when @f@ returns 'True' when
   applied to their respective values. For example, the following
   expressions are all 'True':
  
    > isProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
    > isProperSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
  
   But the following are all 'False':
  
    > isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
    > isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
    > isProperSubmapOfBy (<)  (fromList [(1,1)])       (fromList [(1,1),(2,2)])
  -}
  isProperSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Bool
  isProperSubmapOfBy predicate t1 t2
    = case submapCmp predicate t1 t2 of
        LT -> True
        _  -> False
  
  submapCmp :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Ordering
  submapCmp predicate t1@(Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
    | shorter m1 m2  = GT
    | shorter m2 m1  = submapCmpLt
    | p1 == p2       = submapCmpEq
    | otherwise      = GT  -- disjoint
    where
      submapCmpLt | nomatch p1 p2 m2  = GT
                  | zero p1 m2        = submapCmp predicate t1 l2
                  | otherwise         = submapCmp predicate t1 r2
      submapCmpEq = case (submapCmp predicate l1 l2, submapCmp predicate r1 r2) of
                      (GT,_ ) -> GT
                      (_ ,GT) -> GT
                      (EQ,EQ) -> EQ
                      _       -> LT
  
  submapCmp _         (Bin _ _ _ _) _  = GT
  submapCmp predicate (Tip kx x) (Tip ky y)
    | (kx == ky) && predicate x y = EQ
    | otherwise                   = GT  -- disjoint
  submapCmp predicate (Tip k x) t
    = case lookup k t of
       Just y | predicate x y -> LT
       _                      -> GT -- disjoint
  submapCmp _    Nil Nil = EQ
  submapCmp _    Nil _   = LT
  
  -- | /O(n+m)/. Is this a submap?
  -- Defined as (@'isSubmapOf' = 'isSubmapOfBy' (==)@).
  isSubmapOf :: Eq a => IntMap a -> IntMap a -> Bool
  isSubmapOf m1 m2
    = isSubmapOfBy (==) m1 m2
  
  {- | /O(n+m)/.
   The expression (@'isSubmapOfBy' f m1 m2@) returns 'True' if
   all keys in @m1@ are in @m2@, and when @f@ returns 'True' when
   applied to their respective values. For example, the following
   expressions are all 'True':
  
    > isSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
    > isSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
    > isSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
  
   But the following are all 'False':
  
    > isSubmapOfBy (==) (fromList [(1,2)]) (fromList [(1,1),(2,2)])
    > isSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
    > isSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
  -}
  isSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Bool
  isSubmapOfBy predicate t1@(Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
    | shorter m1 m2  = False
    | shorter m2 m1  = match p1 p2 m2 && (if zero p1 m2 then isSubmapOfBy predicate t1 l2
                                                        else isSubmapOfBy predicate t1 r2)
    | otherwise      = (p1==p2) && isSubmapOfBy predicate l1 l2 && isSubmapOfBy predicate r1 r2
  isSubmapOfBy _         (Bin _ _ _ _) _ = False
  isSubmapOfBy predicate (Tip k x) t     = case lookup k t of
                                           Just y  -> predicate x y
                                           Nothing -> False
  isSubmapOfBy _         Nil _           = True
  
  {--------------------------------------------------------------------
    Mapping
  --------------------------------------------------------------------}
  -- | /O(n)/. Map a function over all values in the map.
  --
  -- > map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]
  
  map :: (a -> b) -> IntMap a -> IntMap b
  map f t
    = case t of
        Bin p m l r -> Bin p m (map f l) (map f r)
        Tip k x     -> Tip k (f x)
        Nil         -> Nil
  
  -- | /O(n)/. Map a function over all values in the map.
  --
  -- > let f key x = (show key) ++ ":" ++ x
  -- > mapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")]
  
  mapWithKey :: (Key -> a -> b) -> IntMap a -> IntMap b
  mapWithKey f t
    = case t of
        Bin p m l r -> Bin p m (mapWithKey f l) (mapWithKey f r)
        Tip k x     -> Tip k (f k x)
        Nil         -> Nil
  
  -- | /O(n)/.
  -- @'traverseWithKey' f s == 'fromList' <$> 'traverse' (\(k, v) -> (,) k <$> f k v) ('toList' m)@
  -- That is, behaves exactly like a regular 'traverse' except that the traversing
  -- function also has access to the key associated with a value.
  --
  -- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
  -- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing
  {-# INLINE traverseWithKey #-}
  traverseWithKey :: Applicative t => (Key -> a -> t b) -> IntMap a -> t (IntMap b)
  traverseWithKey f = go
    where
      go Nil = pure Nil
      go (Tip k v) = Tip k <$> f k v
      go (Bin p m l r) = Bin p m <$> go l <*> 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 -> IntMap b -> (a,IntMap c)
  mapAccum f = mapAccumWithKey (\a' _ x -> f a' x)
  
  -- | /O(n)/. The function @'mapAccumWithKey'@ threads an accumulating
  -- argument through the map in ascending order of keys.
  --
  -- > let f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")
  -- > mapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])
  
  mapAccumWithKey :: (a -> Key -> b -> (a,c)) -> a -> IntMap b -> (a,IntMap c)
  mapAccumWithKey f a t
    = mapAccumL f a t
  
  -- | /O(n)/. The function @'mapAccumL'@ threads an accumulating
  -- argument through the map in ascending order of keys.
  mapAccumL :: (a -> Key -> b -> (a,c)) -> a -> IntMap b -> (a,IntMap c)
  mapAccumL f a t
    = case t of
        Bin p m l r -> let (a1,l') = mapAccumL f a l
                           (a2,r') = mapAccumL f a1 r
                       in (a2,Bin p m l' r')
        Tip k x     -> let (a',x') = f a k x in (a',Tip k x')
        Nil         -> (a,Nil)
  
  -- | /O(n)/. The function @'mapAccumR'@ threads an accumulating
  -- argument through the map in descending order of keys.
  mapAccumRWithKey :: (a -> Key -> b -> (a,c)) -> a -> IntMap b -> (a,IntMap c)
  mapAccumRWithKey f a t
    = case t of
        Bin p m l r -> let (a1,r') = mapAccumRWithKey f a r
                           (a2,l') = mapAccumRWithKey f a1 l
                       in (a2,Bin p m l' r')
        Tip k x     -> let (a',x') = f a k x in (a',Tip k x')
        Nil         -> (a,Nil)
  
  -- | /O(n*min(n,W))/.
  -- @'mapKeys' f s@ is the map obtained by applying @f@ to each key of @s@.
  --
  -- The size of the result may be smaller if @f@ maps two or more distinct
  -- keys to the same new key.  In this case the value at the greatest of the
  -- original keys is retained.
  --
  -- > mapKeys (+ 1) (fromList [(5,"a"), (3,"b")])                        == fromList [(4, "b"), (6, "a")]
  -- > mapKeys (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
  -- > mapKeys (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"
  
  mapKeys :: (Key->Key) -> IntMap a -> IntMap a
  mapKeys f = fromList . foldrWithKey (\k x xs -> (f k, x) : xs) []
  
  -- | /O(n*min(n,W))/.
  -- @'mapKeysWith' c f s@ is the map obtained by applying @f@ to each key of @s@.
  --
  -- The size of the result may be smaller if @f@ maps two or more distinct
  -- keys to the same new key.  In this case the associated values will be
  -- combined using @c@.
  --
  -- > mapKeysWith (++) (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
  -- > mapKeysWith (++) (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"
  
  mapKeysWith :: (a -> a -> a) -> (Key->Key) -> IntMap a -> IntMap a
  mapKeysWith c f = fromListWith c . foldrWithKey (\k x xs -> (f k, x) : xs) []
  
  -- | /O(n*min(n,W))/.
  -- @'mapKeysMonotonic' f s == 'mapKeys' f s@, but works only when @f@
  -- is strictly monotonic.
  -- That is, for any values @x@ and @y@, if @x@ < @y@ then @f x@ < @f y@.
  -- /The precondition is not checked./
  -- Semi-formally, we have:
  --
  -- > and [x < y ==> f x < f y | x <- ls, y <- ls]
  -- >                     ==> mapKeysMonotonic f s == mapKeys f s
  -- >     where ls = keys s
  --
  -- This means that @f@ maps distinct original keys to distinct resulting keys.
  -- This function has slightly better performance than 'mapKeys'.
  --
  -- > mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]
  
  mapKeysMonotonic :: (Key->Key) -> IntMap a -> IntMap a
  mapKeysMonotonic f = fromDistinctAscList . foldrWithKey (\k x xs -> (f k, x) : xs) []
  
  {--------------------------------------------------------------------
    Filter
  --------------------------------------------------------------------}
  -- | /O(n)/. Filter all values that satisfy some predicate.
  --
  -- > filter (> "a") (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
  -- > filter (> "x") (fromList [(5,"a"), (3,"b")]) == empty
  -- > filter (< "a") (fromList [(5,"a"), (3,"b")]) == empty
  
  filter :: (a -> Bool) -> IntMap a -> IntMap a
  filter p m
    = filterWithKey (\_ x -> p x) m
  
  -- | /O(n)/. Filter all keys\/values that satisfy some predicate.
  --
  -- > filterWithKey (\k _ -> k > 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
  
  filterWithKey :: (Key -> a -> Bool) -> IntMap a -> IntMap a
  filterWithKey predicate t
    = case t of
        Bin p m l r
          -> bin p m (filterWithKey predicate l) (filterWithKey predicate r)
        Tip k x
          | predicate k x -> t
          | otherwise     -> Nil
        Nil -> Nil
  
  -- | /O(n)/. Partition the map according to some predicate. The first
  -- map contains all elements that satisfy the predicate, the second all
  -- elements that fail the predicate. See also 'split'.
  --
  -- > partition (> "a") (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
  -- > partition (< "x") (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
  -- > partition (> "x") (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
  
  partition :: (a -> Bool) -> IntMap a -> (IntMap a,IntMap a)
  partition p m
    = partitionWithKey (\_ x -> p x) m
  
  -- | /O(n)/. Partition the map according to some predicate. The first
  -- map contains all elements that satisfy the predicate, the second all
  -- elements that fail the predicate. See also 'split'.
  --
  -- > partitionWithKey (\ k _ -> k > 3) (fromList [(5,"a"), (3,"b")]) == (singleton 5 "a", singleton 3 "b")
  -- > partitionWithKey (\ k _ -> k < 7) (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
  -- > partitionWithKey (\ k _ -> k > 7) (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
  
  partitionWithKey :: (Key -> a -> Bool) -> IntMap a -> (IntMap a,IntMap a)
  partitionWithKey predicate t
    = case t of
        Bin p m l r
          -> let (l1,l2) = partitionWithKey predicate l
                 (r1,r2) = partitionWithKey predicate r
             in (bin p m l1 r1, bin p m l2 r2)
        Tip k x
          | predicate k x -> (t,Nil)
          | otherwise     -> (Nil,t)
        Nil -> (Nil,Nil)
  
  -- | /O(n)/. Map values and collect the 'Just' results.
  --
  -- > let f x = if x == "a" then Just "new a" else Nothing
  -- > mapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a"
  
  mapMaybe :: (a -> Maybe b) -> IntMap a -> IntMap b
  mapMaybe f = 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 :: (Key -> a -> Maybe b) -> IntMap a -> IntMap b
  mapMaybeWithKey f (Bin p m l r)
    = bin p m (mapMaybeWithKey f l) (mapMaybeWithKey f r)
  mapMaybeWithKey f (Tip k x) = case f k x of
    Just y  -> Tip k y
    Nothing -> Nil
  mapMaybeWithKey _ Nil = Nil
  
  -- | /O(n)/. Map values and separate the 'Left' and 'Right' results.
  --
  -- > let f a = if a < "c" then Left a else Right a
  -- > mapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
  -- >     == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])
  -- >
  -- > mapEither (\ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
  -- >     == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
  
  mapEither :: (a -> Either b c) -> IntMap a -> (IntMap b, IntMap c)
  mapEither f m
    = mapEitherWithKey (\_ x -> f x) m
  
  -- | /O(n)/. Map keys\/values and separate the 'Left' and 'Right' results.
  --
  -- > let f k a = if k < 5 then Left (k * 2) else Right (a ++ a)
  -- > mapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
  -- >     == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])
  -- >
  -- > mapEitherWithKey (\_ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
  -- >     == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])
  
  mapEitherWithKey :: (Key -> a -> Either b c) -> IntMap a -> (IntMap b, IntMap c)
  mapEitherWithKey f (Bin p m l r)
    = (bin p m l1 r1, bin p m l2 r2)
    where
      (l1,l2) = mapEitherWithKey f l
      (r1,r2) = mapEitherWithKey f r
  mapEitherWithKey f (Tip k x) = case f k x of
    Left y  -> (Tip k y, Nil)
    Right z -> (Nil, Tip k z)
  mapEitherWithKey _ Nil = (Nil, Nil)
  
  -- | /O(min(n,W))/. The expression (@'split' k map@) is a pair @(map1,map2)@
  -- where all keys in @map1@ are lower than @k@ and all keys in
  -- @map2@ larger than @k@. Any key equal to @k@ is found in neither @map1@ nor @map2@.
  --
  -- > split 2 (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3,"b"), (5,"a")])
  -- > split 3 (fromList [(5,"a"), (3,"b")]) == (empty, singleton 5 "a")
  -- > split 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
  -- > split 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", empty)
  -- > split 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], empty)
  
  split :: Key -> IntMap a -> (IntMap a, IntMap a)
  split k t =
    case t of Bin _ m l r | m < 0 -> if k >= 0 -- handle negative numbers.
                                        then case go k l of (lt, gt) -> (union r lt, gt)
                                        else case go k r of (lt, gt) -> (lt, union gt l)
              _ -> go k t
    where
      go k' t'@(Bin p m l r) | nomatch k' p m = if k' > p then (t', Nil) else (Nil, t')
                             | zero k' m = case go k' l of (lt, gt) -> (lt, union gt r)
                             | otherwise = case go k' r of (lt, gt) -> (union l lt, gt)
      go k' t'@(Tip ky _) | k' > ky   = (t', Nil)
                          | k' < ky   = (Nil, t')
                          | otherwise = (Nil, Nil)
      go _ Nil = (Nil, Nil)
  
  -- | /O(min(n,W))/. Performs a 'split' but also returns whether the pivot
  -- key was found in the original map.
  --
  -- > splitLookup 2 (fromList [(5,"a"), (3,"b")]) == (empty, Nothing, fromList [(3,"b"), (5,"a")])
  -- > splitLookup 3 (fromList [(5,"a"), (3,"b")]) == (empty, Just "b", singleton 5 "a")
  -- > splitLookup 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Nothing, singleton 5 "a")
  -- > splitLookup 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Just "a", empty)
  -- > splitLookup 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], Nothing, empty)
  
  splitLookup :: Key -> IntMap a -> (IntMap a, Maybe a, IntMap a)
  splitLookup k t =
    case t of Bin _ m l r | m < 0 -> if k >= 0 -- handle negative numbers.
                                        then case go k l of (lt, fnd, gt) -> (union r lt, fnd, gt)
                                        else case go k r of (lt, fnd, gt) -> (lt, fnd, union gt l)
              _ -> go k t
    where
      go k' t'@(Bin p m l r) | nomatch k' p m = if k' > p then (t', Nothing, Nil) else (Nil, Nothing, t')
                             | zero k' m = case go k' l of (lt, fnd, gt) -> (lt, fnd, union gt r)
                             | otherwise = case go k' r of (lt, fnd, gt) -> (union l lt, fnd, gt)
      go k' t'@(Tip ky y) | k' > ky   = (t', Nothing, Nil)
                          | k' < ky   = (Nil, Nothing, t')
                          | otherwise = (Nil, Just y, Nil)
      go _ Nil = (Nil, Nothing, Nil)
  
  {--------------------------------------------------------------------
    Fold
  --------------------------------------------------------------------}
  -- | /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 -> IntMap a -> b
  foldr f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
    case t of Bin _ m l r | m < 0 -> go (go z l) r -- put negative numbers before
                          | otherwise -> go (go z r) l
              _ -> go z t
    where
      go z' Nil           = z'
      go z' (Tip _ x)     = f x z'
      go z' (Bin _ _ l r) = go (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 -> IntMap a -> b
  foldr' f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
    case t of Bin _ m l r | m < 0 -> go (go z l) r -- put negative numbers before
                          | otherwise -> go (go z r) l
              _ -> go z t
    where
      STRICT_1_OF_2(go)
      go z' Nil           = z'
      go z' (Tip _ x)     = f x z'
      go z' (Bin _ _ l r) = go (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 -> IntMap b -> a
  foldl f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
    case t of Bin _ m l r | m < 0 -> go (go z r) l -- put negative numbers before
                          | otherwise -> go (go z l) r
              _ -> go z t
    where
      go z' Nil           = z'
      go z' (Tip _ x)     = f z' x
      go z' (Bin _ _ l r) = go (go z' l) 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 -> IntMap b -> a
  foldl' f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
    case t of Bin _ m l r | m < 0 -> go (go z r) l -- put negative numbers before
                          | otherwise -> go (go z l) r
              _ -> go z t
    where
      STRICT_1_OF_2(go)
      go z' Nil           = z'
      go z' (Tip _ x)     = f z' x
      go z' (Bin _ _ l r) = go (go z' l) 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 :: (Int -> a -> b -> b) -> b -> IntMap a -> b
  foldrWithKey f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
    case t of Bin _ m l r | m < 0 -> go (go z l) r -- put negative numbers before
                          | otherwise -> go (go z r) l
              _ -> go z t
    where
      go z' Nil           = z'
      go z' (Tip kx x)    = f kx x z'
      go z' (Bin _ _ l r) = go (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' :: (Int -> a -> b -> b) -> b -> IntMap a -> b
  foldrWithKey' f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
    case t of Bin _ m l r | m < 0 -> go (go z l) r -- put negative numbers before
                          | otherwise -> go (go z r) l
              _ -> go z t
    where
      STRICT_1_OF_2(go)
      go z' Nil           = z'
      go z' (Tip kx x)    = f kx x z'
      go z' (Bin _ _ l r) = go (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 -> Int -> b -> a) -> a -> IntMap b -> a
  foldlWithKey f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
    case t of Bin _ m l r | m < 0 -> go (go z r) l -- put negative numbers before
                          | otherwise -> go (go z l) r
              _ -> go z t
    where
      go z' Nil           = z'
      go z' (Tip kx x)    = f z' kx x
      go z' (Bin _ _ l r) = go (go z' l) 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 -> Int -> b -> a) -> a -> IntMap b -> a
  foldlWithKey' f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
    case t of Bin _ m l r | m < 0 -> go (go z r) l -- put negative numbers before
                          | otherwise -> go (go z l) r
              _ -> go z t
    where
      STRICT_1_OF_2(go)
      go z' Nil           = z'
      go z' (Tip kx x)    = f z' kx x
      go z' (Bin _ _ l r) = go (go z' l) 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 :: IntMap 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  :: IntMap a -> [Key]
  keys = foldrWithKey (\k _ ks -> k : ks) []
  
  -- | /O(n)/. An alias for 'toAscList'. Returns all key\/value pairs in the
  -- map in ascending key order. Subject to list fusion.
  --
  -- > assocs (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
  -- > assocs empty == []
  
  assocs :: IntMap a -> [(Key,a)]
  assocs = toAscList
  
  -- | /O(n*min(n,W))/. The set of all keys of the map.
  --
  -- > keysSet (fromList [(5,"a"), (3,"b")]) == Data.IntSet.fromList [3,5]
  -- > keysSet empty == Data.IntSet.empty
  
  keysSet :: IntMap a -> IntSet.IntSet
  keysSet Nil = IntSet.Nil
  keysSet (Tip kx _) = IntSet.singleton kx
  keysSet (Bin p m l r)
    | m .&. IntSet.suffixBitMask == 0 = IntSet.Bin p m (keysSet l) (keysSet r)
    | otherwise = IntSet.Tip (p .&. IntSet.prefixBitMask) (computeBm (computeBm 0 l) r)
    where STRICT_1_OF_2(computeBm)
          computeBm acc (Bin _ _ l' r') = computeBm (computeBm acc l') r'
          computeBm acc (Tip kx _) = acc .|. IntSet.bitmapOf kx
          computeBm _   Nil = error "Data.IntSet.keysSet: Nil"
  
  -- | /O(n)/. Build a map from a set of keys and a function which for each key
  -- computes its value.
  --
  -- > fromSet (\k -> replicate k 'a') (Data.IntSet.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")]
  -- > fromSet undefined Data.IntSet.empty == empty
  
  fromSet :: (Key -> a) -> IntSet.IntSet -> IntMap a
  fromSet _ IntSet.Nil = Nil
  fromSet f (IntSet.Bin p m l r) = Bin p m (fromSet f l) (fromSet f r)
  fromSet f (IntSet.Tip kx bm) = buildTree f kx bm (IntSet.suffixBitMask + 1)
    where -- This is slightly complicated, as we to convert the dense
          -- representation of IntSet into tree representation of IntMap.
          --
          -- We are given a nonzero bit mask 'bmask' of 'bits' bits with prefix 'prefix'.
          -- We split bmask into halves corresponding to left and right subtree.
          -- If they are both nonempty, we create a Bin node, otherwise exactly
          -- one of them is nonempty and we construct the IntMap from that half.
          buildTree g prefix bmask bits = prefix `seq` bmask `seq` case bits of
            0 -> Tip prefix (g prefix)
            _ -> case intFromNat ((natFromInt bits) `shiftRL` 1) of
                   bits2 | bmask .&. ((1 `shiftLL` bits2) - 1) == 0 ->
                             buildTree g (prefix + bits2) (bmask `shiftRL` bits2) bits2
                         | (bmask `shiftRL` bits2) .&. ((1 `shiftLL` bits2) - 1) == 0 ->
                             buildTree g prefix bmask bits2
                         | otherwise ->
                             Bin prefix bits2 (buildTree g prefix bmask bits2) (buildTree g (prefix + bits2) (bmask `shiftRL` bits2) bits2)
  
  {--------------------------------------------------------------------
    Lists
  --------------------------------------------------------------------}
  -- | /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 :: IntMap a -> [(Key,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 :: IntMap a -> [(Key,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 :: IntMap a -> [(Key,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 :: (Key -> a -> b -> b) -> b -> IntMap a -> b
  foldrFB = foldrWithKey
  {-# INLINE[0] foldrFB #-}
  foldlFB :: (a -> Key -> b -> a) -> a -> IntMap 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 "IntMap.elems" [~1] forall m . elems m = build (\c n -> foldrFB (\_ x xs -> c x xs) n m) #-}
  {-# RULES "IntMap.elemsBack" [1] foldrFB (\_ x xs -> x : xs) [] = elems #-}
  {-# RULES "IntMap.keys" [~1] forall m . keys m = build (\c n -> foldrFB (\k _ xs -> c k xs) n m) #-}
  {-# RULES "IntMap.keysBack" [1] foldrFB (\k _ xs -> k : xs) [] = keys #-}
  {-# RULES "IntMap.toAscList" [~1] forall m . toAscList m = build (\c n -> foldrFB (\k x xs -> c (k,x) xs) n m) #-}
  {-# RULES "IntMap.toAscListBack" [1] foldrFB (\k x xs -> (k, x) : xs) [] = toAscList #-}
  {-# RULES "IntMap.toDescList" [~1] forall m . toDescList m = build (\c n -> foldlFB (\xs k x -> c (k,x) xs) n m) #-}
  {-# RULES "IntMap.toDescListBack" [1] foldlFB (\xs k x -> (k, x) : xs) [] = toDescList #-}
  #endif
  
  
  -- | /O(n*min(n,W))/. Create a map from a list of key\/value pairs.
  --
  -- > fromList [] == empty
  -- > fromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")]
  -- > fromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")]
  
  fromList :: [(Key,a)] -> IntMap a
  fromList xs
    = foldlStrict ins empty xs
    where
      ins t (k,x)  = insert k x t
  
  -- | /O(n*min(n,W))/. Create a map from a list of key\/value pairs with a combining function. See also 'fromAscListWith'.
  --
  -- > fromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"c")] == fromList [(3, "ab"), (5, "cba")]
  -- > fromListWith (++) [] == empty
  
  fromListWith :: (a -> a -> a) -> [(Key,a)] -> IntMap a
  fromListWith f xs
    = fromListWithKey (\_ x y -> f x y) xs
  
  -- | /O(n*min(n,W))/. Build a map from a list of key\/value pairs with a combining function. See also fromAscListWithKey'.
  --
  -- > let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
  -- > fromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"c")] == fromList [(3, "3:a|b"), (5, "5:c|5:b|a")]
  -- > fromListWithKey f [] == empty
  
  fromListWithKey :: (Key -> a -> a -> a) -> [(Key,a)] -> IntMap a
  fromListWithKey f xs
    = foldlStrict ins empty xs
    where
      ins t (k,x) = insertWithKey f k x t
  
  -- | /O(n)/. Build a map from a list of key\/value pairs where
  -- the keys are in ascending order.
  --
  -- > fromAscList [(3,"b"), (5,"a")]          == fromList [(3, "b"), (5, "a")]
  -- > fromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")]
  
  fromAscList :: [(Key,a)] -> IntMap a
  fromAscList xs
    = fromAscListWithKey (\_ x _ -> x) xs
  
  -- | /O(n)/. Build a map from a list of key\/value pairs where
  -- the keys are in ascending order, with a combining function on equal keys.
  -- /The precondition (input list is ascending) is not checked./
  --
  -- > fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]
  
  fromAscListWith :: (a -> a -> a) -> [(Key,a)] -> IntMap a
  fromAscListWith f xs
    = fromAscListWithKey (\_ x y -> f x y) xs
  
  -- | /O(n)/. Build a map from a list of key\/value pairs where
  -- the keys are in ascending order, with a combining function on equal keys.
  -- /The precondition (input list is ascending) is not checked./
  --
  -- > let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
  -- > fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "5:b|a")]
  
  fromAscListWithKey :: (Key -> a -> a -> a) -> [(Key,a)] -> IntMap a
  fromAscListWithKey _ []         = Nil
  fromAscListWithKey f (x0 : xs0) = fromDistinctAscList (combineEq x0 xs0)
    where
      -- [combineEq f xs] combines equal elements with function [f] in an ordered list [xs]
      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
  
  -- | /O(n)/. Build a map from a list of key\/value pairs where
  -- the keys are in ascending order and all distinct.
  -- /The precondition (input list is strictly ascending) is not checked./
  --
  -- > fromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]
  
  fromDistinctAscList :: [(Key,a)] -> IntMap a
  fromDistinctAscList []         = Nil
  fromDistinctAscList (z0 : zs0) = work z0 zs0 Nada
    where
      work (kx,vx) []            stk = finish kx (Tip kx vx) stk
      work (kx,vx) (z@(kz,_):zs) stk = reduce z zs (branchMask kx kz) kx (Tip kx vx) stk
  
      reduce :: (Key,a) -> [(Key,a)] -> Mask -> Prefix -> IntMap a -> Stack a -> IntMap a
      reduce z zs _ px tx Nada = work z zs (Push px tx Nada)
      reduce z zs m px tx stk@(Push py ty stk') =
          let mxy = branchMask px py
              pxy = mask px mxy
          in  if shorter m mxy
                   then reduce z zs m pxy (Bin pxy mxy ty tx) stk'
                   else work z zs (Push px tx stk)
  
      finish _  t  Nada = t
      finish px tx (Push py ty stk) = finish p (join py ty px tx) stk
          where m = branchMask px py
                p = mask px m
  
  data Stack a = Push {-# UNPACK #-} !Prefix !(IntMap a) !(Stack a) | Nada
  
  
  {--------------------------------------------------------------------
    Eq
  --------------------------------------------------------------------}
  instance Eq a => Eq (IntMap a) where
    t1 == t2  = equal t1 t2
    t1 /= t2  = nequal t1 t2
  
  equal :: Eq a => IntMap a -> IntMap a -> Bool
  equal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
    = (m1 == m2) && (p1 == p2) && (equal l1 l2) && (equal r1 r2)
  equal (Tip kx x) (Tip ky y)
    = (kx == ky) && (x==y)
  equal Nil Nil = True
  equal _   _   = False
  
  nequal :: Eq a => IntMap a -> IntMap a -> Bool
  nequal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
    = (m1 /= m2) || (p1 /= p2) || (nequal l1 l2) || (nequal r1 r2)
  nequal (Tip kx x) (Tip ky y)
    = (kx /= ky) || (x/=y)
  nequal Nil Nil = False
  nequal _   _   = True
  
  {--------------------------------------------------------------------
    Ord
  --------------------------------------------------------------------}
  
  instance Ord a => Ord (IntMap a) where
      compare m1 m2 = compare (toList m1) (toList m2)
  
  {--------------------------------------------------------------------
    Functor
  --------------------------------------------------------------------}
  
  instance Functor IntMap where
      fmap = map
  
  {--------------------------------------------------------------------
    Show
  --------------------------------------------------------------------}
  
  instance Show a => Show (IntMap a) where
    showsPrec d m   = showParen (d > 10) $
      showString "fromList " . shows (toList m)
  
  {--------------------------------------------------------------------
    Read
  --------------------------------------------------------------------}
  instance (Read e) => Read (IntMap e) where
  #ifdef __GLASGOW_HASKELL__
    readPrec = parens $ prec 10 $ do
      Ident "fromList" <- lexP
      xs <- readPrec
      return (fromList xs)
  
    readListPrec = readListPrecDefault
  #else
    readsPrec p = readParen (p > 10) $ \ r -> do
      ("fromList",s) <- lex r
      (xs,t) <- reads s
      return (fromList xs,t)
  #endif
  
  {--------------------------------------------------------------------
    Typeable
  --------------------------------------------------------------------}
  
  #include "Typeable.h"
  INSTANCE_TYPEABLE1(IntMap,intMapTc,"IntMap")
  
  {--------------------------------------------------------------------
    Helpers
  --------------------------------------------------------------------}
  {--------------------------------------------------------------------
    Join
  --------------------------------------------------------------------}
  join :: Prefix -> IntMap a -> Prefix -> IntMap a -> IntMap a
  join p1 t1 p2 t2
    | zero p1 m = Bin p m t1 t2
    | otherwise = Bin p m t2 t1
    where
      m = branchMask p1 p2
      p = mask p1 m
  {-# INLINE join #-}
  
  {--------------------------------------------------------------------
    @bin@ assures that we never have empty trees within a tree.
  --------------------------------------------------------------------}
  bin :: Prefix -> Mask -> IntMap a -> IntMap a -> IntMap a
  bin _ _ l Nil = l
  bin _ _ Nil r = r
  bin p m l r   = Bin p m l r
  {-# INLINE bin #-}
  
  
  {--------------------------------------------------------------------
    Endian independent bit twiddling
  --------------------------------------------------------------------}
  zero :: Key -> Mask -> Bool
  zero i m
    = (natFromInt i) .&. (natFromInt m) == 0
  {-# INLINE zero #-}
  
  nomatch,match :: Key -> Prefix -> Mask -> Bool
  nomatch i p m
    = (mask i m) /= p
  {-# INLINE nomatch #-}
  
  match i p m
    = (mask i m) == p
  {-# INLINE match #-}
  
  mask :: Key -> Mask -> Prefix
  mask i m
    = maskW (natFromInt i) (natFromInt m)
  {-# INLINE mask #-}
  
  
  {--------------------------------------------------------------------
    Big endian operations
  --------------------------------------------------------------------}
  maskW :: Nat -> Nat -> Prefix
  maskW i m
    = intFromNat (i .&. (complement (m-1) `xor` m))
  {-# INLINE maskW #-}
  
  shorter :: Mask -> Mask -> Bool
  shorter m1 m2
    = (natFromInt m1) > (natFromInt m2)
  {-# INLINE shorter #-}
  
  branchMask :: Prefix -> Prefix -> Mask
  branchMask p1 p2
    = intFromNat (highestBitMask (natFromInt p1 `xor` natFromInt p2))
  {-# INLINE branchMask #-}
  
  {----------------------------------------------------------------------
    Finding the highest bit (mask) in a word [x] can be done efficiently in
    three ways:
    * convert to a floating point value and the mantissa tells us the
      [log2(x)] that corresponds with the highest bit position. The mantissa
      is retrieved either via the standard C function [frexp] or by some bit
      twiddling on IEEE compatible numbers (float). Note that one needs to
      use at least [double] precision for an accurate mantissa of 32 bit
      numbers.
    * use bit twiddling, a logarithmic sequence of bitwise or's and shifts (bit).
    * use processor specific assembler instruction (asm).
  
    The most portable way would be [bit], but is it efficient enough?
    I have measured the cycle counts of the different methods on an AMD
    Athlon-XP 1800 (~ Pentium III 1.8Ghz) using the RDTSC instruction:
  
    highestBitMask: method  cycles
                    --------------
                     frexp   200
                     float    33
                     bit      11
                     asm      12
  
    highestBit:     method  cycles
                    --------------
                     frexp   195
                     float    33
                     bit      11
                     asm      11
  
    Wow, the bit twiddling is on today's RISC like machines even faster
    than a single CISC instruction (BSR)!
  ----------------------------------------------------------------------}
  
  {----------------------------------------------------------------------
    [highestBitMask] returns a word where only the highest bit is set.
    It is found by first setting all bits in lower positions than the
    highest bit and than taking an exclusive or with the original value.
    Allthough the function may look expensive, GHC compiles this into
    excellent C code that subsequently compiled into highly efficient
    machine code. The algorithm is derived from Jorg Arndt's FXT library.
  ----------------------------------------------------------------------}
  highestBitMask :: Nat -> Nat
  highestBitMask x0
    = case (x0 .|. shiftRL x0 1) of
       x1 -> case (x1 .|. shiftRL x1 2) of
        x2 -> case (x2 .|. shiftRL x2 4) of
         x3 -> case (x3 .|. shiftRL x3 8) of
          x4 -> case (x4 .|. shiftRL x4 16) of
  #if !(defined(__GLASGOW_HASKELL__) && WORD_SIZE_IN_BITS==32)
           x5 -> case (x5 .|. shiftRL x5 32) of   -- for 64 bit platforms
  #endif
            x6 -> (x6 `xor` (shiftRL x6 1))
  {-# INLINE highestBitMask #-}
  
  
  {--------------------------------------------------------------------
    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 #-}
  
  {--------------------------------------------------------------------
    Debugging
  --------------------------------------------------------------------}
  -- | /O(n)/. Show the tree that implements the map. The tree is shown
  -- in a compressed, hanging format.
  showTree :: Show a => IntMap a -> String
  showTree s
    = showTreeWith True False s
  
  
  {- | /O(n)/. The expression (@'showTreeWith' hang wide map@) shows
   the tree that implements the map. If @hang@ is
   'True', a /hanging/ tree is shown otherwise a rotated tree is shown. If
   @wide@ is 'True', an extra wide version is shown.
  -}
  showTreeWith :: Show a => Bool -> Bool -> IntMap a -> String
  showTreeWith hang wide t
    | hang      = (showsTreeHang wide [] t) ""
    | otherwise = (showsTree wide [] [] t) ""
  
  showsTree :: Show a => Bool -> [String] -> [String] -> IntMap a -> ShowS
  showsTree wide lbars rbars t
    = case t of
        Bin p m l r
            -> showsTree wide (withBar rbars) (withEmpty rbars) r .
               showWide wide rbars .
               showsBars lbars . showString (showBin p m) . showString "\n" .
               showWide wide lbars .
               showsTree wide (withEmpty lbars) (withBar lbars) l
        Tip k x
            -> showsBars lbars . showString " " . shows k . showString ":=" . shows x . showString "\n"
        Nil -> showsBars lbars . showString "|\n"
  
  showsTreeHang :: Show a => Bool -> [String] -> IntMap a -> ShowS
  showsTreeHang wide bars t
    = case t of
        Bin p m l r
            -> showsBars bars . showString (showBin p m) . showString "\n" .
               showWide wide bars .
               showsTreeHang wide (withBar bars) l .
               showWide wide bars .
               showsTreeHang wide (withEmpty bars) r
        Tip k x
            -> showsBars bars . showString " " . shows k . showString ":=" . shows x . showString "\n"
        Nil -> showsBars bars . showString "|\n"
  
  showBin :: Prefix -> Mask -> String
  showBin _ _
    = "*" -- ++ show (p,m)
  
  showWide :: Bool -> [String] -> String -> String
  showWide wide bars
    | wide      = showString (concat (reverse bars)) . showString "|\n"
    | otherwise = id
  
  showsBars :: [String] -> ShowS
  showsBars bars
    = case bars of
        [] -> id
        _  -> showString (concat (reverse (tail bars))) . showString node
  
  node :: String
  node           = "+--"
  
  withBar, withEmpty :: [String] -> [String]
  withBar bars   = "|  ":bars
  withEmpty bars = "   ":bars
  