    {-# OPTIONS -cpp -fglasgow-exts #-}
    -----------------------------------------------------------------------------
    -- |
    -- Module      :  Data.IntSet
    -- Copyright   :  (c) Daan Leijen 2002
    -- License     :  BSD-style
    -- Maintainer  :  libraries@haskell.org
    -- Stability   :  provisional
    -- Portability :  portable
    --
    -- An efficient implementation of integer sets.
    --
    -- Since many function names (but not the type name) clash with
    -- "Prelude" names, this module is usually imported @qualified@, e.g.
    --
    -- >  import Data.IntSet (IntSet)
    -- >  import qualified Data.IntSet as IntSet
    --
    -- The implementation is based on /big-endian patricia trees/.  This data
    -- structure performs especially well on binary operations like 'union'
    -- and 'intersection'.  However, my benchmarks show that it is also
    -- (much) faster on insertions and deletions when compared to a generic
    -- size-balanced set implementation (see "Data.Set").
    --
    --    * Chris Okasaki and Andy Gill,  \"/Fast Mergeable Integer Maps/\",
    --      Workshop on ML, September 1998, pages 77-86,
    --      <http://citeseer.ist.psu.edu/okasaki98fast.html>
    --
    --    * D.R. Morrison, \"/PATRICIA -- Practical Algorithm To Retrieve
    --      Information Coded In Alphanumeric/\", Journal of the ACM, 15(4),
    --      October 1968, pages 514-534.
    --
    -- Many operations have a worst-case complexity of /O(min(n,W))/.
    -- This means that the operation can become linear in the number of
    -- elements with a maximum of /W/ -- the number of bits in an 'Int'
    -- (32 or 64).
    -----------------------------------------------------------------------------

    module Data.IntSet  (
                -- * Set type
                  IntSet          -- instance Eq,Show

                -- * Operators
                , (\\)

                -- * Query
                , null
                , size
                , member
                , notMember
                , isSubsetOf
                , isProperSubsetOf

                -- * Construction
                , empty
                , singleton
                , insert
                , delete

                -- * Combine
                , union, unions
                , difference
                , intersection

                -- * Filter
                , filter
                , partition
                , split
                , splitMember

                -- * Min\/Max
                , findMin
                , findMax
                , deleteMin
                , deleteMax
                , deleteFindMin
                , deleteFindMax
                , maxView
                , minView

                -- * Map
    	    , map

                -- * Fold
                , fold

                -- * Conversion
                -- ** List
                , elems
                , toList
                , fromList

                -- ** Ordered list
                , toAscList
                , fromAscList
                , fromDistinctAscList

                -- * Debugging
                , showTree
                , showTreeWith
                ) where

    import Prelude hiding (lookup,filter,foldr,foldl,null,map)
    import Data.Bits

    import qualified Data.List as List
    import Data.Monoid (Monoid(..))
    import Data.Typeable

    {-
    -- just for testing
    import Test.QuickCheck
    import List (nub,sort)
    import qualified List
    import qualified Data.Set as Set
    -}

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

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

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

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

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

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

    shiftRL :: Nat -> Int -> 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(n+m)/. See 'difference'.
    (\\) :: IntSet -> IntSet -> IntSet
    m1 \\ m2 = difference m1 m2

    {--------------------------------------------------------------------
      Types
    --------------------------------------------------------------------}
    -- | A set of integers.
    data IntSet = Nil
                | Tip {-# UNPACK #-} !Int
                | Bin {-# UNPACK #-} !Prefix {-# UNPACK #-} !Mask !IntSet !IntSet
    -- Invariant: Nil is never found as a child of Bin.
    -- Invariant: The Mask is a power of 2.  It is the largest bit position at which
    --            two elements of the set differ.
    -- Invariant: Prefix is the common high-order bits that all elements share to
    --            the left of the Mask bit.
    -- Invariant: In Bin prefix mask left right, left consists of the elements that
    --            don't have the mask bit set; right is all the elements that do.

    type Prefix = Int
    type Mask   = Int

    instance Monoid IntSet where
        mempty  = empty
        mappend = union
        mconcat = unions

    #if __GLASGOW_HASKELL__

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

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

    instance Data IntSet where
      gfoldl f z is = z fromList `f` (toList is)
      toConstr _    = error "toConstr"
      gunfold _ _   = error "gunfold"
      dataTypeOf _  = mkNorepType "Data.IntSet.IntSet"

    #endif

    {--------------------------------------------------------------------
      Query
    --------------------------------------------------------------------}
    -- | /O(1)/. Is the set empty?
    null :: IntSet -> Bool
    null Nil   = True
    null other = False

    -- | /O(n)/. Cardinality of the set.
    size :: IntSet -> Int
    size t
      = case t of
          Bin p m l r -> size l + size r
          Tip y -> 1
          Nil   -> 0

    -- | /O(min(n,W))/. Is the value a member of the set?
    member :: Int -> IntSet -> Bool
    member x t
      = case t of
          Bin p m l r
            | nomatch x p m -> False
            | zero x m      -> member x l
            | otherwise     -> member x r
          Tip y -> (x==y)
          Nil   -> False

    -- | /O(min(n,W))/. Is the element not in the set?
    notMember :: Int -> IntSet -> Bool
    notMember k = not . member k

    -- 'lookup' is used by 'intersection' for left-biasing
    lookup :: Int -> IntSet -> Maybe Int
    lookup k t
      = let nk = natFromInt k  in seq nk (lookupN nk t)

    lookupN :: Nat -> IntSet -> Maybe 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
            | (k == natFromInt kx)  -> Just kx
            | otherwise             -> Nothing
          Nil -> Nothing

    {--------------------------------------------------------------------
      Construction
    --------------------------------------------------------------------}
    -- | /O(1)/. The empty set.
    empty :: IntSet
    empty
      = Nil

    -- | /O(1)/. A set of one element.
    singleton :: Int -> IntSet
    singleton x
      = Tip x

    {--------------------------------------------------------------------
      Insert
    --------------------------------------------------------------------}
    -- | /O(min(n,W))/. Add a value to the set. When the value is already
    -- an element of the set, it is replaced by the new one, ie. 'insert'
    -- is left-biased.
    insert :: Int -> IntSet -> IntSet
    insert x t
      = case t of
          Bin p m l r
            | nomatch x p m -> join x (Tip x) p t
            | zero x m      -> Bin p m (insert x l) r
            | otherwise     -> Bin p m l (insert x r)
          Tip y
            | x==y          -> Tip x
            | otherwise     -> join x (Tip x) y t
          Nil -> Tip x

    -- right-biased insertion, used by 'union'
    insertR :: Int -> IntSet -> IntSet
    insertR x t
      = case t of
          Bin p m l r
            | nomatch x p m -> join x (Tip x) p t
            | zero x m      -> Bin p m (insert x l) r
            | otherwise     -> Bin p m l (insert x r)
          Tip y
            | x==y          -> t
            | otherwise     -> join x (Tip x) y t
          Nil -> Tip x

    -- | /O(min(n,W))/. Delete a value in the set. Returns the
    -- original set when the value was not present.
    delete :: Int -> IntSet -> IntSet
    delete x t
      = case t of
          Bin p m l r
            | nomatch x p m -> t
            | zero x m      -> bin p m (delete x l) r
            | otherwise     -> bin p m l (delete x r)
          Tip y
            | x==y          -> Nil
            | otherwise     -> t
          Nil -> Nil

    {--------------------------------------------------------------------
      Union
    --------------------------------------------------------------------}
    -- | The union of a list of sets.
    unions :: [IntSet] -> IntSet
    unions xs
      = foldlStrict union empty xs

    -- | /O(n+m)/. The union of two sets.
    union :: IntSet -> IntSet -> IntSet
    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 x) t = insert x t
    union t (Tip x) = insertR x t  -- right bias
    union Nil t     = t
    union t Nil     = t

    {--------------------------------------------------------------------
      Difference
    --------------------------------------------------------------------}
    -- | /O(n+m)/. Difference between two sets.
    difference :: IntSet -> IntSet -> IntSet
    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 x) t2
      | member x t2  = Nil
      | otherwise    = t1

    difference Nil t     = Nil
    difference t (Tip x) = delete x t
    difference t Nil     = t

    {--------------------------------------------------------------------
      Intersection
    --------------------------------------------------------------------}
    -- | /O(n+m)/. The intersection of two sets.
    intersection :: IntSet -> IntSet -> IntSet
    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 x) t2
      | member x t2  = t1
      | otherwise    = Nil
    intersection t (Tip x)
      = case lookup x t of
          Just y  -> Tip y
          Nothing -> Nil
    intersection Nil t = Nil
    intersection t Nil = Nil

    {--------------------------------------------------------------------
      Subset
    --------------------------------------------------------------------}
    -- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
    isProperSubsetOf :: IntSet -> IntSet -> Bool
    isProperSubsetOf t1 t2
      = case subsetCmp t1 t2 of
          LT -> True
          ge -> False

    subsetCmp t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
      | shorter m1 m2  = GT
      | shorter m2 m1  = case subsetCmpLt of
                           GT -> GT
                           _ -> LT
      | p1 == p2       = subsetCmpEq
      | otherwise      = GT  -- disjoint
      where
        subsetCmpLt | nomatch p1 p2 m2  = GT
                    | zero p1 m2        = subsetCmp t1 l2
                    | otherwise         = subsetCmp t1 r2
        subsetCmpEq = case (subsetCmp l1 l2, subsetCmp r1 r2) of
                        (GT,_ ) -> GT
                        (_ ,GT) -> GT
                        (EQ,EQ) -> EQ
                        other   -> LT

    subsetCmp (Bin p m l r) t  = GT
    subsetCmp (Tip x) (Tip y)
      | x==y       = EQ
      | otherwise  = GT  -- disjoint
    subsetCmp (Tip x) t
      | member x t = LT
      | otherwise  = GT  -- disjoint
    subsetCmp Nil Nil = EQ
    subsetCmp Nil t   = LT

    -- | /O(n+m)/. Is this a subset?
    -- @(s1 `isSubsetOf` s2)@ tells whether @s1@ is a subset of @s2@.

    isSubsetOf :: IntSet -> IntSet -> Bool
    isSubsetOf 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 isSubsetOf t1 l2
                                                          else isSubsetOf t1 r2)
      | otherwise      = (p1==p2) && isSubsetOf l1 l2 && isSubsetOf r1 r2
    isSubsetOf (Bin p m l r) t  = False
    isSubsetOf (Tip x) t        = member x t
    isSubsetOf Nil t            = True

    {--------------------------------------------------------------------
      Filter
    --------------------------------------------------------------------}
    -- | /O(n)/. Filter all elements that satisfy some predicate.
    filter :: (Int -> Bool) -> IntSet -> IntSet
    filter pred t
      = case t of
          Bin p m l r
            -> bin p m (filter pred l) (filter pred r)
          Tip x
            | pred x    -> t
            | otherwise -> Nil
          Nil -> Nil

    -- | /O(n)/. partition the set according to some predicate.
    partition :: (Int -> Bool) -> IntSet -> (IntSet,IntSet)
    partition pred t
      = case t of
          Bin p m l r
            -> let (l1,l2) = partition pred l
                   (r1,r2) = partition pred r
               in (bin p m l1 r1, bin p m l2 r2)
          Tip x
            | pred x    -> (t,Nil)
            | otherwise -> (Nil,t)
          Nil -> (Nil,Nil)

    -- | /O(min(n,W))/. The expression (@'split' x set@) is a pair @(set1,set2)@
    -- where all elements in @set1@ are lower than @x@ and all elements in
    -- @set2@ larger than @x@.
    --
    -- > split 3 (fromList [1..5]) == (fromList [1,2], fromList [3,4])
    split :: Int -> IntSet -> (IntSet,IntSet)
    split x t
      = case t of
          Bin p m l r
            | m < 0       -> if x >= 0 then let (lt,gt) = split' x l in (union r lt, gt)
                                       else let (lt,gt) = split' x r in (lt, union gt l)
                                       -- handle negative numbers.
            | otherwise   -> split' x t
          Tip y
            | x>y         -> (t,Nil)
            | x<y         -> (Nil,t)
            | otherwise   -> (Nil,Nil)
          Nil             -> (Nil, Nil)

    split' :: Int -> IntSet -> (IntSet,IntSet)
    split' x t
      = case t of
          Bin p m l r
            | match x p m -> if zero x m then let (lt,gt) = split' x l in (lt,union gt r)
                                         else let (lt,gt) = split' x r in (union l lt,gt)
            | otherwise   -> if x < p then (Nil, t)
                                      else (t, Nil)
          Tip y
            | x>y       -> (t,Nil)
            | x<y       -> (Nil,t)
            | otherwise -> (Nil,Nil)
          Nil -> (Nil,Nil)

    -- | /O(min(n,W))/. Performs a 'split' but also returns whether the pivot
    -- element was found in the original set.
    splitMember :: Int -> IntSet -> (IntSet,Bool,IntSet)
    splitMember x t
      = case t of
          Bin p m l r
            | m < 0       -> if x >= 0 then let (lt,found,gt) = splitMember' x l in (union r lt, found, gt)
                                       else let (lt,found,gt) = splitMember' x r in (lt, found, union gt l)
                                       -- handle negative numbers.
            | otherwise   -> splitMember' x t
          Tip y
            | x>y       -> (t,False,Nil)
            | x<y       -> (Nil,False,t)
            | otherwise -> (Nil,True,Nil)
          Nil -> (Nil,False,Nil)

    splitMember' :: Int -> IntSet -> (IntSet,Bool,IntSet)
    splitMember' x t
      = case t of
          Bin p m l r
             | match x p m ->  if zero x m then let (lt,found,gt) = splitMember x l in (lt,found,union gt r)
                                           else let (lt,found,gt) = splitMember x r in (union l lt,found,gt)
             | otherwise   -> if x < p then (Nil, False, t)
                                       else (t, False, Nil)
          Tip y
            | x>y       -> (t,False,Nil)
            | x<y       -> (Nil,False,t)
            | otherwise -> (Nil,True,Nil)
          Nil -> (Nil,False,Nil)

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

    -- | /O(min(n,W))/. Retrieves the maximal key of the set, and the set stripped from that element
    -- @fail@s (in the monad) when passed an empty set.
    maxView :: (Monad m) => IntSet -> m (Int, IntSet)
    maxView t
        = case t of
            Bin p m l r | m < 0 -> let (result,t') = maxViewUnsigned l in return (result, bin p m t' r)
            Bin p m l r         -> let (result,t') = maxViewUnsigned r in return (result, bin p m l t')
            Tip y -> return (y,Nil)
            Nil -> fail "maxView: empty set has no maximal element"

    maxViewUnsigned :: IntSet -> (Int, IntSet)
    maxViewUnsigned t
        = case t of
            Bin p m l r -> let (result,t') = maxViewUnsigned r in (result, bin p m l t')
            Tip y -> (y, Nil)

    -- | /O(min(n,W))/. Retrieves the minimal key of the set, and the set stripped from that element
    -- @fail@s (in the monad) when passed an empty set.
    minView :: (Monad m) => IntSet -> m (Int, IntSet)
    minView t
        = case t of
            Bin p m l r | m < 0 -> let (result,t') = minViewUnsigned r in return (result, bin p m l t')
            Bin p m l r         -> let (result,t') = minViewUnsigned l in return (result, bin p m t' r)
            Tip y -> return (y, Nil)
            Nil -> fail "minView: empty set has no minimal element"

    minViewUnsigned :: IntSet -> (Int, IntSet)
    minViewUnsigned t
        = case t of
            Bin p m l r -> let (result,t') = minViewUnsigned l in (result, bin p m t' r)
            Tip y -> (y, Nil)

    -- Duplicate the Identity monad here because base < mtl.
    newtype Identity a = Identity { runIdentity :: a }
    instance Monad Identity where
    	return a = Identity a
    	m >>= k  = k (runIdentity m)

    -- | /O(min(n,W))/. Delete and find the minimal element.
    --
    -- > deleteFindMin set = (findMin set, deleteMin set)
    deleteFindMin :: IntSet -> (Int, IntSet)
    deleteFindMin = runIdentity . minView

    -- | /O(min(n,W))/. Delete and find the maximal element.
    --
    -- > deleteFindMax set = (findMax set, deleteMax set)
    deleteFindMax :: IntSet -> (Int, IntSet)
    deleteFindMax = runIdentity . maxView

    -- | /O(min(n,W))/. The minimal element of a set.
    findMin :: IntSet -> Int
    findMin = fst . runIdentity . minView

    -- | /O(min(n,W))/. The maximal element of a set.
    findMax :: IntSet -> Int
    findMax = fst . runIdentity . maxView

    -- | /O(min(n,W))/. Delete the minimal element.
    deleteMin :: IntSet -> IntSet
    deleteMin = snd . runIdentity . minView

    -- | /O(min(n,W))/. Delete the maximal element.
    deleteMax :: IntSet -> IntSet
    deleteMax = snd . runIdentity . maxView

    {----------------------------------------------------------------------
      Map
    ----------------------------------------------------------------------}

    -- | /O(n*min(n,W))/.
    -- @'map' f s@ is the set obtained by applying @f@ to each element of @s@.
    --
    -- It's worth noting that the size of the result may be smaller if,
    -- for some @(x,y)@, @x \/= y && f x == f y@

    map :: (Int->Int) -> IntSet -> IntSet
    map f = fromList . List.map f . toList

    {--------------------------------------------------------------------
      Fold
    --------------------------------------------------------------------}
    -- | /O(n)/. Fold over the elements of a set in an unspecified order.
    --
    -- > sum set   == fold (+) 0 set
    -- > elems set == fold (:) [] set
    fold :: (Int -> b -> b) -> b -> IntSet -> b
    fold 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 p m l r -> foldr f z t
          Tip x       -> f x z
          Nil         -> z

    foldr :: (Int -> b -> b) -> b -> IntSet -> b
    foldr f z t
      = case t of
          Bin p m l r -> foldr f (foldr f z r) l
          Tip x       -> f x z
          Nil         -> z

    {--------------------------------------------------------------------
      List variations
    --------------------------------------------------------------------}
    -- | /O(n)/. The elements of a set. (For sets, this is equivalent to toList)
    elems :: IntSet -> [Int]
    elems s
      = toList s

    {--------------------------------------------------------------------
      Lists
    --------------------------------------------------------------------}
    -- | /O(n)/. Convert the set to a list of elements.
    toList :: IntSet -> [Int]
    toList t
      = fold (:) [] t

    -- | /O(n)/. Convert the set to an ascending list of elements.
    toAscList :: IntSet -> [Int]
    toAscList t = toList t

    -- | /O(n*min(n,W))/. Create a set from a list of integers.
    fromList :: [Int] -> IntSet
    fromList xs
      = foldlStrict ins empty xs
      where
        ins t x  = insert x t

    -- | /O(n*min(n,W))/. Build a set from an ascending list of elements.
    fromAscList :: [Int] -> IntSet
    fromAscList xs
      = fromList xs

    -- | /O(n*min(n,W))/. Build a set from an ascending list of distinct elements.
    fromDistinctAscList :: [Int] -> IntSet
    fromDistinctAscList xs
      = fromList xs

    {--------------------------------------------------------------------
      Eq
    --------------------------------------------------------------------}
    instance Eq IntSet where
      t1 == t2  = equal t1 t2
      t1 /= t2  = nequal t1 t2

    equal :: IntSet -> IntSet -> Bool
    equal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
      = (m1 == m2) && (p1 == p2) && (equal l1 l2) && (equal r1 r2)
    equal (Tip x) (Tip y)
      = (x==y)
    equal Nil Nil = True
    equal t1 t2   = False

    nequal :: IntSet -> IntSet -> Bool
    nequal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
      = (m1 /= m2) || (p1 /= p2) || (nequal l1 l2) || (nequal r1 r2)
    nequal (Tip x) (Tip y)
      = (x/=y)
    nequal Nil Nil = False
    nequal t1 t2   = True

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

    instance Ord IntSet where
        compare s1 s2 = compare (toAscList s1) (toAscList s2)
        -- tentative implementation. See if more efficient exists.

    {--------------------------------------------------------------------
      Show
    --------------------------------------------------------------------}
    instance Show IntSet where
      showsPrec p xs = showParen (p > 10) $
        showString "fromList " . shows (toList xs)

    showSet :: [Int] -> ShowS
    showSet []
      = showString "{}"
    showSet (x:xs)
      = showChar '{' . shows x . showTail xs
      where
        showTail []     = showChar '}'
        showTail (x:xs) = showChar ',' . shows x . showTail xs

    {--------------------------------------------------------------------
      Read
    --------------------------------------------------------------------}
    instance Read IntSet 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_TYPEABLE0(IntSet,intSetTc,"IntSet")

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

    {- | /O(n)/. The expression (@'showTreeWith' hang wide map@) shows
     the tree that implements the set. If @hang@ is
     'True', a /hanging/ tree is shown otherwise a rotated tree is shown. If
     @wide@ is 'True', an extra wide version is shown.
    -}
    showTreeWith :: Bool -> Bool -> IntSet -> String
    showTreeWith hang wide t
      | hang      = (showsTreeHang wide [] t) ""
      | otherwise = (showsTree wide [] [] t) ""

    showsTree :: Bool -> [String] -> [String] -> IntSet -> 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 x
              -> showsBars lbars . showString " " . shows x . showString "\n"
          Nil -> showsBars lbars . showString "|\n"

    showsTreeHang :: Bool -> [String] -> IntSet -> 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 x
              -> showsBars bars . 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 -> IntSet -> Prefix -> IntSet -> IntSet
    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 -> IntSet -> IntSet -> IntSet
    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 :: Int -> Mask -> Bool
    zero i m
      = (natFromInt i) .&. (natFromInt m) == 0

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

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

    -- Suppose a is largest such that 2^a divides 2*m.
    -- Then mask i m is i with the low a bits zeroed out.
    mask :: Int -> 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] -> IntSet
    testTree xs   = fromList 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 IntSet where
      arbitrary = do{ xs <- arbitrary
                    ; return (fromList xs)
                    }

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

    prop_InsertDelete :: Int -> IntSet -> Property
    prop_InsertDelete k t
      = not (member k t) ==> delete k (insert k t) == t

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

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

    prop_UnionComm :: IntSet -> IntSet -> Bool
    prop_UnionComm t1 t2
      = (union t1 t2 == union t2 t1)

    prop_Diff :: [Int] -> [Int] -> Bool
    prop_Diff xs ys
      =  toAscList (difference (fromList xs) (fromList ys))
        == List.sort ((List.\\) (nub xs)  (nub ys))

    prop_Int :: [Int] -> [Int] -> Bool
    prop_Int xs ys
      =  toAscList (intersection (fromList xs) (fromList ys))
        == List.sort (nub ((List.intersect) (xs)  (ys)))

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

    prop_List :: [Int] -> Bool
    prop_List xs
      = (sort (nub xs) == toAscList (fromList xs))

    {--------------------------------------------------------------------
      Bin invariants
    --------------------------------------------------------------------}
    powersOf2 :: IntSet
    powersOf2 = fromList [2^i | i <- [0..63]]

    -- Check the invariant that the mask is a power of 2.
    prop_MaskPow2 :: IntSet -> Bool
    prop_MaskPow2 (Bin _ msk left right) = member msk powersOf2 && prop_MaskPow2 left && prop_MaskPow2 right
    prop_MaskPow2 _ = True

    -- Check that the prefix satisfies its invariant.
    prop_Prefix :: IntSet -> Bool
    prop_Prefix s@(Bin prefix msk left right) = all (\elem -> match elem prefix msk) (toList s) && prop_Prefix left && prop_Prefix right
    prop_Prefix _ = True

    -- Check that the left elements don't have the mask bit set, and the right
    -- ones do.
    prop_LeftRight :: IntSet -> Bool
    prop_LeftRight (Bin _ msk left right) = and [x .&. msk == 0 | x <- toList left] && and [x .&. msk == msk | x <- toList right]
    prop_LeftRight _ = True

    {--------------------------------------------------------------------
      IntSet operations are like Set operations
    --------------------------------------------------------------------}
    toSet :: IntSet -> Set.Set Int
    toSet = Set.fromList . toList

    -- Check that IntSet.isProperSubsetOf is the same as Set.isProperSubsetOf.
    prop_isProperSubsetOf :: IntSet -> IntSet -> Bool
    prop_isProperSubsetOf a b = isProperSubsetOf a b == Set.isProperSubsetOf (toSet a) (toSet b)

    -- In the above test, isProperSubsetOf almost always returns False (since a
    -- random set is almost never a subset of another random set).  So this second
    -- test checks the True case.
    prop_isProperSubsetOf2 :: IntSet -> IntSet -> Bool
    prop_isProperSubsetOf2 a b = isProperSubsetOf a c == (a /= c) where
      c = union a b
    -}