content: {-# LANGUAGE CPP #-}
  #if __GLASGOW_HASKELL__
  {-# LANGUAGE MagicHash, BangPatterns, DeriveDataTypeable, StandaloneDeriving #-}
  #endif
  #if !defined(TESTING) && __GLASGOW_HASKELL__ >= 703
  {-# LANGUAGE Trustworthy #-}
  #endif
  -----------------------------------------------------------------------------
  -- |
  -- Module      :  Data.IntSet.Base
  -- Copyright   :  (c) Daan Leijen 2002
  --                (c) Joachim Breitner 2011
  -- License     :  BSD-style
  -- Maintainer  :  libraries@haskell.org
  -- Stability   :  provisional
  -- Portability :  portable
  --
  -- 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.Base (
              -- * Set type
                IntSet(..)      -- 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
  
              -- * 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 Prelude hiding (filter,foldr,foldl,null,map)
  import Data.Bits
  
  import qualified Data.List as List
  import Data.Monoid (Monoid(..))
  import Data.Maybe (fromMaybe)
  import Data.Typeable
  import Control.DeepSeq (NFData)
  
  #if __GLASGOW_HASKELL__
  import Text.Read
  import Data.Data (Data(..), mkNoRepType)
  #endif
  
  #if __GLASGOW_HASKELL__
  import GHC.Exts ( Word(..), Int(..), build )
  import GHC.Prim ( uncheckedShiftL#, uncheckedShiftRL#, indexInt8OffAddr# )
  #else
  import Data.Word
  #endif
  
  -- On GHC, include MachDeps.h to get WORD_SIZE_IN_BITS macro.
  #if defined(__GLASGOW_HASKELL__)
  #include "MachDeps.h"
  #endif
  
  -- Use macros to define strictness of functions.
  -- STRICT_x_OF_y denotes an y-ary function strict in the x-th parameter.
  -- We do not use BangPatterns, because they are not in any standard and we
  -- want the compilers to be compiled by as many compilers as possible.
  #define STRICT_1_OF_2(fn) fn arg _ | arg `seq` False = undefined
  #define STRICT_2_OF_2(fn) fn _ arg | arg `seq` False = undefined
  #define STRICT_1_OF_3(fn) fn arg _ _ | arg `seq` False = undefined
  #define STRICT_2_OF_3(fn) fn _ arg _ | arg `seq` False = undefined
  
  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 #-}
  
  -- Right and left logical shifts.
  shiftRL, shiftLL :: Nat -> Int -> Nat
  #if __GLASGOW_HASKELL__
  {--------------------------------------------------------------------
    GHC: use unboxing to get @shiftRL@ and @shiftLL@ inlined.
  --------------------------------------------------------------------}
  shiftRL (W# x) (I# i) = W# (uncheckedShiftRL# x i)
  shiftLL (W# x) (I# i) = W# (uncheckedShiftL#  x i)
  #else
  shiftRL x i   = shiftR x i
  shiftLL x i   = shiftL x i
  #endif
  {-# INLINE shiftRL #-}
  {-# INLINE shiftLL #-}
  
  {--------------------------------------------------------------------
    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
  
  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 _   = False
  {-# INLINE null #-}
  
  -- | /O(n)/. Cardinality of the set.
  size :: IntSet -> Int
  size t
    = case t of
        Bin _ _ l r -> size l + size r
        Tip _ bm -> bitcount 0 bm
        Nil   -> 0
  
  -- | /O(min(n,W))/. Is the value a member of the set?
  
  -- See Note: Local 'go' functions and capturing]
  member :: Int -> IntSet -> Bool
  member x = x `seq` 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 :: Int -> 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 :: Int -> IntSet -> Maybe Int
  lookupLT x t = x `seq` 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 :: Int -> IntSet -> Maybe Int
  lookupGT x t = x `seq` 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 :: Int -> IntSet -> Maybe Int
  lookupLE x t = x `seq` 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 :: Int -> IntSet -> Maybe Int
  lookupGE x t = x `seq` 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 Int
  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 Int
  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 :: Int -> 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 :: Int -> IntSet -> IntSet
  insert x = x `seq` insertBM (prefixOf x) (bitmapOf x)
  
  -- Helper function for insert and union.
  insertBM :: Prefix -> BitMap -> IntSet -> IntSet
  insertBM kx bm t = kx `seq` bm `seq`
    case t of
      Bin p m l r
        | nomatch kx p m -> join 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)
      Tip kx' bm'
        | kx' == kx -> Tip kx' (bm .|. bm')
        | otherwise -> join kx (Tip kx bm) kx' t
      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 :: Int -> IntSet -> IntSet
  delete x = x `seq` 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 = kx `seq` bm `seq`
    case t of
      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)
      Tip kx' bm'
        | kx' == kx -> tip kx (bm' .&. complement bm)
        | 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 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 :: (Int -> 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 :: (Int -> Bool) -> IntSet -> (IntSet,IntSet)
  partition predicate t
    = case t of
        Bin p m l r
          -> let (l1,l2) = partition predicate l
                 (r1,r2) = partition 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 :: Int -> IntSet -> (IntSet,IntSet)
  split x t =
    case t of Bin _ m l r | m < 0 -> if x >= 0 then case go x l of (lt, gt) -> (union lt r, gt)
                                               else case go x r of (lt, gt) -> (lt, union gt l)
              _ -> 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, 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 :: Int -> 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) -> (union lt r, fnd, gt)
                                               else case go x r of (lt, fnd, gt) -> (lt, fnd, union gt l)
              _ -> 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 = (tip kx' (bm .&. lowerBitmap), (bm .&. bitmapOfx') /= 0, tip kx' (bm .&. higherBitmap))
                                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 (Int, 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 (Int, 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 -> (Int, 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 -> (Int, IntSet)
  deleteFindMax = fromMaybe (error "deleteFindMax: empty set has no maximal element") . maxView
  
  
  -- | /O(min(n,W))/. The minimal element of the set.
  findMin :: IntSet -> Int
  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 -> Int
  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.
  deleteMin :: IntSet -> IntSet
  deleteMin = maybe Nil snd . minView
  
  -- | /O(min(n,W))/. Delete the maximal element.
  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 :: (Int->Int) -> 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 :: (Int -> 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 :: (Int -> 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' :: (Int -> 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
      STRICT_1_OF_2(go)
      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 -> Int -> 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
      STRICT_1_OF_2(go)
      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 -> Int -> 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
      STRICT_1_OF_2(go)
      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 -> [Int]
  elems
    = toAscList
  
  {--------------------------------------------------------------------
    Lists
  --------------------------------------------------------------------}
  -- | /O(n)/. Convert the set to a list of elements. Subject to list fusion.
  toList :: IntSet -> [Int]
  toList
    = toAscList
  
  -- | /O(n)/. Convert the set to an ascending list of elements. Subject to list
  -- fusion.
  toAscList :: IntSet -> [Int]
  toAscList = foldr (:) []
  
  -- | /O(n)/. Convert the set to a descending list of elements. Subject to list
  -- fusion.
  toDescList :: IntSet -> [Int]
  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 :: (Int -> b -> b) -> b -> IntSet -> b
  foldrFB = foldr
  {-# INLINE[0] foldrFB #-}
  foldlFB :: (a -> Int -> 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 :: [Int] -> 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 :: [Int] -> 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 :: [Int] -> 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 (join 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
  --------------------------------------------------------------------}
  
  #include "Typeable.h"
  INSTANCE_TYPEABLE0(IntSet,intSetTc,"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
  
  {--------------------------------------------------------------------
    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 bars
    = case bars of
        [] -> id
        _  -> 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
  --------------------------------------------------------------------}
  {--------------------------------------------------------------------
    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
  {-# INLINE join #-}
  
  {--------------------------------------------------------------------
    @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
  suffixBitMask = bitSize (undefined::Word) - 1
  {-# 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 #-}
  
  {----------------------------------------------------------------------
    Finding the highest bit (mask) in a word [x] can be done efficiently in
    three ways:
    * convert to a floating point value and the mantissa tells us the
      [log2(x)] that corresponds with the highest bit position. The mantissa
      is retrieved either via the standard C function [frexp] or by some bit
      twiddling on IEEE compatible numbers (float). Note that one needs to
      use at least [double] precision for an accurate mantissa of 32 bit
      numbers.
    * use bit twiddling, a logarithmic sequence of bitwise or's and shifts (bit).
    * use processor specific assembler instruction (asm).
  
    The most portable way would be [bit], but is it efficient enough?
    I have measured the cycle counts of the different methods on an AMD
    Athlon-XP 1800 (~ Pentium III 1.8Ghz) using the RDTSC instruction:
  
    highestBitMask: method  cycles
                    --------------
                     frexp   200
                     float    33
                     bit      11
                     asm      12
  
    highestBit:     method  cycles
                    --------------
                     frexp   195
                     float    33
                     bit      11
                     asm      11
  
    Wow, the bit twiddling is on today's RISC like machines even faster
    than a single CISC instruction (BSR)!
  ----------------------------------------------------------------------}
  
  {----------------------------------------------------------------------
    [highestBitMask] returns a word where only the highest bit is set.
    It is found by first setting all bits in lower positions than the
    highest bit and than taking an exclusive or with the original value.
    Allthough the function may look expensive, GHC compiles this into
    excellent C code that subsequently compiled into highly efficient
    machine code. The algorithm is derived from Jorg Arndt's FXT library.
  ----------------------------------------------------------------------}
  highestBitMask :: Nat -> Nat
  highestBitMask x0
    = case (x0 .|. shiftRL x0 1) of
       x1 -> case (x1 .|. shiftRL x1 2) of
        x2 -> case (x2 .|. shiftRL x2 4) of
         x3 -> case (x3 .|. shiftRL x3 8) of
          x4 -> case (x4 .|. shiftRL x4 16) of
  #if !(defined(__GLASGOW_HASKELL__) && WORD_SIZE_IN_BITS==32)
           x5 -> case (x5 .|. shiftRL x5 32) of   -- for 64 bit platforms
  #endif
            x6 -> (x6 `xor` (shiftRL x6 1))
  {-# INLINE highestBitMask #-}
  
  {----------------------------------------------------------------------
    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 bm acc | bm == 0 = acc
                    | otherwise = case lowestBitMask bm of
                                    bitmask -> bitmask `seq` case indexOfTheOnlyBit bitmask of
                                      bi -> bi `seq` go (bm `xor` bitmask) ((f acc) $! (prefix+bi))
  
  foldl'Bits prefix f z bitmap = go bitmap z
    where STRICT_2_OF_2(go)
          go bm acc | bm == 0 = acc
                    | otherwise = case lowestBitMask bm of
                                    bitmask -> bitmask `seq` case indexOfTheOnlyBit bitmask of
                                      bi -> bi `seq` go (bm `xor` bitmask) ((f acc) $! (prefix+bi))
  
  foldrBits prefix f z bitmap = go (revNat bitmap) z
    where go bm acc | bm == 0 = acc
                    | otherwise = case lowestBitMask bm of
                                    bitmask -> bitmask `seq` case indexOfTheOnlyBit bitmask of
                                      bi -> bi `seq` go (bm `xor` bitmask) ((f $! (prefix+(WORD_SIZE_IN_BITS-1)-bi)) acc)
  
  foldr'Bits prefix f z bitmap = go (revNat bitmap) z
    where STRICT_2_OF_2(go)
          go bm acc | bm == 0 = acc
                    | otherwise = case lowestBitMask bm of
                                    bitmask -> bitmask `seq` case indexOfTheOnlyBit bitmask of
                                      bi -> bi `seq` go (bm `xor` bitmask) ((f $! (prefix+(WORD_SIZE_IN_BITS-1)-bi)) acc)
  
  #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 STRICT_1_OF_3(go)
          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 STRICT_1_OF_3(go)
          STRICT_2_OF_3(go)
          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 STRICT_1_OF_2(go)
          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 STRICT_1_OF_2(go)
          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
  bitcount a0 x0 = go a0 x0
    where go a 0 = a
          go a x = go (a + 1) (x .&. (x-1))
  {-# INLINE bitcount #-}
  
  
  {--------------------------------------------------------------------
    Utilities
  --------------------------------------------------------------------}
  foldlStrict :: (a -> b -> a) -> a -> [b] -> a
  foldlStrict f = go
    where
      go z []     = z
      go z (x:xs) = let z' = f z x in z' `seq` go z' xs
  {-# INLINE foldlStrict #-}
  