content: {-# LANGUAGE CPP,NoBangPatterns,MagicHash #-}
  -----------------------------------------------------------------------------
  -- |
  -- Module      :  Data.IntBag
  -- Copyright   :  (c) Daan Leijen 2002
  -- Copyright   :  (c) John Meacham 2007
  -- License     :  BSD-style
  -- Maintainer  :  libraries@haskell.org
  -- Stability   :  provisional
  -- Portability :  portable
  --
  -- An efficient implementation of maps from integer keys to integers.
  --
  -- modified from Data.IntMap
  --
  
  module Util.IntBag  (
              -- * Map type
                IntBag, Key          -- instance Eq,Show
  
              -- * Operators
              , (!)
  
              --, (\\)
  
              -- * Query
              , null
              , size
  --            , member
  --            , notMember
  --	    , lookup
  --            , findWithDefault
  
              -- * Construction
              , empty
              , singleton
              , msingleton
  
              -- ** Insertion
              , insert
  --            , insertWith, insertWithKey, insertLookupWithKey
  
              -- ** Delete\/Update
              , delete
  --            , adjust
  --            , adjustWithKey
  --            , update
  --            , updateWithKey
  --            , updateLookupWithKey
  --            , alter
  --
              -- * Combine
  
              -- ** Union
              , union
  --            , unionWith
  --            , unionWithKey
  --            , unions
  --            , unionsWith
  
              -- ** Difference
  --            , difference
  --            , differenceWith
  --            , differenceWithKey
  --
  --            -- ** Intersection
  --            , intersection
  --            , intersectionWith
  --            , intersectionWithKey
  
              -- * Traversal
              -- ** Map
  --            , map
  --            , mapWithKey
  --            , mapAccum
  --            , mapAccumWithKey
  --
              -- ** Fold
              , fold
              , foldWithKey
  
              -- * Conversion
  --            , elems
  --            , keys
  --	    , keysSet
              , assocs
  
              -- ** Lists
              , toList
              , fromList
  --            , fromListWith
  --            , fromListWithKey
  
              -- ** Ordered lists
              , toAscList
  --            , fromAscList
  --            , fromAscListWith
  --            , fromAscListWithKey
  --            , fromDistinctAscList
  
              -- * Filter
  --            , filter
  --            , filterWithKey
  --            , partition
  --            , partitionWithKey
  --
  --            , mapMaybe
  --            , mapMaybeWithKey
  --            , mapEither
  --            , mapEitherWithKey
  --
  --            , split
  --            , splitLookup
  --
              -- * Submap
  --            , isSubmapOf, isSubmapOfBy
  --            , isProperSubmapOf, isProperSubmapOfBy
  
              -- * Debugging
  --            , showTree
  --            , showTreeWith
              ) where
  
  
  import Prelude hiding (lookup,map,filter,foldr,foldl,null)
  import Data.Bits
  import Data.Monoid (Monoid(..))
  
  
  
  #if __GLASGOW_HASKELL__ >= 503
  import GHC.Word
  import GHC.Exts (  Int(..), shiftRL# )
  #elif __GLASGOW_HASKELL__
  import Word
  import GlaExts ( Word(..), Int(..), shiftRL# )
  #else
  import Data.Word
  #endif
  
  --infixl 9 \\{-This comment teaches CPP correct behaviour -}
  
  -- A "Nat" is a natural machine word (an unsigned Int)
  type Nat = Word
  
  natFromInt :: Key -> Nat
  natFromInt i = fromIntegral i
  
  intFromNat :: Nat -> Key
  intFromNat w = fromIntegral w
  
  shiftRL :: Nat -> Key -> Nat
  #if __GLASGOW_HASKELL__
  {--------------------------------------------------------------------
    GHC: use unboxing to get @shiftRL@ inlined.
  --------------------------------------------------------------------}
  shiftRL (W# x) (I# i)
    = W# (shiftRL# x i)
  #else
  shiftRL x i   = shiftR x i
  #endif
  
  {--------------------------------------------------------------------
    Operators
  --------------------------------------------------------------------}
  
  -- | /O(min(n,W))/. Find the value at a key.
  -- Calls 'error' when the element can not be found.
  
  (!) :: IntBag-> Key -> Int
  m ! k    = find' k m
  
  -- | /O(n+m)/. See 'difference'.
  --(\\) :: IntBag -> IntBag -> IntBag
  --m1 \\ m2 = difference m1 m2
  
  {--------------------------------------------------------------------
    Types
  --------------------------------------------------------------------}
  -- | A map of integers to values @a@.
  data IntBag = Nil
                | Tip {-# UNPACK #-} !Key {-# UNPACK #-} !Int
                | Bin {-# UNPACK #-} !Prefix {-# UNPACK #-} !Mask !IntBag !IntBag
  
  type Prefix = Int
  type Mask   = Int
  type Key    = Int
  
  instance Monoid IntBag where
      mempty  = empty
      mappend = union
      mconcat = unions
  
  
  
  {--------------------------------------------------------------------
    Query
  --------------------------------------------------------------------}
  -- | /O(1)/. Is the map empty?
  null :: IntBag -> Bool
  null Nil   = True
  null other = False
  
  -- | /O(n)/. Number of elements in the map.
  size :: IntBag -> Int
  size t
    = case t of
        Bin p m l r -> size l + size r
        Tip k x -> x
        Nil     -> 0
  
  -- | /O(min(n,W))/. Is the key a member of the map?
  --member :: Key -> IntBag -> Bool
  --member k m
  --  = case lookup k m of
  --      Nothing -> False
  --      Just x  -> True
  
  -- | /O(log n)/. Is the key not a member of the map?
  --notMember :: Key -> IntBag -> Bool
  --notMember k m = not $ member k m
  
  -- | /O(min(n,W))/. Lookup the value at a key in the map.
  --lookup :: (Monad m) => Key -> IntBag -> m Int
  --lookup k t = case lookup' k t of
  --    Just x -> return x
  --    Nothing -> fail "Data.IntBag.lookup: Key not found"
  --
  --lookup' :: Key -> IntBag -> Maybe Int
  --lookup' k t
  --  = let nk = natFromInt k  in seq nk (lookupN nk t)
  --
  --
  lookupN :: Nat -> IntBag -> Int
  lookupN k t
    = case t of
        Bin p m l r
          | zeroN k (natFromInt m) -> lookupN k l
          | otherwise              -> lookupN k r
        Tip kx x
          | (k == natFromInt kx)  -> x
          | otherwise             -> 0
        Nil -> 0
  
  find' :: Key -> IntBag -> Int
  find' k m  = lookupN (natFromInt k) m
  
  
  -- | /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 :: Int -> Key -> IntBag -> Int
  --findWithDefault def k m
  --  = case lookup k m of
  --      Nothing -> def
  --      Just x  -> x
  
  {--------------------------------------------------------------------
    Construction
  --------------------------------------------------------------------}
  -- | /O(1)/. The empty map.
  empty :: IntBag
  empty = Nil
  
  -- | /O(1)/. A map of one element.
  singleton :: Key -> IntBag
  singleton k = Tip k 1
  
  msingleton :: Key -> Int -> IntBag
  msingleton k x | x > 0 = Tip k x
                 | otherwise = Nil
  
  {--------------------------------------------------------------------
    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
  -- added to the supplied value, i.e. 'insert' is equivalent to
  -- @'insertWith' 'const'@.
  insert :: Key -> Int -> IntBag -> IntBag
  insert k x t | k `seq` x < 0 = delete k (negate x) t
  insert _ 0 t = t
  insert k x t = f t where
      f t = case t of
        Bin p m l r
          | nomatch k p m -> join k (Tip k x) p t
          | zero k m      -> Bin p m (f l) r
          | otherwise     -> Bin p m l (f r)
        Tip ky y
          | k==ky         -> Tip k (x + y)
          | otherwise     -> join k (Tip k x) ky t
        Nil -> Tip k x
  
  
  
  {--------------------------------------------------------------------
    Deletion
    [delete] is the inlined version of [deleteWith (\k x -> Nothing)]
  --------------------------------------------------------------------}
  -- | /O(min(n,W))/. Delete a key and its value from the map. When the key is not
  -- a member of the map, the original map is returned.
  delete :: Key -> Int -> IntBag -> IntBag
  delete k 0 t | k `seq` True = t
  delete k x t | x < 0 = insert k (negate x) t
  delete k x t = f t where
      f t = case t of
        Bin p m l r
          | nomatch k p m -> t
          | zero k m      -> bin p m (f l) r
          | otherwise     -> bin p m l (f r)
        Tip ky y
          | k==ky         -> if y < x then Nil else Tip ky (y - x)
          | 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 ::  (a -> a) -> Key -> IntBag -> IntBag
  --adjust f k m
  --  = adjustWithKey (\k 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.
  --adjustWithKey ::  (Key -> a -> a) -> Key -> IntBag -> IntBag
  --adjustWithKey f k m
  --  = updateWithKey (\k x -> Just (f k x)) k m
  --
  ---- | /O(min(n,W))/. The expression (@'update' f k map@) updates the value @x@
  ---- at @k@ (if it is in the map). If (@f x@) is 'Nothing', the element is
  ---- deleted. If it is (@'Just' y@), the key @k@ is bound to the new value @y@.
  --update ::  (a -> Maybe a) -> Key -> IntBag -> IntBag
  --update f k m
  --  = updateWithKey (\k x -> f x) k m
  --
  ---- | /O(min(n,W))/. The expression (@'update' f k map@) updates the value @x@
  ---- at @k@ (if it is in the map). If (@f k x@) is 'Nothing', the element is
  ---- deleted. If it is (@'Just' y@), the key @k@ is bound to the new value @y@.
  --updateWithKey ::  (Key -> a -> Maybe a) -> Key -> IntBag -> IntBag
  --updateWithKey f k t
  --  = case t of
  --      Bin p m l r
  --        | nomatch k p m -> t
  --        | zero k m      -> bin p m (updateWithKey f k l) r
  --        | otherwise     -> bin p m l (updateWithKey f k r)
  --      Tip ky y
  --        | k==ky         -> case (f k y) of
  --                             Just y' -> Tip ky y'
  --                             Nothing -> Nil
  --        | otherwise     -> t
  --      Nil -> Nil
  --
  ---- | /O(min(n,W))/. Lookup and update.
  --updateLookupWithKey ::  (Key -> a -> Maybe a) -> Key -> IntBag -> (Maybe a,IntBag)
  --updateLookupWithKey f k t
  --  = case t of
  --      Bin p m l r
  --        | nomatch k p m -> (Nothing,t)
  --        | zero k m      -> let (found,l') = updateLookupWithKey f k l in (found,bin p m l' r)
  --        | otherwise     -> let (found,r') = updateLookupWithKey f k r in (found,bin p m l r')
  --      Tip ky y
  --        | k==ky         -> case (f k y) of
  --                             Just y' -> (Just y,Tip ky y')
  --                             Nothing -> (Just y,Nil)
  --        | otherwise     -> (Nothing,t)
  --      Nil -> (Nothing,Nil)
  --
  --
  --
  ---- | /O(log n)/. The expression (@'alter' f k map@) alters the value @x@ at @k@, or absence thereof.
  ---- 'alter' can be used to insert, delete, or update a value in a 'Map'.
  ---- In short : @'lookup' k ('alter' f k m) = f ('lookup' k m)@
  --alter f k t
  --  = case t of
  --      Bin p m l r
  --        | nomatch k p m -> case f Nothing of
  --                             Nothing -> t
  --                             Just x -> join k (Tip k x) p t
  --        | zero k m      -> bin p m (alter f k l) r
  --        | otherwise     -> bin p m l (alter f k r)
  --      Tip ky y
  --        | k==ky         -> case f (Just y) of
  --                             Just x -> Tip ky x
  --                             Nothing -> Nil
  --        | otherwise     -> case f Nothing of
  --                             Just x -> join k (Tip k x) ky t
  --                             Nothing -> Tip ky y
  --      Nil               -> case f Nothing of
  --                             Just x -> Tip k x
  --                             Nothing -> Nil
  --
  
  {--------------------------------------------------------------------
    Union
  --------------------------------------------------------------------}
  -- | The union of a list of maps.
  unions :: [IntBag] -> IntBag
  unions xs
    = foldlStrict union empty xs
  
  -- | The union of a list of maps, with a combining operation
  --unionsWith :: (Int->Int->Int) -> [IntBag] -> IntBag
  --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 :: IntBag -> IntBag -> IntBag
  union t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
    | shorter m1 m2  = union1
    | shorter m2 m1  = union2
    | p1 == p2       = Bin p1 m1 (union l1 l2) (union r1 r2)
    | otherwise      = join p1 t1 p2 t2
    where
      union1  | nomatch p2 p1 m1  = join p1 t1 p2 t2
              | zero p2 m1        = Bin p1 m1 (union l1 t2) r1
              | otherwise         = Bin p1 m1 l1 (union r1 t2)
  
      union2  | nomatch p1 p2 m2  = join p1 t1 p2 t2
              | zero p1 m2        = Bin p2 m2 (union t1 l2) r2
              | otherwise         = Bin p2 m2 l2 (union t1 r2)
  
  union (Tip k x) t = insert k x t
  union t (Tip k x) = insert k x t
  --union t (Tip k x) = insertWith (\x y -> y) k x t  -- right bias
  union Nil t       = t
  union t Nil       = t
  
  -- | /O(n+m)/. The union with a combining function.
  --unionWith :: (a -> a -> a) -> IntMap a -> IntMap a -> IntMap a
  --unionWith f m1 m2
  --  = unionWithKey (\k x y -> f x y) m1 m2
  --
  ---- | /O(n+m)/. The union with a combining function.
  --unionWithKey :: (Key -> a -> a -> a) -> IntMap a -> IntMap a -> IntMap a
  --unionWithKey f t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
  --  | shorter m1 m2  = union1
  --  | shorter m2 m1  = union2
  --  | p1 == p2       = Bin p1 m1 (unionWithKey f l1 l2) (unionWithKey f r1 r2)
  --  | otherwise      = join p1 t1 p2 t2
  --  where
  --    union1  | nomatch p2 p1 m1  = join p1 t1 p2 t2
  --            | zero p2 m1        = Bin p1 m1 (unionWithKey f l1 t2) r1
  --            | otherwise         = Bin p1 m1 l1 (unionWithKey f r1 t2)
  --
  --    union2  | nomatch p1 p2 m2  = join p1 t1 p2 t2
  --            | zero p1 m2        = Bin p2 m2 (unionWithKey f t1 l2) r2
  --            | otherwise         = Bin p2 m2 l2 (unionWithKey f t1 r2)
  --
  --unionWithKey f (Tip k x) t = insertWithKey f k x t
  --unionWithKey f t (Tip k x) = insertWithKey (\k x y -> f k y x) k x t  -- right bias
  --unionWithKey f Nil t  = t
  --unionWithKey f t Nil  = t
  
  {--------------------------------------------------------------------
    Difference
  --------------------------------------------------------------------}
  -- | /O(n+m)/. Difference between two maps (based on keys).
  --difference :: IntMap a -> IntMap b -> IntMap a
  --difference t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
  --  | shorter m1 m2  = difference1
  --  | shorter m2 m1  = difference2
  --  | p1 == p2       = bin p1 m1 (difference l1 l2) (difference r1 r2)
  --  | otherwise      = t1
  --  where
  --    difference1 | nomatch p2 p1 m1  = t1
  --                | zero p2 m1        = bin p1 m1 (difference l1 t2) r1
  --                | otherwise         = bin p1 m1 l1 (difference r1 t2)
  --
  --    difference2 | nomatch p1 p2 m2  = t1
  --                | zero p1 m2        = difference t1 l2
  --                | otherwise         = difference t1 r2
  --
  --difference t1@(Tip k x) t2
  --  | member k t2  = Nil
  --  | otherwise    = t1
  --
  --difference Nil t       = Nil
  --difference t (Tip k x) = delete k t
  --difference t Nil       = t
  --
  ---- | /O(n+m)/. Difference with a combining function.
  --differenceWith :: (a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap a
  --differenceWith f m1 m2
  --  = differenceWithKey (\k 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@.
  --differenceWithKey :: (Key -> a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap a
  --differenceWithKey f t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
  --  | shorter m1 m2  = difference1
  --  | shorter m2 m1  = difference2
  --  | p1 == p2       = bin p1 m1 (differenceWithKey f l1 l2) (differenceWithKey f r1 r2)
  --  | otherwise      = t1
  --  where
  --    difference1 | nomatch p2 p1 m1  = t1
  --                | zero p2 m1        = bin p1 m1 (differenceWithKey f l1 t2) r1
  --                | otherwise         = bin p1 m1 l1 (differenceWithKey f r1 t2)
  --
  --    difference2 | nomatch p1 p2 m2  = t1
  --                | zero p1 m2        = differenceWithKey f t1 l2
  --                | otherwise         = differenceWithKey f t1 r2
  --
  --differenceWithKey f t1@(Tip k x) t2
  --  = case lookup k t2 of
  --      Just y  -> case f k x y of
  --                   Just y' -> Tip k y'
  --                   Nothing -> Nil
  --      Nothing -> t1
  --
  --differenceWithKey f Nil t       = Nil
  --differenceWithKey f t (Tip k y) = updateWithKey (\k x -> f k x y) k t
  --differenceWithKey f t Nil       = t
  --
  
  {--------------------------------------------------------------------
    Intersection
  --------------------------------------------------------------------}
  ---- | /O(n+m)/. The (left-biased) intersection of two maps (based on keys).
  --intersection :: IntBag -> IntBag -> IntBag
  --intersection t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
  --  | shorter m1 m2  = intersection1
  --  | shorter m2 m1  = intersection2
  --  | p1 == p2       = bin p1 m1 (intersection l1 l2) (intersection r1 r2)
  --  | otherwise      = Nil
  --  where
  --    intersection1 | nomatch p2 p1 m1  = Nil
  --                  | zero p2 m1        = intersection l1 t2
  --                  | otherwise         = intersection r1 t2
  --
  --    intersection2 | nomatch p1 p2 m2  = Nil
  --                  | zero p1 m2        = intersection t1 l2
  --                  | otherwise         = intersection t1 r2
  --
  --intersection t1@(Tip k x) t2
  --  | member k t2  = t1
  --  | otherwise    = Nil
  --intersection t (Tip k x)
  --  = case lookup k t of
  --      Just y  -> Tip k y
  --      Nothing -> Nil
  --intersection Nil t = Nil
  --intersection t Nil = Nil
  --
  ---- | /O(n+m)/. The intersection with a combining function.
  --intersectionWith :: (a -> b -> a) -> IntMap a -> IntMap b -> IntMap a
  --intersectionWith f m1 m2
  --  = intersectionWithKey (\k x y -> f x y) m1 m2
  --
  ---- | /O(n+m)/. The intersection with a combining function.
  --intersectionWithKey :: (Key -> a -> b -> a) -> IntMap a -> IntMap b -> IntMap a
  --intersectionWithKey f t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
  --  | shorter m1 m2  = intersection1
  --  | shorter m2 m1  = intersection2
  --  | p1 == p2       = bin p1 m1 (intersectionWithKey f l1 l2) (intersectionWithKey f r1 r2)
  --  | otherwise      = Nil
  --  where
  --    intersection1 | nomatch p2 p1 m1  = Nil
  --                  | zero p2 m1        = intersectionWithKey f l1 t2
  --                  | otherwise         = intersectionWithKey f r1 t2
  --
  --    intersection2 | nomatch p1 p2 m2  = Nil
  --                  | zero p1 m2        = intersectionWithKey f t1 l2
  --                  | otherwise         = intersectionWithKey f t1 r2
  --
  --intersectionWithKey f t1@(Tip k x) t2
  --  = case lookup k t2 of
  --      Just y  -> Tip k (f k x y)
  --      Nothing -> Nil
  --intersectionWithKey f t1 (Tip k y)
  --  = case lookup k t1 of
  --      Just x  -> Tip k (f k x y)
  --      Nothing -> Nil
  --intersectionWithKey f Nil t = Nil
  --intersectionWithKey f t Nil = Nil
  
  
  {--------------------------------------------------------------------
    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 pred t1 t2
  --  = case submapCmp pred t1 t2 of
  --      LT -> True
  --      ge -> False
  --
  --submapCmp pred t1@(Bin p1 m1 l1 r1) t2@(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 pred t1 l2
  --                | otherwise         = submapCmp pred t1 r2
  --    submapCmpEq = case (submapCmp pred l1 l2, submapCmp pred r1 r2) of
  --                    (GT,_ ) -> GT
  --                    (_ ,GT) -> GT
  --                    (EQ,EQ) -> EQ
  --                    other   -> LT
  --
  --submapCmp pred (Bin p m l r) t  = GT
  --submapCmp pred (Tip kx x) (Tip ky y)
  --  | (kx == ky) && pred x y = EQ
  --  | otherwise              = GT  -- disjoint
  --submapCmp pred (Tip k x) t
  --  = case lookup k t of
  --     Just y  | pred x y -> LT
  --     other   -> GT -- disjoint
  --submapCmp pred Nil Nil = EQ
  --submapCmp pred Nil t   = 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 pred t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
    | shorter m1 m2  = False
    | shorter m2 m1  = match p1 p2 m2 && (if zero p1 m2 then isSubmapOfBy pred t1 l2
                                                        else isSubmapOfBy pred t1 r2)
    | otherwise      = (p1==p2) && isSubmapOfBy pred l1 l2 && isSubmapOfBy pred r1 r2
  isSubmapOfBy pred (Bin p m l r) t  = False
  isSubmapOfBy pred (Tip k x) t      = case lookup k t of
                                     Just y  -> pred x y
                                     Nothing -> False
  isSubmapOfBy pred Nil t            = True
  
  {--------------------------------------------------------------------
    Mapping
  --------------------------------------------------------------------}
  -- | /O(n)/. Map a function over all values in the map.
  map :: (a -> b) -> IntMap a -> IntMap b
  map f m
    = mapWithKey (\k x -> f x) m
  
  -- | /O(n)/. Map a function over all values in the map.
  mapWithKey :: (Key -> a -> b) -> IntMap a -> IntMap b
  mapWithKey f t
    = case t of
        Bin p m l r -> Bin p m (mapWithKey f l) (mapWithKey f r)
        Tip k x     -> Tip k (f k x)
        Nil         -> Nil
  
  -- | /O(n)/. The function @'mapAccum'@ threads an accumulating
  -- argument through the map in ascending order of keys.
  mapAccum :: (a -> b -> (a,c)) -> a -> IntMap b -> (a,IntMap c)
  mapAccum f a m
    = mapAccumWithKey (\a k x -> f a x) a m
  
  -- | /O(n)/. The function @'mapAccumWithKey'@ threads an accumulating
  -- argument through the map in ascending order of keys.
  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 throught the map in descending order of keys.
  mapAccumR :: (a -> Key -> b -> (a,c)) -> a -> IntMap b -> (a,IntMap c)
  mapAccumR f a t
    = case t of
        Bin p m l r -> let (a1,r') = mapAccumR f a r
                           (a2,l') = mapAccumR f a1 l
                       in (a2,Bin p m l' r')
        Tip k x     -> let (a',x') = f a k x in (a',Tip k x')
        Nil         -> (a,Nil)
  
  {--------------------------------------------------------------------
    Filter
  --------------------------------------------------------------------}
  -- | /O(n)/. Filter all values that satisfy some predicate.
  filter :: (a -> Bool) -> IntMap a -> IntMap a
  filter p m
    = filterWithKey (\k x -> p x) m
  
  -- | /O(n)/. Filter all keys\/values that satisfy some predicate.
  filterWithKey :: (Key -> a -> Bool) -> IntMap a -> IntMap a
  filterWithKey pred t
    = case t of
        Bin p m l r
          -> bin p m (filterWithKey pred l) (filterWithKey pred r)
        Tip k x
          | pred 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 -> Bool) -> IntMap a -> (IntMap a,IntMap a)
  partition p m
    = partitionWithKey (\k 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 :: (Key -> a -> Bool) -> IntMap a -> (IntMap a,IntMap a)
  partitionWithKey pred t
    = case t of
        Bin p m l r
          -> let (l1,l2) = partitionWithKey pred l
                 (r1,r2) = partitionWithKey pred r
             in (bin p m l1 r1, bin p m l2 r2)
        Tip k x
          | pred k x  -> (t,Nil)
          | otherwise -> (Nil,t)
        Nil -> (Nil,Nil)
  
  -- | /O(n)/. Map values and collect the 'Just' results.
  mapMaybe :: (a -> Maybe b) -> IntMap a -> IntMap b
  mapMaybe f m
    = mapMaybeWithKey (\k x -> f x) m
  
  -- | /O(n)/. Map keys\/values and collect the 'Just' results.
  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 f Nil = Nil
  
  -- | /O(n)/. Map values and separate the 'Left' and 'Right' results.
  mapEither :: (a -> Either b c) -> IntMap a -> (IntMap b, IntMap c)
  mapEither f m
    = mapEitherWithKey (\k x -> f x) m
  
  -- | /O(n)/. Map keys\/values and separate the 'Left' and 'Right' results.
  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 f Nil = (Nil, Nil)
  
  -- | /O(log n)/. The expression (@'split' k map@) is a pair @(map1,map2)@
  -- where all keys in @map1@ are lower than @k@ and all keys in
  -- @map2@ larger than @k@. Any key equal to @k@ is found in neither @map1@ nor @map2@.
  split :: Key -> IntMap a -> (IntMap a,IntMap a)
  split k t
    = case t of
        Bin p m l r
            | m < 0 -> (if k >= 0 -- handle negative numbers.
                        then let (lt,gt) = split' k l in (union r lt, gt)
                        else let (lt,gt) = split' k r in (lt, union gt l))
            | otherwise   -> split' k t
        Tip ky y
          | k>ky      -> (t,Nil)
          | k<ky      -> (Nil,t)
          | otherwise -> (Nil,Nil)
        Nil -> (Nil,Nil)
  
  split' :: Key -> IntMap a -> (IntMap a,IntMap a)
  split' k t
    = case t of
        Bin p m l r
          | nomatch k p m -> if k>p then (t,Nil) else (Nil,t)
          | zero k m  -> let (lt,gt) = split k l in (lt,union gt r)
          | otherwise -> let (lt,gt) = split k r in (union l lt,gt)
        Tip ky y
          | k>ky      -> (t,Nil)
          | k<ky      -> (Nil,t)
          | otherwise -> (Nil,Nil)
        Nil -> (Nil,Nil)
  
  -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
  -- key was found in the original map.
  splitLookup :: Key -> IntMap a -> (IntMap a,Maybe a,IntMap a)
  splitLookup k t
    = case t of
        Bin p m l r
            | m < 0 -> (if k >= 0 -- handle negative numbers.
                        then let (lt,found,gt) = splitLookup' k l in (union r lt,found, gt)
                        else let (lt,found,gt) = splitLookup' k r in (lt,found, union gt l))
            | otherwise   -> splitLookup' k t
        Tip ky y
          | k>ky      -> (t,Nothing,Nil)
          | k<ky      -> (Nil,Nothing,t)
          | otherwise -> (Nil,Just y,Nil)
        Nil -> (Nil,Nothing,Nil)
  
  splitLookup' :: Key -> IntMap a -> (IntMap a,Maybe a,IntMap a)
  splitLookup' k t
    = case t of
        Bin p m l r
          | nomatch k p m -> if k>p then (t,Nothing,Nil) else (Nil,Nothing,t)
          | zero k m  -> let (lt,found,gt) = splitLookup k l in (lt,found,union gt r)
          | otherwise -> let (lt,found,gt) = splitLookup k r in (union l lt,found,gt)
        Tip ky y
          | k>ky      -> (t,Nothing,Nil)
          | k<ky      -> (Nil,Nothing,t)
          | otherwise -> (Nil,Just y,Nil)
        Nil -> (Nil,Nothing,Nil)
        -}
  
  {--------------------------------------------------------------------
    Fold
  --------------------------------------------------------------------}
  -- | /O(n)/. Fold the values in the map, such that
  -- @'fold' f z == 'Prelude.foldr' f z . 'elems'@.
  -- For example,
  --
  -- > elems map = fold (:) [] map
  --
  fold :: (Int -> b -> b) -> b -> IntBag -> b
  fold f z t
    = foldWithKey (\k x y -> f x y) z t
  
  -- | /O(n)/. Fold the keys and values in the map, such that
  -- @'foldWithKey' f z == 'Prelude.foldr' ('uncurry' f) z . 'toAscList'@.
  -- For example,
  --
  -- > keys map = foldWithKey (\k x ks -> k:ks) [] map
  --
  foldWithKey :: (Key -> Int -> b -> b) -> b -> IntBag -> b
  foldWithKey f z t
    = foldr f z t
  
  foldr :: (Key -> Int -> b -> b) -> b -> IntBag -> b
  foldr f z t
    = case t of
        Bin 0 m l r | m < 0 -> foldr' f (foldr' f z l) r  -- put negative numbers before.
        Bin _ _ _ _ -> foldr' f z t
        Tip k x     -> f k x z
        Nil         -> z
  
  foldr' :: (Key -> Int -> b -> b) -> b -> IntBag -> b
  foldr' f z t
    = case t of
        Bin p m l r -> foldr' f (foldr' f z r) l
        Tip k x     -> f k x z
        Nil         -> z
  
  
  
  {--------------------------------------------------------------------
    List variations
  --------------------------------------------------------------------}
  -- | /O(n)/.
  -- Return all elements of the map in the ascending order of their keys.
  --elems :: IntMap a -> [a]
  --elems m
  --  = foldWithKey (\k x xs -> x:xs) [] m
  --
  ---- | /O(n)/. Return all keys of the map in ascending order.
  --keys  :: IntMap a -> [Key]
  --keys m
  --  = foldWithKey (\k x ks -> k:ks) [] m
  --
  ---- | /O(n*min(n,W))/. The set of all keys of the map.
  --keysSet :: IntMap a -> IntSet.IntSet
  --keysSet m = IntSet.fromDistinctAscList (keys m)
  
  
  -- | /O(n)/. Return all key\/value pairs in the map in ascending key order.
  assocs :: IntBag -> [(Key,Int)]
  assocs m = toList m
  
  
  {--------------------------------------------------------------------
    Lists
  --------------------------------------------------------------------}
  -- | /O(n)/. Convert the map to a list of key\/value pairs.
  toList :: IntBag -> [(Key,Int)]
  toList t
    = foldWithKey (\k x xs -> (k,x):xs) [] t
  
  -- | /O(n)/. Convert the map to a list of key\/value pairs where the
  -- keys are in ascending order.
  toAscList :: IntBag -> [(Key,Int)]
  toAscList t
    = -- NOTE: the following algorithm only works for big-endian trees
      let (pos,neg) = span (\(k,x) -> k >=0) (foldr (\k x xs -> (k,x):xs) [] t) in neg ++ pos
  
  -- | /O(n*min(n,W))/. Create a map from a list of key\/value pairs.
  fromList :: [(Key,Int)] -> IntBag
  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 :: (a -> a -> a) -> [(Key,a)] -> IntMap a
  --fromListWith f xs
  --  = fromListWithKey (\k 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'.
  --fromListWithKey :: (Key -> a -> a -> a) -> [(Key,a)] -> IntMap a
  --fromListWithKey f xs
  --  = foldlStrict ins empty xs
  --  where
  --    ins t (k,x) = insertWithKey f k x t
  --
  ---- | /O(n*min(n,W))/. Build a map from a list of key\/value pairs where
  ---- the keys are in ascending order.
  --fromAscList :: [(Key,a)] -> IntMap a
  --fromAscList xs
  --  = fromList xs
  --
  ---- | /O(n*min(n,W))/. Build a map from a list of key\/value pairs where
  ---- the keys are in ascending order, with a combining function on equal keys.
  --fromAscListWith :: (a -> a -> a) -> [(Key,a)] -> IntMap a
  --fromAscListWith f xs
  --  = fromListWith f xs
  --
  ---- | /O(n*min(n,W))/. Build a map from a list of key\/value pairs where
  ---- the keys are in ascending order, with a combining function on equal keys.
  --fromAscListWithKey :: (Key -> a -> a -> a) -> [(Key,a)] -> IntMap a
  --fromAscListWithKey f xs
  --  = fromListWithKey f xs
  --
  ---- | /O(n*min(n,W))/. Build a map from a list of key\/value pairs where
  ---- the keys are in ascending order and all distinct.
  --fromDistinctAscList :: [(Key,a)] -> IntMap a
  --fromDistinctAscList xs
  --  = fromList xs
  
  
  {--------------------------------------------------------------------
    Eq
  --------------------------------------------------------------------}
  instance Eq IntBag where
    t1 == t2  = equal t1 t2
    t1 /= t2  = nequal t1 t2
  
  equal :: IntBag -> IntBag -> 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 t1 t2   = False
  
  nequal :: IntBag -> IntBag -> 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 t1 t2   = True
  
  {--------------------------------------------------------------------
    Ord
  --------------------------------------------------------------------}
  
  instance Ord IntBag where
      compare m1 m2 = compare (toList m1) (toList m2)
  
  
  {--------------------------------------------------------------------
    Show
  --------------------------------------------------------------------}
  
  instance Show IntBag where
    showsPrec d m   = showParen (d > 10) $
      showString "fromList " . shows (toList m)
  
  --showMap :: (Show a) => [(Key,a)] -> ShowS
  --showMap []
  --  = showString "{}"
  --showMap (x:xs)
  --  = showChar '{' . showElem x . showTail xs
  --  where
  --    showTail []     = showChar '}'
  --    showTail (x:xs) = showChar ',' . showElem x . showTail xs
  --
  --    showElem (k,x)  = shows k . showString ":=" . shows x
  --
  --{--------------------------------------------------------------------
  --  Read
  ----------------------------------------------------------------------}
  --instance (Read e) => Read (IntMap e) where
  -- #ifdef __GLASGOW_HASKELL__
  --  readPrec = parens $ prec 10 $ do
  --    Ident "fromList" <- lexP
  --    xs <- readPrec
  --    return (fromList xs)
  --
  --  readListPrec = readListPrecDefault
  -- #else
  --  readsPrec p = readParen (p > 10) $ \ r -> do
  --    ("fromList",s) <- lex r
  --    (xs,t) <- reads s
  --    return (fromList xs,t)
  -- #endif
  --
  --{--------------------------------------------------------------------
  --  Typeable
  ----------------------------------------------------------------------}
  --
  -- #include "Typeable.h"
  --INSTANCE_TYPEABLE1(IntMap,intMapTc,"IntMap")
  --
  --{--------------------------------------------------------------------
  --  Debugging
  ----------------------------------------------------------------------}
  ---- | /O(n)/. Show the tree that implements the map. The tree is shown
  ---- in a compressed, hanging format.
  --showTree :: Show a => IntMap a -> String
  --showTree s
  --  = showTreeWith True False s
  --
  --
  --{- | /O(n)/. The expression (@'showTreeWith' hang wide map@) shows
  -- the tree that implements the map. If @hang@ is
  -- 'True', a /hanging/ tree is shown otherwise a rotated tree is shown. If
  -- @wide@ is 'True', an extra wide version is shown.
  ---}
  --showTreeWith :: Show a => Bool -> Bool -> IntMap a -> String
  --showTreeWith hang wide t
  --  | hang      = (showsTreeHang wide [] t) ""
  --  | otherwise = (showsTree wide [] [] t) ""
  --
  --showsTree :: Show a => Bool -> [String] -> [String] -> IntMap a -> ShowS
  --showsTree wide lbars rbars t
  --  = case t of
  --      Bin p m l r
  --          -> showsTree wide (withBar rbars) (withEmpty rbars) r .
  --             showWide wide rbars .
  --             showsBars lbars . showString (showBin p m) . showString "\n" .
  --             showWide wide lbars .
  --             showsTree wide (withEmpty lbars) (withBar lbars) l
  --      Tip k x
  --          -> showsBars lbars . showString " " . shows k . showString ":=" . shows x . showString "\n"
  --      Nil -> showsBars lbars . showString "|\n"
  --
  --showsTreeHang :: Show a => Bool -> [String] -> IntMap a -> ShowS
  --showsTreeHang wide bars t
  --  = case t of
  --      Bin p m l r
  --          -> showsBars bars . showString (showBin p m) . showString "\n" .
  --             showWide wide bars .
  --             showsTreeHang wide (withBar bars) l .
  --             showWide wide bars .
  --             showsTreeHang wide (withEmpty bars) r
  --      Tip k x
  --          -> showsBars bars . showString " " . shows k . showString ":=" . shows x . showString "\n"
  --      Nil -> showsBars bars . showString "|\n"
  --
  --showBin p m
  --  = "*" -- ++ show (p,m)
  --
  --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           = "+--"
  --withBar bars   = "|  ":bars
  --withEmpty bars = "   ":bars
  --
  --
  {--------------------------------------------------------------------
    Helpers
  --------------------------------------------------------------------}
  {--------------------------------------------------------------------
    Join
  --------------------------------------------------------------------}
  join :: Prefix -> IntBag -> Prefix -> IntBag -> IntBag
  join p1 t1 p2 t2
    | zero p1 m = Bin p m t1 t2
    | otherwise = Bin p m t2 t1
    where
      m = branchMask p1 p2
      p = mask p1 m
  
  {--------------------------------------------------------------------
    @bin@ assures that we never have empty trees within a tree.
  --------------------------------------------------------------------}
  bin :: Prefix -> Mask -> IntBag -> IntBag -> IntBag
  bin p m l Nil = l
  bin p m Nil r = r
  bin p m l r   = Bin p m l r
  
  
  {--------------------------------------------------------------------
    Endian independent bit twiddling
  --------------------------------------------------------------------}
  zero :: Key -> Mask -> Bool
  zero i m
    = (natFromInt i) .&. (natFromInt m) == 0
  
  nomatch,_match :: Key -> Prefix -> Mask -> Bool
  nomatch i p m
    = (mask i m) /= p
  
  _match i p m
    = (mask i m) == p
  
  mask :: Key -> Mask -> Prefix
  mask i m
    = maskW (natFromInt i) (natFromInt m)
  
  
  zeroN :: Nat -> Nat -> Bool
  zeroN i m = (i .&. m) == 0
  
  {--------------------------------------------------------------------
    Big endian operations
  --------------------------------------------------------------------}
  maskW :: Nat -> Nat -> Prefix
  maskW i m
    = intFromNat (i .&. (complement (m-1) `xor` m))
  
  shorter :: Mask -> Mask -> Bool
  shorter m1 m2
    = (natFromInt m1) > (natFromInt m2)
  
  branchMask :: Prefix -> Prefix -> Mask
  branchMask p1 p2
    = intFromNat (highestBitMask (natFromInt p1 `xor` natFromInt p2))
  
  {----------------------------------------------------------------------
    Finding the highest bit (mask) in a word [x] can be done efficiently in
    three ways:
    * convert to a floating point value and the mantissa tells us the
      [log2(x)] that corresponds with the highest bit position. The mantissa
      is retrieved either via the standard C function [frexp] or by some bit
      twiddling on IEEE compatible numbers (float). Note that one needs to
      use at least [double] precision for an accurate mantissa of 32 bit
      numbers.
    * use bit twiddling, a logarithmic sequence of bitwise or's and shifts (bit).
    * use processor specific assembler instruction (asm).
  
    The most portable way would be [bit], but is it efficient enough?
    I have measured the cycle counts of the different methods on an AMD
    Athlon-XP 1800 (~ Pentium III 1.8Ghz) using the RDTSC instruction:
  
    highestBitMask: method  cycles
                    --------------
                     frexp   200
                     float    33
                     bit      11
                     asm      12
  
    highestBit:     method  cycles
                    --------------
                     frexp   195
                     float    33
                     bit      11
                     asm      11
  
    Wow, the bit twiddling is on today's RISC like machines even faster
    than a single CISC instruction (BSR)!
  ----------------------------------------------------------------------}
  
  {----------------------------------------------------------------------
    [highestBitMask] returns a word where only the highest bit is set.
    It is found by first setting all bits in lower positions than the
    highest bit and than taking an exclusive or with the original value.
    Allthough the function may look expensive, GHC compiles this into
    excellent C code that subsequently compiled into highly efficient
    machine code. The algorithm is derived from Jorg Arndt's FXT library.
  ----------------------------------------------------------------------}
  highestBitMask :: Nat -> Nat
  highestBitMask x
    = case (x .|. shiftRL x 1) of
       x -> case (x .|. shiftRL x 2) of
        x -> case (x .|. shiftRL x 4) of
         x -> case (x .|. shiftRL x 8) of
          x -> case (x .|. shiftRL x 16) of
           x -> case (x .|. shiftRL x 32) of   -- for 64 bit platforms
            x -> (x `xor` (shiftRL x 1))
  
  
  {--------------------------------------------------------------------
    Utilities
  --------------------------------------------------------------------}
  foldlStrict f z xs
    = case xs of
        []     -> z
        (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
  
  {-
  {--------------------------------------------------------------------
    Testing
  --------------------------------------------------------------------}
  testTree :: [Int] -> IntMap Int
  testTree xs   = fromList [(x,x*x*30696 `mod` 65521) | x <- xs]
  test1 = testTree [1..20]
  test2 = testTree [30,29..10]
  test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
  
  {--------------------------------------------------------------------
    QuickCheck
  --------------------------------------------------------------------}
  qcheck prop
    = check config prop
    where
      config = Config
        { configMaxTest = 500
        , configMaxFail = 5000
        , configSize    = \n -> (div n 2 + 3)
        , configEvery   = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
        }
  
  
  {--------------------------------------------------------------------
    Arbitrary, reasonably balanced trees
  --------------------------------------------------------------------}
  instance Arbitrary a => Arbitrary (IntMap a) where
    arbitrary = do{ ks <- arbitrary
                  ; xs <- mapM (\k -> do{ x <- arbitrary; return (k,x)}) ks
                  ; return (fromList xs)
                  }
  
  
  {--------------------------------------------------------------------
    Single, Insert, Delete
  --------------------------------------------------------------------}
  prop_Single :: Key -> Int -> Bool
  prop_Single k x
    = (insert k x empty == singleton k x)
  
  prop_InsertDelete :: Key -> Int -> IntMap Int -> Property
  prop_InsertDelete k x t
    = not (member k t) ==> delete k (insert k x t) == t
  
  prop_UpdateDelete :: Key -> IntMap Int -> Bool
  prop_UpdateDelete k t
    = update (const Nothing) k t == delete k t
  
  
  {--------------------------------------------------------------------
    Union
  --------------------------------------------------------------------}
  prop_UnionInsert :: Key -> Int -> IntMap Int -> Bool
  prop_UnionInsert k x t
    = union (singleton k x) t == insert k x t
  
  prop_UnionAssoc :: IntMap Int -> IntMap Int -> IntMap Int -> Bool
  prop_UnionAssoc t1 t2 t3
    = union t1 (union t2 t3) == union (union t1 t2) t3
  
  prop_UnionComm :: IntMap Int -> IntMap Int -> Bool
  prop_UnionComm t1 t2
    = (union t1 t2 == unionWith (\x y -> y) t2 t1)
  
  
  prop_Diff :: [(Key,Int)] -> [(Key,Int)] -> Bool
  prop_Diff xs ys
    =  List.sort (keys (difference (fromListWith (+) xs) (fromListWith (+) ys)))
      == List.sort ((List.\\) (nub (Prelude.map fst xs))  (nub (Prelude.map fst ys)))
  
  prop_Int :: [(Key,Int)] -> [(Key,Int)] -> Bool
  prop_Int xs ys
    =  List.sort (keys (intersection (fromListWith (+) xs) (fromListWith (+) ys)))
      == List.sort (nub ((List.intersect) (Prelude.map fst xs)  (Prelude.map fst ys)))
  
  {--------------------------------------------------------------------
    Lists
  --------------------------------------------------------------------}
  prop_Ordered
    = forAll (choose (5,100)) $ \n ->
      let xs = [(x,()) | x <- [0..n::Int]]
      in fromAscList xs == fromList xs
  
  prop_List :: [Key] -> Bool
  prop_List xs
    = (sort (nub xs) == [x | (x,()) <- toAscList (fromList [(x,()) | x <- xs])])
  -}
  