Data/IntSet/Internal.hs

    {-# LANGUAGE CPP #-}
    {-# LANGUAGE BangPatterns #-}
    #if __GLASGOW_HASKELL__
    {-# LANGUAGE MagicHash, DeriveDataTypeable, StandaloneDeriving #-}
    #endif
    #if !defined(TESTING) && __GLASGOW_HASKELL__ >= 703
    {-# LANGUAGE Trustworthy #-}
    #endif
    #if __GLASGOW_HASKELL__ >= 708
    {-# LANGUAGE TypeFamilies #-}
    #endif

    #include "containers.h"

    -----------------------------------------------------------------------------
    -- |
    -- Module      :  Data.IntSet.Internal
    -- Copyright   :  (c) Daan Leijen 2002
    --                (c) Joachim Breitner 2011
    -- License     :  BSD-style
    -- Maintainer  :  libraries@haskell.org
    -- Portability :  portable
    --
    -- = WARNING
    --
    -- This module is considered __internal__.
    --
    -- The Package Versioning Policy __does not apply__.
    --
    -- This contents of this module may change __in any way whatsoever__
    -- and __without any warning__ between minor versions of this package.
    --
    -- Authors importing this module are expected to track development
    -- closely.
    --
    -- = Description
    --
    -- An efficient implementation of integer sets.
    --
    -- These modules are intended to be imported qualified, to avoid name
    -- clashes with Prelude functions, 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.
    --
    -- Additionally, this implementation places bitmaps in the leaves of the tree.
    -- Their size is the natural size of a machine word (32 or 64 bits) and greatly
    -- reduce memory footprint and execution times for dense sets, e.g. sets where
    -- it is likely that many values lie close to each other. The asymptotics are
    -- not affected by this optimization.
    --
    -- 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).
    -----------------------------------------------------------------------------

    -- [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 IntSet 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.IntSet.Internal (
        -- * Set type
          IntSet(..), Key -- instance Eq,Show

        -- * Operators
        , (\\)

        -- * Query
        , null
        , size
        , member
        , notMember
        , lookupLT
        , lookupGT
        , lookupLE
        , lookupGE
        , isSubsetOf
        , isProperSubsetOf

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

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

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

        -- * Map
        , map

        -- * Folds
        , foldr
        , foldl
        -- ** Strict folds
        , foldr'
        , foldl'
        -- ** Legacy folds
        , fold

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

        -- * Conversion

        -- ** List
        , elems
        , toList
        , fromList

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

        -- * Debugging
        , showTree
        , showTreeWith

        -- * Internals
        , match
        , suffixBitMask
        , prefixBitMask
        , bitmapOf
        ) where

    import Control.DeepSeq (NFData(rnf))
    import Data.Bits
    import qualified Data.List as List
    import Data.Maybe (fromMaybe)
    #if !MIN_VERSION_base(4,8,0)
    import Data.Monoid (Monoid(..))
    import Data.Word (Word)
    #endif
    #if MIN_VERSION_base(4,9,0)
    import Data.Semigroup (Semigroup((<>), stimes), stimesIdempotentMonoid)
    #endif
    import Data.Typeable
    import Prelude hiding (filter, foldr, foldl, null, map)

    import Utils.Containers.Internal.BitUtil
    import Utils.Containers.Internal.StrictFold
    import Utils.Containers.Internal.StrictPair

    #if __GLASGOW_HASKELL__
    import Data.Data (Data(..), Constr, mkConstr, constrIndex, Fixity(Prefix), DataType, mkDataType)
    import Text.Read
    #endif

    #if __GLASGOW_HASKELL__
    import GHC.Exts (Int(..), build)
    #if __GLASGOW_HASKELL__ >= 708
    import qualified GHC.Exts as GHCExts
    #endif
    import GHC.Prim (indexInt8OffAddr#)
    #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
    {-# INLINE natFromInt #-}

    intFromNat :: Nat -> Int
    intFromNat w = fromIntegral w
    {-# INLINE intFromNat #-}

    {--------------------------------------------------------------------
      Operators
    --------------------------------------------------------------------}
    -- | /O(n+m)/. See 'difference'.
    (\\) :: IntSet -> IntSet -> IntSet
    m1 \\ m2 = difference m1 m2

    {--------------------------------------------------------------------
      Types
    --------------------------------------------------------------------}

    -- | A set of integers.

    -- See Note: Order of constructors
    data IntSet = 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.
                | Tip {-# UNPACK #-} !Prefix {-# UNPACK #-} !BitMap
    -- Invariant: The Prefix is zero for all but the last 5 (on 32 bit arches) or 6
    --            bits (on 64 bit arches). The values of the map represented by a tip
    --            are the prefix plus the indices of the set bits in the bit map.
                | Nil

    -- A number stored in a set is stored as
    -- * Prefix (all but last 5-6 bits) and
    -- * BitMap (last 5-6 bits stored as a bitmask)
    --   Last 5-6 bits are called a Suffix.

    type Prefix = Int
    type Mask   = Int
    type BitMap = Word
    type Key    = Int

    instance Monoid IntSet where
        mempty  = empty
        mconcat = unions
    #if !(MIN_VERSION_base(4,9,0))
        mappend = union
    #else
        mappend = (<>)

    instance Semigroup IntSet where
        (<>)    = union
        stimes  = stimesIdempotentMonoid
    #endif

    #if __GLASGOW_HASKELL__

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

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

    instance Data IntSet where
      gfoldl f z is = z fromList `f` (toList is)
      toConstr _     = fromListConstr
      gunfold k z c  = case constrIndex c of
        1 -> k (z fromList)
        _ -> error "gunfold"
      dataTypeOf _   = intSetDataType

    fromListConstr :: Constr
    fromListConstr = mkConstr intSetDataType "fromList" [] Prefix

    intSetDataType :: DataType
    intSetDataType = mkDataType "Data.IntSet.Internal.IntSet" [fromListConstr]

    #endif

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

    -- | /O(n)/. Cardinality of the set.
    size :: IntSet -> Int
    size = go 0
      where
        go !acc (Bin _ _ l r) = go (go acc l) r
        go acc (Tip _ bm) = acc + bitcount 0 bm
        go acc Nil = acc

    -- | /O(min(n,W))/. Is the value a member of the set?

    -- See Note: Local 'go' functions and capturing]
    member :: Key -> IntSet -> Bool
    member !x = go
      where
        go (Bin p m l r)
          | nomatch x p m = False
          | zero x m      = go l
          | otherwise     = go r
        go (Tip y bm) = prefixOf x == y && bitmapOf x .&. bm /= 0
        go Nil = False

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

    -- | /O(log n)/. Find largest element smaller than the given one.
    --
    -- > lookupLT 3 (fromList [3, 5]) == Nothing
    -- > lookupLT 5 (fromList [3, 5]) == Just 3

    -- See Note: Local 'go' functions and capturing.
    lookupLT :: Key -> IntSet -> Maybe Key
    lookupLT !x t = case t of
        Bin _ m l r | m < 0 -> if x >= 0 then go r l else go Nil r
        _ -> go Nil t
      where
        go def (Bin p m l r) | nomatch x p m = if x < p then unsafeFindMax def else unsafeFindMax r
                             | zero x m  = go def l
                             | otherwise = go l r
        go def (Tip kx bm) | prefixOf x > kx = Just $ kx + highestBitSet bm
                           | prefixOf x == kx && maskLT /= 0 = Just $ kx + highestBitSet maskLT
                           | otherwise = unsafeFindMax def
                           where maskLT = (bitmapOf x - 1) .&. bm
        go def Nil = unsafeFindMax def

    -- | /O(log n)/. Find smallest element greater than the given one.
    --
    -- > lookupGT 4 (fromList [3, 5]) == Just 5
    -- > lookupGT 5 (fromList [3, 5]) == Nothing

    -- See Note: Local 'go' functions and capturing.
    lookupGT :: Key -> IntSet -> Maybe Key
    lookupGT !x t = case t of
        Bin _ m l r | m < 0 -> if x >= 0 then go Nil l else go l r
        _ -> go Nil t
      where
        go def (Bin p m l r) | nomatch x p m = if x < p then unsafeFindMin l else unsafeFindMin def
                             | zero x m  = go r l
                             | otherwise = go def r
        go def (Tip kx bm) | prefixOf x < kx = Just $ kx + lowestBitSet bm
                           | prefixOf x == kx && maskGT /= 0 = Just $ kx + lowestBitSet maskGT
                           | otherwise = unsafeFindMin def
                           where maskGT = (- ((bitmapOf x) `shiftLL` 1)) .&. bm
        go def Nil = unsafeFindMin def

    -- | /O(log n)/. Find largest element smaller or equal to the given one.
    --
    -- > lookupLE 2 (fromList [3, 5]) == Nothing
    -- > lookupLE 4 (fromList [3, 5]) == Just 3
    -- > lookupLE 5 (fromList [3, 5]) == Just 5

    -- See Note: Local 'go' functions and capturing.
    lookupLE :: Key -> IntSet -> Maybe Key
    lookupLE !x t = case t of
        Bin _ m l r | m < 0 -> if x >= 0 then go r l else go Nil r
        _ -> go Nil t
      where
        go def (Bin p m l r) | nomatch x p m = if x < p then unsafeFindMax def else unsafeFindMax r
                             | zero x m  = go def l
                             | otherwise = go l r
        go def (Tip kx bm) | prefixOf x > kx = Just $ kx + highestBitSet bm
                           | prefixOf x == kx && maskLE /= 0 = Just $ kx + highestBitSet maskLE
                           | otherwise = unsafeFindMax def
                           where maskLE = (((bitmapOf x) `shiftLL` 1) - 1) .&. bm
        go def Nil = unsafeFindMax def

    -- | /O(log n)/. Find smallest element greater or equal to the given one.
    --
    -- > lookupGE 3 (fromList [3, 5]) == Just 3
    -- > lookupGE 4 (fromList [3, 5]) == Just 5
    -- > lookupGE 6 (fromList [3, 5]) == Nothing

    -- See Note: Local 'go' functions and capturing.
    lookupGE :: Key -> IntSet -> Maybe Key
    lookupGE !x t = case t of
        Bin _ m l r | m < 0 -> if x >= 0 then go Nil l else go l r
        _ -> go Nil t
      where
        go def (Bin p m l r) | nomatch x p m = if x < p then unsafeFindMin l else unsafeFindMin def
                             | zero x m  = go r l
                             | otherwise = go def r
        go def (Tip kx bm) | prefixOf x < kx = Just $ kx + lowestBitSet bm
                           | prefixOf x == kx && maskGE /= 0 = Just $ kx + lowestBitSet maskGE
                           | otherwise = unsafeFindMin def
                           where maskGE = (- (bitmapOf x)) .&. bm
        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 :: IntSet -> Maybe Key
    unsafeFindMin Nil = Nothing
    unsafeFindMin (Tip kx bm) = Just $ kx + lowestBitSet bm
    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 :: IntSet -> Maybe Key
    unsafeFindMax Nil = Nothing
    unsafeFindMax (Tip kx bm) = Just $ kx + highestBitSet bm
    unsafeFindMax (Bin _ _ _ r) = unsafeFindMax r

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

    -- | /O(1)/. A set of one element.
    singleton :: Key -> IntSet
    singleton x
      = Tip (prefixOf x) (bitmapOf x)
    {-# INLINE singleton #-}

    {--------------------------------------------------------------------
      Insert
    --------------------------------------------------------------------}
    -- | /O(min(n,W))/. Add a value to the set. There is no left- or right bias for
    -- IntSets.
    insert :: Key -> IntSet -> IntSet
    insert !x = insertBM (prefixOf x) (bitmapOf x)

    -- Helper function for insert and union.
    insertBM :: Prefix -> BitMap -> IntSet -> IntSet
    insertBM !kx !bm t@(Bin p m l r)
      | nomatch kx p m = link kx (Tip kx bm) p t
      | zero kx m      = Bin p m (insertBM kx bm l) r
      | otherwise      = Bin p m l (insertBM kx bm r)
    insertBM kx bm t@(Tip kx' bm')
      | kx' == kx = Tip kx' (bm .|. bm')
      | otherwise = link kx (Tip kx bm) kx' t
    insertBM kx bm Nil = Tip kx bm

    -- | /O(min(n,W))/. Delete a value in the set. Returns the
    -- original set when the value was not present.
    delete :: Key -> IntSet -> IntSet
    delete !x = deleteBM (prefixOf x) (bitmapOf x)

    -- Deletes all values mentioned in the BitMap from the set.
    -- Helper function for delete and difference.
    deleteBM :: Prefix -> BitMap -> IntSet -> IntSet
    deleteBM !kx !bm t@(Bin p m l r)
      | nomatch kx p m = t
      | zero kx m      = bin p m (deleteBM kx bm l) r
      | otherwise      = bin p m l (deleteBM kx bm r)
    deleteBM kx bm t@(Tip kx' bm')
      | kx' == kx = tip kx (bm' .&. complement bm)
      | otherwise = t
    deleteBM _ _ 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      = link p1 t1 p2 t2
      where
        union1  | nomatch p2 p1 m1  = link 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  = link p1 t1 p2 t2
                | zero p1 m2        = Bin p2 m2 (union t1 l2) r2
                | otherwise         = Bin p2 m2 l2 (union t1 r2)

    union t@(Bin _ _ _ _) (Tip kx bm) = insertBM kx bm t
    union t@(Bin _ _ _ _) Nil = t
    union (Tip kx bm) t = insertBM kx bm t
    union Nil t = 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 t@(Bin _ _ _ _) (Tip kx bm) = deleteBM kx bm t
    difference t@(Bin _ _ _ _) Nil = t

    difference t1@(Tip kx bm) t2 = differenceTip t2
      where differenceTip (Bin p2 m2 l2 r2) | nomatch kx p2 m2 = t1
                                            | zero kx m2 = differenceTip l2
                                            | otherwise = differenceTip r2
            differenceTip (Tip kx2 bm2) | kx == kx2 = tip kx (bm .&. complement bm2)
                                        | otherwise = t1
            differenceTip Nil = t1

    difference Nil _     = Nil

    {--------------------------------------------------------------------
      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@(Bin _ _ _ _) (Tip kx2 bm2) = intersectBM t1
      where intersectBM (Bin p1 m1 l1 r1) | nomatch kx2 p1 m1 = Nil
                                          | zero kx2 m1       = intersectBM l1
                                          | otherwise         = intersectBM r1
            intersectBM (Tip kx1 bm1) | kx1 == kx2 = tip kx1 (bm1 .&. bm2)
                                      | otherwise = Nil
            intersectBM Nil = Nil

    intersection (Bin _ _ _ _) Nil = Nil

    intersection (Tip kx1 bm1) t2 = intersectBM t2
      where intersectBM (Bin p2 m2 l2 r2) | nomatch kx1 p2 m2 = Nil
                                          | zero kx1 m2       = intersectBM l2
                                          | otherwise         = intersectBM r2
            intersectBM (Tip kx2 bm2) | kx1 == kx2 = tip kx1 (bm1 .&. bm2)
                                      | otherwise = Nil
            intersectBM Nil = Nil

    intersection 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
          _  -> False

    subsetCmp :: IntSet -> IntSet -> Ordering
    subsetCmp t1@(Bin p1 m1 l1 r1) (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
                        _       -> LT

    subsetCmp (Bin _ _ _ _) _  = GT
    subsetCmp (Tip kx1 bm1) (Tip kx2 bm2)
      | kx1 /= kx2                  = GT -- disjoint
      | bm1 == bm2                  = EQ
      | bm1 .&. complement bm2 == 0 = LT
      | otherwise                   = GT
    subsetCmp t1@(Tip kx _) (Bin p m l r)
      | nomatch kx p m = GT
      | zero kx m      = case subsetCmp t1 l of GT -> GT ; _ -> LT
      | otherwise      = case subsetCmp t1 r of GT -> GT ; _ -> LT
    subsetCmp (Tip _ _) Nil = GT -- disjoint
    subsetCmp Nil Nil = EQ
    subsetCmp Nil _   = 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) (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 _ _ _ _) _  = False
    isSubsetOf (Tip kx1 bm1) (Tip kx2 bm2) = kx1 == kx2 && bm1 .&. complement bm2 == 0
    isSubsetOf t1@(Tip kx _) (Bin p m l r)
      | nomatch kx p m = False
      | zero kx m      = isSubsetOf t1 l
      | otherwise      = isSubsetOf t1 r
    isSubsetOf (Tip _ _) Nil = False
    isSubsetOf Nil _         = True

    {--------------------------------------------------------------------
      Filter
    --------------------------------------------------------------------}
    -- | /O(n)/. Filter all elements that satisfy some predicate.
    filter :: (Key -> Bool) -> IntSet -> IntSet
    filter predicate t
      = case t of
          Bin p m l r
            -> bin p m (filter predicate l) (filter predicate r)
          Tip kx bm
            -> tip kx (foldl'Bits 0 (bitPred kx) 0 bm)
          Nil -> Nil
      where bitPred kx bm bi | predicate (kx + bi) = bm .|. bitmapOfSuffix bi
                             | otherwise           = bm
            {-# INLINE bitPred #-}

    -- | /O(n)/. partition the set according to some predicate.
    partition :: (Key -> Bool) -> IntSet -> (IntSet,IntSet)
    partition predicate0 t0 = toPair $ go predicate0 t0
      where
        go predicate t
          = case t of
              Bin p m l r
                -> let (l1 :*: l2) = go predicate l
                       (r1 :*: r2) = go predicate r
                   in bin p m l1 r1 :*: bin p m l2 r2
              Tip kx bm
                -> let bm1 = foldl'Bits 0 (bitPred kx) 0 bm
                   in  tip kx bm1 :*: tip kx (bm `xor` bm1)
              Nil -> (Nil :*: Nil)
          where bitPred kx bm bi | predicate (kx + bi) = bm .|. bitmapOfSuffix bi
                                 | otherwise           = bm
                {-# INLINE bitPred #-}

    -- | /O(min(n,W))/. The expression (@'split' x set@) is a pair @(set1,set2)@
    -- where @set1@ comprises the elements of @set@ less than @x@ and @set2@
    -- comprises the elements of @set@ greater than @x@.
    --
    -- > split 3 (fromList [1..5]) == (fromList [1,2], fromList [4,5])
    split :: Key -> IntSet -> (IntSet,IntSet)
    split x t =
      case t of
          Bin _ m l r
              | m < 0 -> if x >= 0  -- handle negative numbers.
                         then case go x l of (lt :*: gt) -> let !lt' = union lt r
                                                            in (lt', gt)
                         else case go x r of (lt :*: gt) -> let !gt' = union gt l
                                                            in (lt, gt')
          _ -> case go x t of
              (lt :*: gt) -> (lt, gt)
      where
        go !x' t'@(Bin p m l r)
            | match x' p m = if zero x' m
                             then case go x' l of
                                 (lt :*: gt) -> lt :*: union gt r
                             else case go x' r of
                                 (lt :*: gt) -> union lt l :*: gt
            | otherwise   = if x' < p then (Nil :*: t')
                            else (t' :*: Nil)
        go x' t'@(Tip kx' bm)
            | kx' > x'          = (Nil :*: t')
              -- equivalent to kx' > prefixOf x'
            | kx' < prefixOf x' = (t' :*: Nil)
            | otherwise = tip kx' (bm .&. lowerBitmap) :*: tip kx' (bm .&. higherBitmap)
                where lowerBitmap = bitmapOf x' - 1
                      higherBitmap = complement (lowerBitmap + bitmapOf x')
        go _ Nil = (Nil :*: Nil)

    -- | /O(min(n,W))/. Performs a 'split' but also returns whether the pivot
    -- element was found in the original set.
    splitMember :: Key -> IntSet -> (IntSet,Bool,IntSet)
    splitMember x t =
      case t of
          Bin _ m l r | m < 0 -> if x >= 0
                                 then case go x l of
                                     (lt, fnd, gt) -> let !lt' = union lt r
                                                      in (lt', fnd, gt)
                                 else case go x r of
                                     (lt, fnd, gt) -> let !gt' = union gt l
                                                      in (lt, fnd, gt')
          _ -> go x t
      where
        go x' t'@(Bin p m l r)
            | match x' p m = if zero x' m
                             then case go x' l of
                                 (lt, fnd, gt) -> (lt, fnd, union gt r)
                             else case go x' r of
                                 (lt, fnd, gt) -> (union lt l, fnd, gt)
            | otherwise   = if x' < p then (Nil, False, t') else (t', False, Nil)
        go x' t'@(Tip kx' bm)
            | kx' > x'          = (Nil, False, t')
              -- equivalent to kx' > prefixOf x'
            | kx' < prefixOf x' = (t', False, Nil)
            | otherwise = let !lt = tip kx' (bm .&. lowerBitmap)
                              !found = (bm .&. bitmapOfx') /= 0
                              !gt = tip kx' (bm .&. higherBitmap)
                          in (lt, found, gt)
                where bitmapOfx' = bitmapOf x'
                      lowerBitmap = bitmapOfx' - 1
                      higherBitmap = complement (lowerBitmap + bitmapOfx')
        go _ Nil = (Nil, False, Nil)

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

    -- | /O(min(n,W))/. Retrieves the maximal key of the set, and the set
    -- stripped of that element, or 'Nothing' if passed an empty set.
    maxView :: IntSet -> Maybe (Key, IntSet)
    maxView 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 kx bm) = case highestBitSet bm of bi -> (kx + bi, tip kx (bm .&. complement (bitmapOfSuffix bi)))
        go Nil = error "maxView Nil"

    -- | /O(min(n,W))/. Retrieves the minimal key of the set, and the set
    -- stripped of that element, or 'Nothing' if passed an empty set.
    minView :: IntSet -> Maybe (Key, IntSet)
    minView 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 kx bm) = case lowestBitSet bm of bi -> (kx + bi, tip kx (bm .&. complement (bitmapOfSuffix bi)))
        go Nil = error "minView Nil"

    -- | /O(min(n,W))/. Delete and find the minimal element.
    --
    -- > deleteFindMin set = (findMin set, deleteMin set)
    deleteFindMin :: IntSet -> (Key, IntSet)
    deleteFindMin = fromMaybe (error "deleteFindMin: empty set has no minimal element") . minView

    -- | /O(min(n,W))/. Delete and find the maximal element.
    --
    -- > deleteFindMax set = (findMax set, deleteMax set)
    deleteFindMax :: IntSet -> (Key, IntSet)
    deleteFindMax = fromMaybe (error "deleteFindMax: empty set has no maximal element") . maxView

    -- | /O(min(n,W))/. The minimal element of the set.
    findMin :: IntSet -> Key
    findMin Nil = error "findMin: empty set has no minimal element"
    findMin (Tip kx bm) = kx + lowestBitSet bm
    findMin (Bin _ m l r)
      |   m < 0   = find r
      | otherwise = find l
        where find (Tip kx bm) = kx + lowestBitSet bm
              find (Bin _ _ l' _) = find l'
              find Nil            = error "findMin Nil"

    -- | /O(min(n,W))/. The maximal element of a set.
    findMax :: IntSet -> Key
    findMax Nil = error "findMax: empty set has no maximal element"
    findMax (Tip kx bm) = kx + highestBitSet bm
    findMax (Bin _ m l r)
      |   m < 0   = find l
      | otherwise = find r
        where find (Tip kx bm) = kx + highestBitSet bm
              find (Bin _ _ _ r') = find r'
              find Nil            = error "findMax Nil"

    -- | /O(min(n,W))/. Delete the minimal element. Returns an empty set if the set is empty.
    --
    -- Note that this is a change of behaviour for consistency with 'Data.Set.Set' &#8211;
    -- versions prior to 0.5 threw an error if the 'IntSet' was already empty.
    deleteMin :: IntSet -> IntSet
    deleteMin = maybe Nil snd . minView

    -- | /O(min(n,W))/. Delete the maximal element. Returns an empty set if the set is empty.
    --
    -- Note that this is a change of behaviour for consistency with 'Data.Set.Set' &#8211;
    -- versions prior to 0.5 threw an error if the 'IntSet' was already empty.
    deleteMax :: IntSet -> IntSet
    deleteMax = maybe Nil snd . 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 :: (Key -> Key) -> IntSet -> IntSet
    map f = fromList . List.map f . toList

    {--------------------------------------------------------------------
      Fold
    --------------------------------------------------------------------}
    -- | /O(n)/. Fold the elements in the set using the given right-associative
    -- binary operator. This function is an equivalent of 'foldr' and is present
    -- for compatibility only.
    --
    -- /Please note that fold will be deprecated in the future and removed./
    fold :: (Key -> b -> b) -> b -> IntSet -> b
    fold = foldr
    {-# INLINE fold #-}

    -- | /O(n)/. Fold the elements in the set using the given right-associative
    -- binary operator, such that @'foldr' f z == 'Prelude.foldr' f z . 'toAscList'@.
    --
    -- For example,
    --
    -- > toAscList set = foldr (:) [] set
    foldr :: (Key -> b -> b) -> b -> IntSet -> 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 kx bm)   = foldrBits kx f z' bm
        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' :: (Key -> b -> b) -> b -> IntSet -> 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 kx bm)   = foldr'Bits kx f z' bm
        go z' (Bin _ _ l r) = go (go z' r) l
    {-# INLINE foldr' #-}

    -- | /O(n)/. Fold the elements in the set using the given left-associative
    -- binary operator, such that @'foldl' f z == 'Prelude.foldl' f z . 'toAscList'@.
    --
    -- For example,
    --
    -- > toDescList set = foldl (flip (:)) [] set
    foldl :: (a -> Key -> a) -> a -> IntSet -> 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 kx bm)   = foldlBits kx f z' bm
        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 -> Key -> a) -> a -> IntSet -> 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 kx bm)   = foldl'Bits kx f z' bm
        go z' (Bin _ _ l r) = go (go z' l) r
    {-# INLINE foldl' #-}

    {--------------------------------------------------------------------
      List variations
    --------------------------------------------------------------------}
    -- | /O(n)/. An alias of 'toAscList'. The elements of a set in ascending order.
    -- Subject to list fusion.
    elems :: IntSet -> [Key]
    elems
      = toAscList

    {--------------------------------------------------------------------
      Lists
    --------------------------------------------------------------------}
    #if __GLASGOW_HASKELL__ >= 708
    instance GHCExts.IsList IntSet where
      type Item IntSet = Key
      fromList = fromList
      toList   = toList
    #endif

    -- | /O(n)/. Convert the set to a list of elements. Subject to list fusion.
    toList :: IntSet -> [Key]
    toList
      = toAscList

    -- | /O(n)/. Convert the set to an ascending list of elements. Subject to list
    -- fusion.
    toAscList :: IntSet -> [Key]
    toAscList = foldr (:) []

    -- | /O(n)/. Convert the set to a descending list of elements. Subject to list
    -- fusion.
    toDescList :: IntSet -> [Key]
    toDescList = foldl (flip (:)) []

    -- List fusion for the list generating functions.
    #if __GLASGOW_HASKELL__
    -- The foldrFB and foldlFB are foldr and foldl equivalents, used for list fusion.
    -- They are important to convert unfused to{Asc,Desc}List back, see mapFB in prelude.
    foldrFB :: (Key -> b -> b) -> b -> IntSet -> b
    foldrFB = foldr
    {-# INLINE[0] foldrFB #-}
    foldlFB :: (a -> Key -> a) -> a -> IntSet -> a
    foldlFB = foldl
    {-# INLINE[0] foldlFB #-}

    -- Inline elems and toList, so that we need to fuse only toAscList.
    {-# INLINE elems #-}
    {-# INLINE toList #-}

    -- The fusion is enabled up to phase 2 included. If it does not succeed,
    -- convert in phase 1 the expanded to{Asc,Desc}List calls back to
    -- 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 to{Asc,Desc}List -- it was forbidden to inline it
    -- before phase 0, otherwise the fusion rules would not fire at all.
    {-# NOINLINE[0] toAscList #-}
    {-# NOINLINE[0] toDescList #-}
    {-# RULES "IntSet.toAscList" [~1] forall s . toAscList s = build (\c n -> foldrFB c n s) #-}
    {-# RULES "IntSet.toAscListBack" [1] foldrFB (:) [] = toAscList #-}
    {-# RULES "IntSet.toDescList" [~1] forall s . toDescList s = build (\c n -> foldlFB (\xs x -> c x xs) n s) #-}
    {-# RULES "IntSet.toDescListBack" [1] foldlFB (\xs x -> x : xs) [] = toDescList #-}
    #endif

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

    -- | /O(n)/. Build a set from an ascending list of elements.
    -- /The precondition (input list is ascending) is not checked./
    fromAscList :: [Key] -> IntSet
    fromAscList [] = Nil
    fromAscList (x0 : xs0) = fromDistinctAscList (combineEq x0 xs0)
      where
        combineEq x' [] = [x']
        combineEq x' (x:xs)
          | x==x'     = combineEq x' xs
          | otherwise = x' : combineEq x xs

    -- | /O(n)/. Build a set from an ascending list of distinct elements.
    -- /The precondition (input list is strictly ascending) is not checked./
    fromDistinctAscList :: [Key] -> IntSet
    fromDistinctAscList []         = Nil
    fromDistinctAscList (z0 : zs0) = work (prefixOf z0) (bitmapOf z0) zs0 Nada
      where
        -- 'work' accumulates all values that go into one tip, before passing this Tip
        -- to 'reduce'
        work kx bm []     stk = finish kx (Tip kx bm) stk
        work kx bm (z:zs) stk | kx == prefixOf z = work kx (bm .|. bitmapOf z) zs stk
        work kx bm (z:zs) stk = reduce z zs (branchMask z kx) kx (Tip kx bm) stk

        reduce z zs _ px tx Nada = work (prefixOf z) (bitmapOf 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 (prefixOf z) (bitmapOf z) zs (Push px tx stk)

        finish _  t  Nada = t
        finish px tx (Push py ty stk) = finish p (link py ty px tx) stk
            where m = branchMask px py
                  p = mask px m

    data Stack = Push {-# UNPACK #-} !Prefix !IntSet !Stack | Nada

    {--------------------------------------------------------------------
      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 kx1 bm1) (Tip kx2 bm2)
      = kx1 == kx2 && bm1 == bm2
    equal Nil Nil = True
    equal _   _   = 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 kx1 bm1) (Tip kx2 bm2)
      = kx1 /= kx2 || bm1 /= bm2
    nequal Nil Nil = False
    nequal _   _   = 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)

    {--------------------------------------------------------------------
      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
    --------------------------------------------------------------------}

    INSTANCE_TYPEABLE0(IntSet)

    {--------------------------------------------------------------------
      NFData
    --------------------------------------------------------------------}

    -- The IntSet constructors consist only of strict fields of Ints and
    -- IntSets, thus the default NFData instance which evaluates to whnf
    -- should suffice
    instance NFData IntSet where rnf x = seq x ()

    {--------------------------------------------------------------------
      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 kx bm
              -> showsBars lbars . showString " " . shows kx . showString " + " .
                                                    showsBitMap bm . 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 kx bm
              -> showsBars bars . showString " " . shows kx . showString " + " .
                                                   showsBitMap bm . 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 [] = id
    showsBars bars = showString (concat (reverse (tail bars))) . showString node

    showsBitMap :: Word -> ShowS
    showsBitMap = showString . showBitMap

    showBitMap :: Word -> String
    showBitMap w = show $ foldrBits 0 (:) [] w

    node :: String
    node           = "+--"

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

    {--------------------------------------------------------------------
      Helpers
    --------------------------------------------------------------------}
    {--------------------------------------------------------------------
      Link
    --------------------------------------------------------------------}
    link :: Prefix -> IntSet -> Prefix -> IntSet -> IntSet
    link 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 link #-}

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

    {--------------------------------------------------------------------
      @tip@ assures that we never have empty bitmaps within a tree.
    --------------------------------------------------------------------}
    tip :: Prefix -> BitMap -> IntSet
    tip _ 0 = Nil
    tip kx bm = Tip kx bm
    {-# INLINE tip #-}

    {----------------------------------------------------------------------
      Functions that generate Prefix and BitMap of a Key or a Suffix.
    ----------------------------------------------------------------------}

    suffixBitMask :: Int
    #if MIN_VERSION_base(4,7,0)
    suffixBitMask = finiteBitSize (undefined::Word) - 1
    #else
    suffixBitMask = bitSize (undefined::Word) - 1
    #endif
    {-# INLINE suffixBitMask #-}

    prefixBitMask :: Int
    prefixBitMask = complement suffixBitMask
    {-# INLINE prefixBitMask #-}

    prefixOf :: Int -> Prefix
    prefixOf x = x .&. prefixBitMask
    {-# INLINE prefixOf #-}

    suffixOf :: Int -> Int
    suffixOf x = x .&. suffixBitMask
    {-# INLINE suffixOf #-}

    bitmapOfSuffix :: Int -> BitMap
    bitmapOfSuffix s = 1 `shiftLL` s
    {-# INLINE bitmapOfSuffix #-}

    bitmapOf :: Int -> BitMap
    bitmapOf x = bitmapOfSuffix (suffixOf x)
    {-# INLINE bitmapOf #-}

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

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

    match i p m
      = (mask i m) == p
    {-# INLINE match #-}

    -- 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)
    {-# 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 #-}

    {----------------------------------------------------------------------
      To get best performance, we provide fast implementations of
      lowestBitSet, highestBitSet and fold[lr][l]Bits for GHC.
      If the intel bsf and bsr instructions ever become GHC primops,
      this code should be reimplemented using these.

      Performance of this code is crucial for folds, toList, filter, partition.

      The signatures of methods in question are placed after this comment.
    ----------------------------------------------------------------------}

    lowestBitSet :: Nat -> Int
    highestBitSet :: Nat -> Int
    foldlBits :: Int -> (a -> Int -> a) -> a -> Nat -> a
    foldl'Bits :: Int -> (a -> Int -> a) -> a -> Nat -> a
    foldrBits :: Int -> (Int -> a -> a) -> a -> Nat -> a
    foldr'Bits :: Int -> (Int -> a -> a) -> a -> Nat -> a

    {-# INLINE lowestBitSet #-}
    {-# INLINE highestBitSet #-}
    {-# INLINE foldlBits #-}
    {-# INLINE foldl'Bits #-}
    {-# INLINE foldrBits #-}
    {-# INLINE foldr'Bits #-}

    #if defined(__GLASGOW_HASKELL__) && (WORD_SIZE_IN_BITS==32 || WORD_SIZE_IN_BITS==64)
    {----------------------------------------------------------------------
      For lowestBitSet we use wordsize-dependant implementation based on
      multiplication and DeBrujn indeces, which was proposed by Edward Kmett
      <http://haskell.org/pipermail/libraries/2011-September/016749.html>

      The core of this implementation is fast indexOfTheOnlyBit,
      which is given a Nat with exactly one bit set, and returns
      its index.

      Lot of effort was put in these implementations, please benchmark carefully
      before changing this code.
    ----------------------------------------------------------------------}

    indexOfTheOnlyBit :: Nat -> Int
    {-# INLINE indexOfTheOnlyBit #-}
    indexOfTheOnlyBit bitmask =
      I# (lsbArray `indexInt8OffAddr#` unboxInt (intFromNat ((bitmask * magic) `shiftRL` offset)))
      where unboxInt (I# i) = i
    #if WORD_SIZE_IN_BITS==32
            magic = 0x077CB531
            offset = 27
            !lsbArray = "\0\1\28\2\29\14\24\3\30\22\20\15\25\17\4\8\31\27\13\23\21\19\16\7\26\12\18\6\11\5\10\9"#
    #else
            magic = 0x07EDD5E59A4E28C2
            offset = 58
            !lsbArray = "\63\0\58\1\59\47\53\2\60\39\48\27\54\33\42\3\61\51\37\40\49\18\28\20\55\30\34\11\43\14\22\4\62\57\46\52\38\26\32\41\50\36\17\19\29\10\13\21\56\45\25\31\35\16\9\12\44\24\15\8\23\7\6\5"#
    #endif
    -- The lsbArray gets inlined to every call site of indexOfTheOnlyBit.
    -- That cannot be easily avoided, as GHC forbids top-level Addr# literal.
    -- One could go around that by supplying getLsbArray :: () -> Addr# marked
    -- as NOINLINE. But the code size of calling it and processing the result
    -- is 48B on 32-bit and 56B on 64-bit architectures -- so the 32B and 64B array
    -- is actually improvement on 32-bit and only a 8B size increase on 64-bit.

    lowestBitMask :: Nat -> Nat
    lowestBitMask x = x .&. negate x
    {-# INLINE lowestBitMask #-}

    -- Reverse the order of bits in the Nat.
    revNat :: Nat -> Nat
    #if WORD_SIZE_IN_BITS==32
    revNat x1 = case ((x1 `shiftRL` 1) .&. 0x55555555) .|. ((x1 .&. 0x55555555) `shiftLL` 1) of
                  x2 -> case ((x2 `shiftRL` 2) .&. 0x33333333) .|. ((x2 .&. 0x33333333) `shiftLL` 2) of
                     x3 -> case ((x3 `shiftRL` 4) .&. 0x0F0F0F0F) .|. ((x3 .&. 0x0F0F0F0F) `shiftLL` 4) of
                       x4 -> case ((x4 `shiftRL` 8) .&. 0x00FF00FF) .|. ((x4 .&. 0x00FF00FF) `shiftLL` 8) of
                         x5 -> ( x5 `shiftRL` 16             ) .|. ( x5               `shiftLL` 16);
    #else
    revNat x1 = case ((x1 `shiftRL` 1) .&. 0x5555555555555555) .|. ((x1 .&. 0x5555555555555555) `shiftLL` 1) of
                  x2 -> case ((x2 `shiftRL` 2) .&. 0x3333333333333333) .|. ((x2 .&. 0x3333333333333333) `shiftLL` 2) of
                     x3 -> case ((x3 `shiftRL` 4) .&. 0x0F0F0F0F0F0F0F0F) .|. ((x3 .&. 0x0F0F0F0F0F0F0F0F) `shiftLL` 4) of
                       x4 -> case ((x4 `shiftRL` 8) .&. 0x00FF00FF00FF00FF) .|. ((x4 .&. 0x00FF00FF00FF00FF) `shiftLL` 8) of
                         x5 -> case ((x5 `shiftRL` 16) .&. 0x0000FFFF0000FFFF) .|. ((x5 .&. 0x0000FFFF0000FFFF) `shiftLL` 16) of
                           x6 -> ( x6 `shiftRL` 32             ) .|. ( x6               `shiftLL` 32);
    #endif

    lowestBitSet x = indexOfTheOnlyBit (lowestBitMask x)

    highestBitSet x = indexOfTheOnlyBit (highestBitMask x)

    foldlBits prefix f z bitmap = go bitmap z
      where go 0 acc = acc
            go bm acc = go (bm `xor` bitmask) ((f acc) $! (prefix+bi))
              where
                !bitmask = lowestBitMask bm
                !bi = indexOfTheOnlyBit bitmask

    foldl'Bits prefix f z bitmap = go bitmap z
      where go 0 acc = acc
            go bm !acc = go (bm `xor` bitmask) ((f acc) $! (prefix+bi))
              where !bitmask = lowestBitMask bm
                    !bi = indexOfTheOnlyBit bitmask

    foldrBits prefix f z bitmap = go (revNat bitmap) z
      where go 0 acc = acc
            go bm acc = go (bm `xor` bitmask) ((f $! (prefix+(WORD_SIZE_IN_BITS-1)-bi)) acc)
              where !bitmask = lowestBitMask bm
                    !bi = indexOfTheOnlyBit bitmask

    foldr'Bits prefix f z bitmap = go (revNat bitmap) z
      where go 0 acc = acc
            go bm !acc = go (bm `xor` bitmask) ((f $! (prefix+(WORD_SIZE_IN_BITS-1)-bi)) acc)
              where !bitmask = lowestBitMask bm
                    !bi = indexOfTheOnlyBit bitmask

    #else
    {----------------------------------------------------------------------
      In general case we use logarithmic implementation of
      lowestBitSet and highestBitSet, which works up to bit sizes of 64.

      Folds are linear scans.
    ----------------------------------------------------------------------}

    lowestBitSet n0 =
        let (n1,b1) = if n0 .&. 0xFFFFFFFF /= 0 then (n0,0)  else (n0 `shiftRL` 32, 32)
            (n2,b2) = if n1 .&. 0xFFFF /= 0     then (n1,b1) else (n1 `shiftRL` 16, 16+b1)
            (n3,b3) = if n2 .&. 0xFF /= 0       then (n2,b2) else (n2 `shiftRL` 8,  8+b2)
            (n4,b4) = if n3 .&. 0xF /= 0        then (n3,b3) else (n3 `shiftRL` 4,  4+b3)
            (n5,b5) = if n4 .&. 0x3 /= 0        then (n4,b4) else (n4 `shiftRL` 2,  2+b4)
            b6      = if n5 .&. 0x1 /= 0        then     b5  else                   1+b5
        in b6

    highestBitSet n0 =
        let (n1,b1) = if n0 .&. 0xFFFFFFFF00000000 /= 0 then (n0 `shiftRL` 32, 32)    else (n0,0)
            (n2,b2) = if n1 .&. 0xFFFF0000 /= 0         then (n1 `shiftRL` 16, 16+b1) else (n1,b1)
            (n3,b3) = if n2 .&. 0xFF00 /= 0             then (n2 `shiftRL` 8,  8+b2)  else (n2,b2)
            (n4,b4) = if n3 .&. 0xF0 /= 0               then (n3 `shiftRL` 4,  4+b3)  else (n3,b3)
            (n5,b5) = if n4 .&. 0xC /= 0                then (n4 `shiftRL` 2,  2+b4)  else (n4,b4)
            b6      = if n5 .&. 0x2 /= 0                then                   1+b5   else     b5
        in b6

    foldlBits prefix f z bm = let lb = lowestBitSet bm
                              in  go (prefix+lb) z (bm `shiftRL` lb)
      where go !_ acc 0 = acc
            go bi acc n | n `testBit` 0 = go (bi + 1) (f acc bi) (n `shiftRL` 1)
                        | otherwise     = go (bi + 1)    acc     (n `shiftRL` 1)

    foldl'Bits prefix f z bm = let lb = lowestBitSet bm
                               in  go (prefix+lb) z (bm `shiftRL` lb)
      where go !_ !acc 0 = acc
            go bi acc n | n `testBit` 0 = go (bi + 1) (f acc bi) (n `shiftRL` 1)
                        | otherwise     = go (bi + 1)    acc     (n `shiftRL` 1)

    foldrBits prefix f z bm = let lb = lowestBitSet bm
                              in  go (prefix+lb) (bm `shiftRL` lb)
      where go !_ 0 = z
            go bi n | n `testBit` 0 = f bi (go (bi + 1) (n `shiftRL` 1))
                    | otherwise     =       go (bi + 1) (n `shiftRL` 1)

    foldr'Bits prefix f z bm = let lb = lowestBitSet bm
                               in  go (prefix+lb) (bm `shiftRL` lb)
      where
            go !_ 0 = z
            go bi n | n `testBit` 0 = f bi $! go (bi + 1) (n `shiftRL` 1)
                    | otherwise     =         go (bi + 1) (n `shiftRL` 1)

    #endif

    {----------------------------------------------------------------------
      [bitcount] as posted by David F. Place to haskell-cafe on April 11, 2006,
      based on the code on
      http://graphics.stanford.edu/~seander/bithacks.html#CountBitsSetKernighan,
      where the following source is given:
        Published in 1988, the C Programming Language 2nd Ed. (by Brian W.
        Kernighan and Dennis M. Ritchie) mentions this in exercise 2-9. On April
        19, 2006 Don Knuth pointed out to me that this method "was first published
        by Peter Wegner in CACM 3 (1960), 322. (Also discovered independently by
        Derrick Lehmer and published in 1964 in a book edited by Beckenbach.)"
    ----------------------------------------------------------------------}

    bitcount :: Int -> Word -> Int
    #if MIN_VERSION_base(4,5,0)
    bitcount a x = a + popCount x
    #else
    bitcount a0 x0 = go a0 x0
      where go a 0 = a
            go a x = go (a + 1) (x .&. (x-1))
    #endif
    {-# INLINE bitcount #-}

    {--------------------------------------------------------------------
      Utilities
    --------------------------------------------------------------------}

    -- | /O(1)/.  Decompose a set into pieces based on the structure of the underlying
    -- tree.  This function is useful for consuming a set in parallel.
    --
    -- No guarantee is made as to the sizes of the pieces; an internal, but
    -- deterministic process determines this.  However, it is guaranteed that the
    -- pieces returned will be in ascending order (all elements in the first submap
    -- less than all elements in the second, and so on).
    --
    -- Examples:
    --
    -- > splitRoot (fromList [1..120]) == [fromList [1..63],fromList [64..120]]
    -- > splitRoot empty == []
    --
    --  Note that the current implementation does not return more than two subsets,
    --  but you should not depend on this behaviour because it can change in the
    --  future without notice. Also, the current version does not continue
    --  splitting all the way to individual singleton sets -- it stops at some
    --  point.
    splitRoot :: IntSet -> [IntSet]
    splitRoot Nil = []
    -- NOTE: we don't currently split below Tip, but we could.
    splitRoot x@(Tip _ _) = [x]
    splitRoot (Bin _ m l r) | m < 0 = [r, l]
                            | otherwise = [l, r]
    {-# INLINE splitRoot #-}