Data/Set/Internal.hs

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

    #include "containers.h"

    -----------------------------------------------------------------------------
    -- |
    -- Module      :  Data.Set.Internal
    -- Copyright   :  (c) Daan Leijen 2002
    -- 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 sets.
    --
    -- These modules are intended to be imported qualified, to avoid name
    -- clashes with Prelude functions, e.g.
    --
    -- >  import Data.Set (Set)
    -- >  import qualified Data.Set as Set
    --
    -- The implementation of 'Set' is based on /size balanced/ binary trees (or
    -- trees of /bounded balance/) as described by:
    --
    --    * Stephen Adams, \"/Efficient sets: a balancing act/\",
    --      Journal of Functional Programming 3(4):553-562, October 1993,
    --      <http://www.swiss.ai.mit.edu/~adams/BB/>.
    --    * J. Nievergelt and E.M. Reingold,
    --      \"/Binary search trees of bounded balance/\",
    --      SIAM journal of computing 2(1), March 1973.
    --
    --  Bounds for 'union', 'intersection', and 'difference' are as given
    --  by
    --
    --    * Guy Blelloch, Daniel Ferizovic, and Yihan Sun,
    --      \"/Just Join for Parallel Ordered Sets/\",
    --      <https://arxiv.org/abs/1602.02120v3>.
    --
    -- Note that the implementation is /left-biased/ -- the elements of a
    -- first argument are always preferred to the second, for example in
    -- 'union' or 'insert'.  Of course, left-biasing can only be observed
    -- when equality is an equivalence relation instead of structural
    -- equality.
    --
    -- /Warning/: The size of the set must not exceed @maxBound::Int@. Violation of
    -- this condition is not detected and if the size limit is exceeded, the
    -- behavior of the set is completely undefined.
    -----------------------------------------------------------------------------

    -- [Note: Using INLINABLE]
    -- ~~~~~~~~~~~~~~~~~~~~~~~
    -- It is crucial to the performance that the functions specialize on the Ord
    -- type when possible. GHC 7.0 and higher does this by itself when it sees th
    -- unfolding of a function -- that is why all public functions are marked
    -- INLINABLE (that exposes the unfolding).

    -- [Note: Using INLINE]
    -- ~~~~~~~~~~~~~~~~~~~~
    -- For other compilers and GHC pre 7.0, we mark some of the functions INLINE.
    -- We mark the functions that just navigate down the tree (lookup, insert,
    -- delete and similar). That navigation code gets inlined and thus specialized
    -- when possible. There is a price to pay -- code growth. The code INLINED is
    -- therefore only the tree navigation, all the real work (rebalancing) is not
    -- INLINED by using a NOINLINE.
    --
    -- All methods marked INLINE have to be nonrecursive -- a 'go' function doing
    -- the real work is provided.

    -- [Note: Type of local 'go' function]
    -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    -- If the local 'go' function uses an Ord class, it sometimes heap-allocates
    -- the Ord dictionary when the 'go' function does not have explicit type.
    -- In that case we give 'go' explicit type. But this slightly decrease
    -- performance, as the resulting 'go' function can float out to top level.

    -- [Note: Local 'go' functions and capturing]
    -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    -- As opposed to IntSet, when 'go' function captures an argument, increased
    -- heap-allocation can occur: sometimes in a polymorphic function, the 'go'
    -- floats out of its enclosing function and then it heap-allocates the
    -- dictionary and the argument. Maybe it floats out too late and strictness
    -- analyzer cannot see that these could be passed on stack.

    -- [Note: Order of constructors]
    -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    -- The order of constructors of Set matters when considering performance.
    -- Currently in GHC 7.0, when type has 2 constructors, a forward conditional
    -- jump is made when successfully matching second constructor. Successful match
    -- of first constructor results in the forward jump not taken.
    -- On GHC 7.0, reordering constructors from Tip | Bin to Bin | Tip
    -- improves the benchmark by up to 10% on x86.

    module Data.Set.Internal (
                -- * Set type
                  Set(..)       -- instance Eq,Ord,Show,Read,Data,Typeable

                -- * Operators
                , (\\)

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

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

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

                -- * Filter
                , filter
                , takeWhileAntitone
                , dropWhileAntitone
                , spanAntitone
                , partition
                , split
                , splitMember
                , splitRoot

                -- * Indexed
                , lookupIndex
                , findIndex
                , elemAt
                , deleteAt
                , take
                , drop
                , splitAt

                -- * Map
                , map
                , mapMonotonic

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

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

                -- * Conversion

                -- ** List
                , elems
                , toList
                , fromList

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

                -- * Debugging
                , showTree
                , showTreeWith
                , valid

                -- Internals (for testing)
                , bin
                , balanced
                , link
                , merge
                ) where

    import Prelude hiding (filter,foldl,foldr,null,map,take,drop,splitAt)
    import qualified Data.List as List
    import Data.Bits (shiftL, shiftR)
    #if !MIN_VERSION_base(4,8,0)
    import Data.Monoid (Monoid(..))
    #endif
    #if MIN_VERSION_base(4,9,0)
    import Data.Semigroup (Semigroup((<>), stimes), stimesIdempotentMonoid)
    import Data.Functor.Classes
    #endif
    import qualified Data.Foldable as Foldable
    import Data.Typeable
    import Control.DeepSeq (NFData(rnf))

    import Utils.Containers.Internal.StrictFold
    import Utils.Containers.Internal.StrictPair
    import Utils.Containers.Internal.PtrEquality

    #if __GLASGOW_HASKELL__
    import GHC.Exts ( build, lazy )
    #if __GLASGOW_HASKELL__ >= 708
    import qualified GHC.Exts as GHCExts
    #endif
    import Text.Read
    import Data.Data
    #endif

    {--------------------------------------------------------------------
      Operators
    --------------------------------------------------------------------}
    infixl 9 \\ --

    -- | /O(m*log(n\/m+1)), m <= n/. See 'difference'.
    (\\) :: Ord a => Set a -> Set a -> Set a
    m1 \\ m2 = difference m1 m2
    #if __GLASGOW_HASKELL__
    {-# INLINABLE (\\) #-}
    #endif

    {--------------------------------------------------------------------
      Sets are size balanced trees
    --------------------------------------------------------------------}
    -- | A set of values @a@.

    -- See Note: Order of constructors
    data Set a    = Bin {-# UNPACK #-} !Size !a !(Set a) !(Set a)
                  | Tip

    type Size     = Int

    #if __GLASGOW_HASKELL__ >= 708
    type role Set nominal
    #endif

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

    instance Ord a => Semigroup (Set a) where
        (<>)    = union
        stimes  = stimesIdempotentMonoid
    #endif

    instance Foldable.Foldable Set where
        fold = go
          where go Tip = mempty
                go (Bin 1 k _ _) = k
                go (Bin _ k l r) = go l `mappend` (k `mappend` go r)
        {-# INLINABLE fold #-}
        foldr = foldr
        {-# INLINE foldr #-}
        foldl = foldl
        {-# INLINE foldl #-}
        foldMap f t = go t
          where go Tip = mempty
                go (Bin 1 k _ _) = f k
                go (Bin _ k l r) = go l `mappend` (f k `mappend` go r)
        {-# INLINE foldMap #-}

    #if MIN_VERSION_base(4,6,0)
        foldl' = foldl'
        {-# INLINE foldl' #-}
        foldr' = foldr'
        {-# INLINE foldr' #-}
    #endif
    #if MIN_VERSION_base(4,8,0)
        length = size
        {-# INLINE length #-}
        null   = null
        {-# INLINE null #-}
        toList = toList
        {-# INLINE toList #-}
        elem = go
          where go !_ Tip = False
                go x (Bin _ y l r) = x == y || go x l || go x r
        {-# INLINABLE elem #-}
        minimum = findMin
        {-# INLINE minimum #-}
        maximum = findMax
        {-# INLINE maximum #-}
        sum = foldl' (+) 0
        {-# INLINABLE sum #-}
        product = foldl' (*) 1
        {-# INLINABLE product #-}
    #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 a, Ord a) => Data (Set a) where
      gfoldl f z set = z fromList `f` (toList set)
      toConstr _     = fromListConstr
      gunfold k z c  = case constrIndex c of
        1 -> k (z fromList)
        _ -> error "gunfold"
      dataTypeOf _   = setDataType
      dataCast1 f    = gcast1 f

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

    setDataType :: DataType
    setDataType = mkDataType "Data.Set.Internal.Set" [fromListConstr]

    #endif

    {--------------------------------------------------------------------
      Query
    --------------------------------------------------------------------}
    -- | /O(1)/. Is this the empty set?
    null :: Set a -> Bool
    null Tip      = True
    null (Bin {}) = False
    {-# INLINE null #-}

    -- | /O(1)/. The number of elements in the set.
    size :: Set a -> Int
    size Tip = 0
    size (Bin sz _ _ _) = sz
    {-# INLINE size #-}

    -- | /O(log n)/. Is the element in the set?
    member :: Ord a => a -> Set a -> Bool
    member = go
      where
        go !_ Tip = False
        go x (Bin _ y l r) = case compare x y of
          LT -> go x l
          GT -> go x r
          EQ -> True
    #if __GLASGOW_HASKELL__
    {-# INLINABLE member #-}
    #else
    {-# INLINE member #-}
    #endif

    -- | /O(log n)/. Is the element not in the set?
    notMember :: Ord a => a -> Set a -> Bool
    notMember a t = not $ member a t
    #if __GLASGOW_HASKELL__
    {-# INLINABLE notMember #-}
    #else
    {-# INLINE notMember #-}
    #endif

    -- | /O(log n)/. Find largest element smaller than the given one.
    --
    -- > lookupLT 3 (fromList [3, 5]) == Nothing
    -- > lookupLT 5 (fromList [3, 5]) == Just 3
    lookupLT :: Ord a => a -> Set a -> Maybe a
    lookupLT = goNothing
      where
        goNothing !_ Tip = Nothing
        goNothing x (Bin _ y l r) | x <= y = goNothing x l
                                  | otherwise = goJust x y r

        goJust !_ best Tip = Just best
        goJust x best (Bin _ y l r) | x <= y = goJust x best l
                                    | otherwise = goJust x y r
    #if __GLASGOW_HASKELL__
    {-# INLINABLE lookupLT #-}
    #else
    {-# INLINE lookupLT #-}
    #endif

    -- | /O(log n)/. Find smallest element greater than the given one.
    --
    -- > lookupGT 4 (fromList [3, 5]) == Just 5
    -- > lookupGT 5 (fromList [3, 5]) == Nothing
    lookupGT :: Ord a => a -> Set a -> Maybe a
    lookupGT = goNothing
      where
        goNothing !_ Tip = Nothing
        goNothing x (Bin _ y l r) | x < y = goJust x y l
                                  | otherwise = goNothing x r

        goJust !_ best Tip = Just best
        goJust x best (Bin _ y l r) | x < y = goJust x y l
                                    | otherwise = goJust x best r
    #if __GLASGOW_HASKELL__
    {-# INLINABLE lookupGT #-}
    #else
    {-# INLINE lookupGT #-}
    #endif

    -- | /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
    lookupLE :: Ord a => a -> Set a -> Maybe a
    lookupLE = goNothing
      where
        goNothing !_ Tip = Nothing
        goNothing x (Bin _ y l r) = case compare x y of LT -> goNothing x l
                                                        EQ -> Just y
                                                        GT -> goJust x y r

        goJust !_ best Tip = Just best
        goJust x best (Bin _ y l r) = case compare x y of LT -> goJust x best l
                                                          EQ -> Just y
                                                          GT -> goJust x y r
    #if __GLASGOW_HASKELL__
    {-# INLINABLE lookupLE #-}
    #else
    {-# INLINE lookupLE #-}
    #endif

    -- | /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
    lookupGE :: Ord a => a -> Set a -> Maybe a
    lookupGE = goNothing
      where
        goNothing !_ Tip = Nothing
        goNothing x (Bin _ y l r) = case compare x y of LT -> goJust x y l
                                                        EQ -> Just y
                                                        GT -> goNothing x r

        goJust !_ best Tip = Just best
        goJust x best (Bin _ y l r) = case compare x y of LT -> goJust x y l
                                                          EQ -> Just y
                                                          GT -> goJust x best r
    #if __GLASGOW_HASKELL__
    {-# INLINABLE lookupGE #-}
    #else
    {-# INLINE lookupGE #-}
    #endif

    {--------------------------------------------------------------------
      Construction
    --------------------------------------------------------------------}
    -- | /O(1)/. The empty set.
    empty  :: Set a
    empty = Tip
    {-# INLINE empty #-}

    -- | /O(1)/. Create a singleton set.
    singleton :: a -> Set a
    singleton x = Bin 1 x Tip Tip
    {-# INLINE singleton #-}

    {--------------------------------------------------------------------
      Insertion, Deletion
    --------------------------------------------------------------------}
    -- | /O(log n)/. Insert an element in a set.
    -- If the set already contains an element equal to the given value,
    -- it is replaced with the new value.

    -- See Note: Type of local 'go' function
    -- See Note: Avoiding worker/wrapper (in Data.Map.Internal)
    insert :: Ord a => a -> Set a -> Set a
    insert x0 = go x0 x0
      where
        go :: Ord a => a -> a -> Set a -> Set a
        go orig !_ Tip = singleton (lazy orig)
        go orig !x t@(Bin sz y l r) = case compare x y of
            LT | l' `ptrEq` l -> t
               | otherwise -> balanceL y l' r
               where !l' = go orig x l
            GT | r' `ptrEq` r -> t
               | otherwise -> balanceR y l r'
               where !r' = go orig x r
            EQ | lazy orig `seq` (orig `ptrEq` y) -> t
               | otherwise -> Bin sz (lazy orig) l r
    #if __GLASGOW_HASKELL__
    {-# INLINABLE insert #-}
    #else
    {-# INLINE insert #-}
    #endif

    #ifndef __GLASGOW_HASKELL__
    lazy :: a -> a
    lazy a = a
    #endif

    -- Insert an element to the set only if it is not in the set.
    -- Used by `union`.

    -- See Note: Type of local 'go' function
    -- See Note: Avoiding worker/wrapper (in Data.Map.Internal)
    insertR :: Ord a => a -> Set a -> Set a
    insertR x0 = go x0 x0
      where
        go :: Ord a => a -> a -> Set a -> Set a
        go orig !_ Tip = singleton (lazy orig)
        go orig !x t@(Bin _ y l r) = case compare x y of
            LT | l' `ptrEq` l -> t
               | otherwise -> balanceL y l' r
               where !l' = go orig x l
            GT | r' `ptrEq` r -> t
               | otherwise -> balanceR y l r'
               where !r' = go orig x r
            EQ -> t
    #if __GLASGOW_HASKELL__
    {-# INLINABLE insertR #-}
    #else
    {-# INLINE insertR #-}
    #endif

    -- | /O(log n)/. Delete an element from a set.

    -- See Note: Type of local 'go' function
    delete :: Ord a => a -> Set a -> Set a
    delete = go
      where
        go :: Ord a => a -> Set a -> Set a
        go !_ Tip = Tip
        go x t@(Bin _ y l r) = case compare x y of
            LT | l' `ptrEq` l -> t
               | otherwise -> balanceR y l' r
               where !l' = go x l
            GT | r' `ptrEq` r -> t
               | otherwise -> balanceL y l r'
               where !r' = go x r
            EQ -> glue l r
    #if __GLASGOW_HASKELL__
    {-# INLINABLE delete #-}
    #else
    {-# INLINE delete #-}
    #endif

    {--------------------------------------------------------------------
      Subset
    --------------------------------------------------------------------}
    -- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
    isProperSubsetOf :: Ord a => Set a -> Set a -> Bool
    isProperSubsetOf s1 s2
        = (size s1 < size s2) && (isSubsetOf s1 s2)
    #if __GLASGOW_HASKELL__
    {-# INLINABLE isProperSubsetOf #-}
    #endif

    -- | /O(n+m)/. Is this a subset?
    -- @(s1 `isSubsetOf` s2)@ tells whether @s1@ is a subset of @s2@.
    isSubsetOf :: Ord a => Set a -> Set a -> Bool
    isSubsetOf t1 t2
      = (size t1 <= size t2) && (isSubsetOfX t1 t2)
    #if __GLASGOW_HASKELL__
    {-# INLINABLE isSubsetOf #-}
    #endif

    isSubsetOfX :: Ord a => Set a -> Set a -> Bool
    isSubsetOfX Tip _ = True
    isSubsetOfX _ Tip = False
    isSubsetOfX (Bin _ x l r) t
      = found && isSubsetOfX l lt && isSubsetOfX r gt
      where
        (lt,found,gt) = splitMember x t
    #if __GLASGOW_HASKELL__
    {-# INLINABLE isSubsetOfX #-}
    #endif

    {--------------------------------------------------------------------
      Minimal, Maximal
    --------------------------------------------------------------------}

    -- We perform call-pattern specialization manually on lookupMin
    -- and lookupMax. Otherwise, GHC doesn't seem to do it, which is
    -- unfortunate if, for example, someone uses findMin or findMax.

    lookupMinSure :: a -> Set a -> a
    lookupMinSure x Tip = x
    lookupMinSure _ (Bin _ x l _) = lookupMinSure x l

    -- | /O(log n)/. The minimal element of a set.
    --
    -- @since 0.5.9

    lookupMin :: Set a -> Maybe a
    lookupMin Tip = Nothing
    lookupMin (Bin _ x l _) = Just $! lookupMinSure x l

    -- | /O(log n)/. The minimal element of a set.
    findMin :: Set a -> a
    findMin t
      | Just r <- lookupMin t = r
      | otherwise = error "Set.findMin: empty set has no minimal element"

    lookupMaxSure :: a -> Set a -> a
    lookupMaxSure x Tip = x
    lookupMaxSure _ (Bin _ x _ r) = lookupMaxSure x r

    -- | /O(log n)/. The maximal element of a set.
    --
    -- @since 0.5.9

    lookupMax :: Set a -> Maybe a
    lookupMax Tip = Nothing
    lookupMax (Bin _ x _ r) = Just $! lookupMaxSure x r

    -- | /O(log n)/. The maximal element of a set.
    findMax :: Set a -> a
    findMax t
      | Just r <- lookupMax t = r
      | otherwise = error "Set.findMax: empty set has no maximal element"

    -- | /O(log n)/. Delete the minimal element. Returns an empty set if the set is empty.
    deleteMin :: Set a -> Set a
    deleteMin (Bin _ _ Tip r) = r
    deleteMin (Bin _ x l r)   = balanceR x (deleteMin l) r
    deleteMin Tip             = Tip

    -- | /O(log n)/. Delete the maximal element. Returns an empty set if the set is empty.
    deleteMax :: Set a -> Set a
    deleteMax (Bin _ _ l Tip) = l
    deleteMax (Bin _ x l r)   = balanceL x l (deleteMax r)
    deleteMax Tip             = Tip

    {--------------------------------------------------------------------
      Union.
    --------------------------------------------------------------------}
    -- | The union of a list of sets: (@'unions' == 'foldl' 'union' 'empty'@).
    unions :: Ord a => [Set a] -> Set a
    unions = foldlStrict union empty
    #if __GLASGOW_HASKELL__
    {-# INLINABLE unions #-}
    #endif

    -- | /O(m*log(n\/m + 1)), m <= n/. The union of two sets, preferring the first set when
    -- equal elements are encountered.
    union :: Ord a => Set a -> Set a -> Set a
    union t1 Tip  = t1
    union t1 (Bin _ x Tip Tip) = insertR x t1
    union (Bin _ x Tip Tip) t2 = insert x t2
    union Tip t2  = t2
    union t1@(Bin _ x l1 r1) t2 = case splitS x t2 of
      (l2 :*: r2)
        | l1l2 `ptrEq` l1 && r1r2 `ptrEq` r1 -> t1
        | otherwise -> link x l1l2 r1r2
        where !l1l2 = union l1 l2
              !r1r2 = union r1 r2
    #if __GLASGOW_HASKELL__
    {-# INLINABLE union #-}
    #endif

    {--------------------------------------------------------------------
      Difference
    --------------------------------------------------------------------}
    -- | /O(m*log(n\/m + 1)), m <= n/. Difference of two sets.
    difference :: Ord a => Set a -> Set a -> Set a
    difference Tip _   = Tip
    difference t1 Tip  = t1
    difference t1 (Bin _ x l2 r2) = case split x t1 of
       (l1, r1)
         | size l1l2 + size r1r2 == size t1 -> t1
         | otherwise -> merge l1l2 r1r2
         where !l1l2 = difference l1 l2
               !r1r2 = difference r1 r2
    #if __GLASGOW_HASKELL__
    {-# INLINABLE difference #-}
    #endif

    {--------------------------------------------------------------------
      Intersection
    --------------------------------------------------------------------}
    -- | /O(m*log(n\/m + 1)), m <= n/. The intersection of two sets.
    -- Elements of the result come from the first set, so for example
    --
    -- > import qualified Data.Set as S
    -- > data AB = A | B deriving Show
    -- > instance Ord AB where compare _ _ = EQ
    -- > instance Eq AB where _ == _ = True
    -- > main = print (S.singleton A `S.intersection` S.singleton B,
    -- >               S.singleton B `S.intersection` S.singleton A)
    --
    -- prints @(fromList [A],fromList [B])@.
    intersection :: Ord a => Set a -> Set a -> Set a
    intersection Tip _ = Tip
    intersection _ Tip = Tip
    intersection t1@(Bin _ x l1 r1) t2
      | b = if l1l2 `ptrEq` l1 && r1r2 `ptrEq` r1
            then t1
            else link x l1l2 r1r2
      | otherwise = merge l1l2 r1r2
      where
        !(l2, b, r2) = splitMember x t2
        !l1l2 = intersection l1 l2
        !r1r2 = intersection r1 r2
    #if __GLASGOW_HASKELL__
    {-# INLINABLE intersection #-}
    #endif

    {--------------------------------------------------------------------
      Filter and partition
    --------------------------------------------------------------------}
    -- | /O(n)/. Filter all elements that satisfy the predicate.
    filter :: (a -> Bool) -> Set a -> Set a
    filter _ Tip = Tip
    filter p t@(Bin _ x l r)
        | p x = if l `ptrEq` l' && r `ptrEq` r'
                then t
                else link x l' r'
        | otherwise = merge l' r'
        where
          !l' = filter p l
          !r' = filter p r

    -- | /O(n)/. Partition the set into two sets, one with all elements that satisfy
    -- the predicate and one with all elements that don't satisfy the predicate.
    -- See also 'split'.
    partition :: (a -> Bool) -> Set a -> (Set a,Set a)
    partition p0 t0 = toPair $ go p0 t0
      where
        go _ Tip = (Tip :*: Tip)
        go p t@(Bin _ x l r) = case (go p l, go p r) of
          ((l1 :*: l2), (r1 :*: r2))
            | p x       -> (if l1 `ptrEq` l && r1 `ptrEq` r
                            then t
                            else link x l1 r1) :*: merge l2 r2
            | otherwise -> merge l1 r1 :*:
                           (if l2 `ptrEq` l && r2 `ptrEq` r
                            then t
                            else link x l2 r2)

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

    -- | /O(n*log n)/.
    -- @'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 :: Ord b => (a->b) -> Set a -> Set b
    map f = fromList . List.map f . toList
    #if __GLASGOW_HASKELL__
    {-# INLINABLE map #-}
    #endif

    -- | /O(n)/. The
    --
    -- @'mapMonotonic' f s == 'map' f s@, but works only when @f@ is strictly increasing.
    -- /The precondition is not checked./
    -- Semi-formally, we have:
    --
    -- > and [x < y ==> f x < f y | x <- ls, y <- ls]
    -- >                     ==> mapMonotonic f s == map f s
    -- >     where ls = toList s

    mapMonotonic :: (a->b) -> Set a -> Set b
    mapMonotonic _ Tip = Tip
    mapMonotonic f (Bin sz x l r) = Bin sz (f x) (mapMonotonic f l) (mapMonotonic f r)

    {--------------------------------------------------------------------
      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 :: (a -> b -> b) -> b -> Set a -> 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 :: (a -> b -> b) -> b -> Set a -> b
    foldr f z = go z
      where
        go z' Tip           = z'
        go z' (Bin _ x l r) = go (f x (go z' r)) l
    {-# INLINE foldr #-}

    -- | /O(n)/. A strict version of 'foldr'. Each application of the operator is
    -- evaluated before using the result in the next application. This
    -- function is strict in the starting value.
    foldr' :: (a -> b -> b) -> b -> Set a -> b
    foldr' f z = go z
      where
        go !z' Tip           = z'
        go z' (Bin _ x l r) = go (f x (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 -> b -> a) -> a -> Set b -> a
    foldl f z = go z
      where
        go z' Tip           = z'
        go z' (Bin _ x l r) = go (f (go z' l) x) r
    {-# INLINE foldl #-}

    -- | /O(n)/. A strict version of 'foldl'. Each application of the operator is
    -- evaluated before using the result in the next application. This
    -- function is strict in the starting value.
    foldl' :: (a -> b -> a) -> a -> Set b -> a
    foldl' f z = go z
      where
        go !z' Tip           = z'
        go z' (Bin _ x l r) = go (f (go z' l) x) r
    {-# INLINE foldl' #-}

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

    {--------------------------------------------------------------------
      Lists
    --------------------------------------------------------------------}
    #if __GLASGOW_HASKELL__ >= 708
    instance (Ord a) => GHCExts.IsList (Set a) where
      type Item (Set a) = a
      fromList = fromList
      toList   = toList
    #endif

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

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

    -- | /O(n)/. Convert the set to a descending list of elements. Subject to list
    -- fusion.
    toDescList :: Set a -> [a]
    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 :: (a -> b -> b) -> b -> Set a -> b
    foldrFB = foldr
    {-# INLINE[0] foldrFB #-}
    foldlFB :: (a -> b -> a) -> a -> Set b -> 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 "Set.toAscList" [~1] forall s . toAscList s = build (\c n -> foldrFB c n s) #-}
    {-# RULES "Set.toAscListBack" [1] foldrFB (:) [] = toAscList #-}
    {-# RULES "Set.toDescList" [~1] forall s . toDescList s = build (\c n -> foldlFB (\xs x -> c x xs) n s) #-}
    {-# RULES "Set.toDescListBack" [1] foldlFB (\xs x -> x : xs) [] = toDescList #-}
    #endif

    -- | /O(n*log n)/. Create a set from a list of elements.
    --
    -- If the elements are ordered, a linear-time implementation is used,
    -- with the performance equal to 'fromDistinctAscList'.

    -- For some reason, when 'singleton' is used in fromList or in
    -- create, it is not inlined, so we inline it manually.
    fromList :: Ord a => [a] -> Set a
    fromList [] = Tip
    fromList [x] = Bin 1 x Tip Tip
    fromList (x0 : xs0) | not_ordered x0 xs0 = fromList' (Bin 1 x0 Tip Tip) xs0
                        | otherwise = go (1::Int) (Bin 1 x0 Tip Tip) xs0
      where
        not_ordered _ [] = False
        not_ordered x (y : _) = x >= y
        {-# INLINE not_ordered #-}

        fromList' t0 xs = foldlStrict ins t0 xs
          where ins t x = insert x t

        go !_ t [] = t
        go _ t [x] = insertMax x t
        go s l xs@(x : xss) | not_ordered x xss = fromList' l xs
                            | otherwise = case create s xss of
                                (r, ys, []) -> go (s `shiftL` 1) (link x l r) ys
                                (r, _,  ys) -> fromList' (link x l r) ys

        -- The create is returning a triple (tree, xs, ys). Both xs and ys
        -- represent not yet processed elements and only one of them can be nonempty.
        -- If ys is nonempty, the keys in ys are not ordered with respect to tree
        -- and must be inserted using fromList'. Otherwise the keys have been
        -- ordered so far.
        create !_ [] = (Tip, [], [])
        create s xs@(x : xss)
          | s == 1 = if not_ordered x xss then (Bin 1 x Tip Tip, [], xss)
                                          else (Bin 1 x Tip Tip, xss, [])
          | otherwise = case create (s `shiftR` 1) xs of
                          res@(_, [], _) -> res
                          (l, [y], zs) -> (insertMax y l, [], zs)
                          (l, ys@(y:yss), _) | not_ordered y yss -> (l, [], ys)
                                             | otherwise -> case create (s `shiftR` 1) yss of
                                                       (r, zs, ws) -> (link y l r, zs, ws)
    #if __GLASGOW_HASKELL__
    {-# INLINABLE fromList #-}
    #endif

    {--------------------------------------------------------------------
      Building trees from ascending/descending lists can be done in linear time.

      Note that if [xs] is ascending that:
        fromAscList xs == fromList xs
    --------------------------------------------------------------------}
    -- | /O(n)/. Build a set from an ascending list in linear time.
    -- /The precondition (input list is ascending) is not checked./
    fromAscList :: Eq a => [a] -> Set a
    fromAscList xs = fromDistinctAscList (combineEq xs)
    #if __GLASGOW_HASKELL__
    {-# INLINABLE fromAscList #-}
    #endif

    -- | /O(n)/. Build a set from a descending list in linear time.
    -- /The precondition (input list is descending) is not checked./
    fromDescList :: Eq a => [a] -> Set a
    fromDescList xs = fromDistinctDescList (combineEq xs)
    #if __GLASGOW_HASKELL__
    {-# INLINABLE fromDescList #-}
    #endif

    -- [combineEq xs] combines equal elements with [const] in an ordered list [xs]
    --
    -- TODO: combineEq allocates an intermediate list. It *should* be better to
    -- make fromAscListBy and fromDescListBy the fundamental operations, and to
    -- implement the rest using those.
    combineEq :: Eq a => [a] -> [a]
    combineEq [] = []
    combineEq (x : xs) = combineEq' x xs
      where
        combineEq' z [] = [z]
        combineEq' z (y:ys)
          | z == y = combineEq' z ys
          | otherwise = z : combineEq' y ys

    -- | /O(n)/. Build a set from an ascending list of distinct elements in linear time.
    -- /The precondition (input list is strictly ascending) is not checked./

    -- For some reason, when 'singleton' is used in fromDistinctAscList or in
    -- create, it is not inlined, so we inline it manually.
    fromDistinctAscList :: [a] -> Set a
    fromDistinctAscList [] = Tip
    fromDistinctAscList (x0 : xs0) = go (1::Int) (Bin 1 x0 Tip Tip) xs0
      where
        go !_ t [] = t
        go s l (x : xs) = case create s xs of
                            (r :*: ys) -> let !t' = link x l r
                                          in go (s `shiftL` 1) t' ys

        create !_ [] = (Tip :*: [])
        create s xs@(x : xs')
          | s == 1 = (Bin 1 x Tip Tip :*: xs')
          | otherwise = case create (s `shiftR` 1) xs of
                          res@(_ :*: []) -> res
                          (l :*: (y:ys)) -> case create (s `shiftR` 1) ys of
                            (r :*: zs) -> (link y l r :*: zs)

    -- | /O(n)/. Build a set from a descending list of distinct elements in linear time.
    -- /The precondition (input list is strictly descending) is not checked./

    -- For some reason, when 'singleton' is used in fromDistinctDescList or in
    -- create, it is not inlined, so we inline it manually.
    fromDistinctDescList :: [a] -> Set a
    fromDistinctDescList [] = Tip
    fromDistinctDescList (x0 : xs0) = go (1::Int) (Bin 1 x0 Tip Tip) xs0
      where
        go !_ t [] = t
        go s r (x : xs) = case create s xs of
                            (l :*: ys) -> let !t' = link x l r
                                          in go (s `shiftL` 1) t' ys

        create !_ [] = (Tip :*: [])
        create s xs@(x : xs')
          | s == 1 = (Bin 1 x Tip Tip :*: xs')
          | otherwise = case create (s `shiftR` 1) xs of
                          res@(_ :*: []) -> res
                          (r :*: (y:ys)) -> case create (s `shiftR` 1) ys of
                            (l :*: zs) -> (link y l r :*: zs)

    {--------------------------------------------------------------------
      Eq converts the set to a list. In a lazy setting, this
      actually seems one of the faster methods to compare two trees
      and it is certainly the simplest :-)
    --------------------------------------------------------------------}
    instance Eq a => Eq (Set a) where
      t1 == t2  = (size t1 == size t2) && (toAscList t1 == toAscList t2)

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

    instance Ord a => Ord (Set a) where
        compare s1 s2 = compare (toAscList s1) (toAscList s2)

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

    #if MIN_VERSION_base(4,9,0)
    instance Eq1 Set where
        liftEq eq m n =
            size m == size n && liftEq eq (toList m) (toList n)

    instance Ord1 Set where
        liftCompare cmp m n =
            liftCompare cmp (toList m) (toList n)

    instance Show1 Set where
        liftShowsPrec sp sl d m =
            showsUnaryWith (liftShowsPrec sp sl) "fromList" d (toList m)
    #endif

    {--------------------------------------------------------------------
      Read
    --------------------------------------------------------------------}
    instance (Read a, Ord a) => Read (Set a) 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/Data
    --------------------------------------------------------------------}

    INSTANCE_TYPEABLE1(Set)

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

    instance NFData a => NFData (Set a) where
        rnf Tip           = ()
        rnf (Bin _ y l r) = rnf y `seq` rnf l `seq` rnf r

    {--------------------------------------------------------------------
      Split
    --------------------------------------------------------------------}
    -- | /O(log n)/. 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 :: Ord a => a -> Set a -> (Set a,Set a)
    split x t = toPair $ splitS x t
    {-# INLINABLE split #-}

    splitS :: Ord a => a -> Set a -> StrictPair (Set a) (Set a)
    splitS _ Tip = (Tip :*: Tip)
    splitS x (Bin _ y l r)
          = case compare x y of
              LT -> let (lt :*: gt) = splitS x l in (lt :*: link y gt r)
              GT -> let (lt :*: gt) = splitS x r in (link y l lt :*: gt)
              EQ -> (l :*: r)
    {-# INLINABLE splitS #-}

    -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
    -- element was found in the original set.
    splitMember :: Ord a => a -> Set a -> (Set a,Bool,Set a)
    splitMember _ Tip = (Tip, False, Tip)
    splitMember x (Bin _ y l r)
       = case compare x y of
           LT -> let (lt, found, gt) = splitMember x l
                     !gt' = link y gt r
                 in (lt, found, gt')
           GT -> let (lt, found, gt) = splitMember x r
                     !lt' = link y l lt
                 in (lt', found, gt)
           EQ -> (l, True, r)
    #if __GLASGOW_HASKELL__
    {-# INLINABLE splitMember #-}
    #endif

    {--------------------------------------------------------------------
      Indexing
    --------------------------------------------------------------------}

    -- | /O(log n)/. Return the /index/ of an element, which is its zero-based
    -- index in the sorted sequence of elements. The index is a number from /0/ up
    -- to, but not including, the 'size' of the set. Calls 'error' when the element
    -- is not a 'member' of the set.
    --
    -- > findIndex 2 (fromList [5,3])    Error: element is not in the set
    -- > findIndex 3 (fromList [5,3]) == 0
    -- > findIndex 5 (fromList [5,3]) == 1
    -- > findIndex 6 (fromList [5,3])    Error: element is not in the set

    -- See Note: Type of local 'go' function
    findIndex :: Ord a => a -> Set a -> Int
    findIndex = go 0
      where
        go :: Ord a => Int -> a -> Set a -> Int
        go !_ !_ Tip  = error "Set.findIndex: element is not in the set"
        go idx x (Bin _ kx l r) = case compare x kx of
          LT -> go idx x l
          GT -> go (idx + size l + 1) x r
          EQ -> idx + size l
    #if __GLASGOW_HASKELL__
    {-# INLINABLE findIndex #-}
    #endif

    -- | /O(log n)/. Lookup the /index/ of an element, which is its zero-based index in
    -- the sorted sequence of elements. The index is a number from /0/ up to, but not
    -- including, the 'size' of the set.
    --
    -- > isJust   (lookupIndex 2 (fromList [5,3])) == False
    -- > fromJust (lookupIndex 3 (fromList [5,3])) == 0
    -- > fromJust (lookupIndex 5 (fromList [5,3])) == 1
    -- > isJust   (lookupIndex 6 (fromList [5,3])) == False

    -- See Note: Type of local 'go' function
    lookupIndex :: Ord a => a -> Set a -> Maybe Int
    lookupIndex = go 0
      where
        go :: Ord a => Int -> a -> Set a -> Maybe Int
        go !_ !_ Tip  = Nothing
        go idx x (Bin _ kx l r) = case compare x kx of
          LT -> go idx x l
          GT -> go (idx + size l + 1) x r
          EQ -> Just $! idx + size l
    #if __GLASGOW_HASKELL__
    {-# INLINABLE lookupIndex #-}
    #endif

    -- | /O(log n)/. Retrieve an element by its /index/, i.e. by its zero-based
    -- index in the sorted sequence of elements. If the /index/ is out of range (less
    -- than zero, greater or equal to 'size' of the set), 'error' is called.
    --
    -- > elemAt 0 (fromList [5,3]) == 3
    -- > elemAt 1 (fromList [5,3]) == 5
    -- > elemAt 2 (fromList [5,3])    Error: index out of range

    elemAt :: Int -> Set a -> a
    elemAt !_ Tip = error "Set.elemAt: index out of range"
    elemAt i (Bin _ x l r)
      = case compare i sizeL of
          LT -> elemAt i l
          GT -> elemAt (i-sizeL-1) r
          EQ -> x
      where
        sizeL = size l

    -- | /O(log n)/. Delete the element at /index/, i.e. by its zero-based index in
    -- the sorted sequence of elements. If the /index/ is out of range (less than zero,
    -- greater or equal to 'size' of the set), 'error' is called.
    --
    -- > deleteAt 0    (fromList [5,3]) == singleton 5
    -- > deleteAt 1    (fromList [5,3]) == singleton 3
    -- > deleteAt 2    (fromList [5,3])    Error: index out of range
    -- > deleteAt (-1) (fromList [5,3])    Error: index out of range

    deleteAt :: Int -> Set a -> Set a
    deleteAt !i t =
      case t of
        Tip -> error "Set.deleteAt: index out of range"
        Bin _ x l r -> case compare i sizeL of
          LT -> balanceR x (deleteAt i l) r
          GT -> balanceL x l (deleteAt (i-sizeL-1) r)
          EQ -> glue l r
          where
            sizeL = size l

    -- | Take a given number of elements in order, beginning
    -- with the smallest ones.
    --
    -- @
    -- take n = 'fromDistinctAscList' . 'Prelude.take' n . 'toAscList'
    -- @
    take :: Int -> Set a -> Set a
    take i m | i >= size m = m
    take i0 m0 = go i0 m0
      where
        go i !_ | i <= 0 = Tip
        go !_ Tip = Tip
        go i (Bin _ x l r) =
          case compare i sizeL of
            LT -> go i l
            GT -> link x l (go (i - sizeL - 1) r)
            EQ -> l
          where sizeL = size l

    -- | Drop a given number of elements in order, beginning
    -- with the smallest ones.
    --
    -- @
    -- drop n = 'fromDistinctAscList' . 'Prelude.drop' n . 'toAscList'
    -- @
    drop :: Int -> Set a -> Set a
    drop i m | i >= size m = Tip
    drop i0 m0 = go i0 m0
      where
        go i m | i <= 0 = m
        go !_ Tip = Tip
        go i (Bin _ x l r) =
          case compare i sizeL of
            LT -> link x (go i l) r
            GT -> go (i - sizeL - 1) r
            EQ -> insertMin x r
          where sizeL = size l

    -- | /O(log n)/. Split a set at a particular index.
    --
    -- @
    -- splitAt !n !xs = ('take' n xs, 'drop' n xs)
    -- @
    splitAt :: Int -> Set a -> (Set a, Set a)
    splitAt i0 m0
      | i0 >= size m0 = (m0, Tip)
      | otherwise = toPair $ go i0 m0
      where
        go i m | i <= 0 = Tip :*: m
        go !_ Tip = Tip :*: Tip
        go i (Bin _ x l r)
          = case compare i sizeL of
              LT -> case go i l of
                      ll :*: lr -> ll :*: link x lr r
              GT -> case go (i - sizeL - 1) r of
                      rl :*: rr -> link x l rl :*: rr
              EQ -> l :*: insertMin x r
          where sizeL = size l

    -- | /O(log n)/. Take while a predicate on the elements holds.
    -- The user is responsible for ensuring that for all elements @j@ and @k@ in the set,
    -- @j \< k ==\> p j \>= p k@. See note at 'spanAntitone'.
    --
    -- @
    -- takeWhileAntitone p = 'fromDistinctAscList' . 'Data.List.takeWhile' p . 'toList'
    -- takeWhileAntitone p = 'filter' p
    -- @

    takeWhileAntitone :: (a -> Bool) -> Set a -> Set a
    takeWhileAntitone _ Tip = Tip
    takeWhileAntitone p (Bin _ x l r)
      | p x = link x l (takeWhileAntitone p r)
      | otherwise = takeWhileAntitone p l

    -- | /O(log n)/. Drop while a predicate on the elements holds.
    -- The user is responsible for ensuring that for all elements @j@ and @k@ in the set,
    -- @j \< k ==\> p j \>= p k@. See note at 'spanAntitone'.
    --
    -- @
    -- dropWhileAntitone p = 'fromDistinctAscList' . 'Data.List.dropWhile' p . 'toList'
    -- dropWhileAntitone p = 'filter' (not . p)
    -- @

    dropWhileAntitone :: (a -> Bool) -> Set a -> Set a
    dropWhileAntitone _ Tip = Tip
    dropWhileAntitone p (Bin _ x l r)
      | p x = dropWhileAntitone p r
      | otherwise = link x (dropWhileAntitone p l) r

    -- | /O(log n)/. Divide a set at the point where a predicate on the elements stops holding.
    -- The user is responsible for ensuring that for all elements @j@ and @k@ in the set,
    -- @j \< k ==\> p j \>= p k@.
    --
    -- @
    -- spanAntitone p xs = ('takeWhileAntitone' p xs, 'dropWhileAntitone' p xs)
    -- spanAntitone p xs = partition p xs
    -- @
    --
    -- Note: if @p@ is not actually antitone, then @spanAntitone@ will split the set
    -- at some /unspecified/ point where the predicate switches from holding to not
    -- holding (where the predicate is seen to hold before the first element and to fail
    -- after the last element).

    spanAntitone :: (a -> Bool) -> Set a -> (Set a, Set a)
    spanAntitone p0 m = toPair (go p0 m)
      where
        go _ Tip = Tip :*: Tip
        go p (Bin _ x l r)
          | p x = let u :*: v = go p r in link x l u :*: v
          | otherwise = let u :*: v = go p l in u :*: link x v r

    {--------------------------------------------------------------------
      Utility functions that maintain the balance properties of the tree.
      All constructors assume that all values in [l] < [x] and all values
      in [r] > [x], and that [l] and [r] are valid trees.

      In order of sophistication:
        [Bin sz x l r]    The type constructor.
        [bin x l r]       Maintains the correct size, assumes that both [l]
                          and [r] are balanced with respect to each other.
        [balance x l r]   Restores the balance and size.
                          Assumes that the original tree was balanced and
                          that [l] or [r] has changed by at most one element.
        [link x l r]      Restores balance and size.

      Furthermore, we can construct a new tree from two trees. Both operations
      assume that all values in [l] < all values in [r] and that [l] and [r]
      are valid:
        [glue l r]        Glues [l] and [r] together. Assumes that [l] and
                          [r] are already balanced with respect to each other.
        [merge l r]       Merges two trees and restores balance.
    --------------------------------------------------------------------}

    {--------------------------------------------------------------------
      Link
    --------------------------------------------------------------------}
    link :: a -> Set a -> Set a -> Set a
    link x Tip r  = insertMin x r
    link x l Tip  = insertMax x l
    link x l@(Bin sizeL y ly ry) r@(Bin sizeR z lz rz)
      | delta*sizeL < sizeR  = balanceL z (link x l lz) rz
      | delta*sizeR < sizeL  = balanceR y ly (link x ry r)
      | otherwise            = bin x l r

    -- insertMin and insertMax don't perform potentially expensive comparisons.
    insertMax,insertMin :: a -> Set a -> Set a
    insertMax x t
      = case t of
          Tip -> singleton x
          Bin _ y l r
              -> balanceR y l (insertMax x r)

    insertMin x t
      = case t of
          Tip -> singleton x
          Bin _ y l r
              -> balanceL y (insertMin x l) r

    {--------------------------------------------------------------------
      [merge l r]: merges two trees.
    --------------------------------------------------------------------}
    merge :: Set a -> Set a -> Set a
    merge Tip r   = r
    merge l Tip   = l
    merge l@(Bin sizeL x lx rx) r@(Bin sizeR y ly ry)
      | delta*sizeL < sizeR = balanceL y (merge l ly) ry
      | delta*sizeR < sizeL = balanceR x lx (merge rx r)
      | otherwise           = glue l r

    {--------------------------------------------------------------------
      [glue l r]: glues two trees together.
      Assumes that [l] and [r] are already balanced with respect to each other.
    --------------------------------------------------------------------}
    glue :: Set a -> Set a -> Set a
    glue Tip r = r
    glue l Tip = l
    glue l@(Bin sl xl ll lr) r@(Bin sr xr rl rr)
      | sl > sr = let !(m :*: l') = maxViewSure xl ll lr in balanceR m l' r
      | otherwise = let !(m :*: r') = minViewSure xr rl rr in balanceL m l r'

    -- | /O(log n)/. Delete and find the minimal element.
    --
    -- > deleteFindMin set = (findMin set, deleteMin set)

    deleteFindMin :: Set a -> (a,Set a)
    deleteFindMin t
      | Just r <- minView t = r
      | otherwise = (error "Set.deleteFindMin: can not return the minimal element of an empty set", Tip)

    -- | /O(log n)/. Delete and find the maximal element.
    --
    -- > deleteFindMax set = (findMax set, deleteMax set)
    deleteFindMax :: Set a -> (a,Set a)
    deleteFindMax t
      | Just r <- maxView t = r
      | otherwise = (error "Set.deleteFindMax: can not return the maximal element of an empty set", Tip)

    minViewSure :: a -> Set a -> Set a -> StrictPair a (Set a)
    minViewSure = go
      where
        go x Tip r = x :*: r
        go x (Bin _ xl ll lr) r =
          case go xl ll lr of
            xm :*: l' -> xm :*: balanceR x l' r

    -- | /O(log n)/. Retrieves the minimal key of the set, and the set
    -- stripped of that element, or 'Nothing' if passed an empty set.
    minView :: Set a -> Maybe (a, Set a)
    minView Tip = Nothing
    minView (Bin _ x l r) = Just $! toPair $ minViewSure x l r

    maxViewSure :: a -> Set a -> Set a -> StrictPair a (Set a)
    maxViewSure = go
      where
        go x l Tip = x :*: l
        go x l (Bin _ xr rl rr) =
          case go xr rl rr of
            xm :*: r' -> xm :*: balanceL x l r'

    -- | /O(log n)/. Retrieves the maximal key of the set, and the set
    -- stripped of that element, or 'Nothing' if passed an empty set.
    maxView :: Set a -> Maybe (a, Set a)
    maxView Tip = Nothing
    maxView (Bin _ x l r) = Just $! toPair $ maxViewSure x l r

    {--------------------------------------------------------------------
      [balance x l r] balances two trees with value x.
      The sizes of the trees should balance after decreasing the
      size of one of them. (a rotation).

      [delta] is the maximal relative difference between the sizes of
              two trees, it corresponds with the [w] in Adams' paper.
      [ratio] is the ratio between an outer and inner sibling of the
              heavier subtree in an unbalanced setting. It determines
              whether a double or single rotation should be performed
              to restore balance. It is correspondes with the inverse
              of $\alpha$ in Adam's article.

      Note that according to the Adam's paper:
      - [delta] should be larger than 4.646 with a [ratio] of 2.
      - [delta] should be larger than 3.745 with a [ratio] of 1.534.

      But the Adam's paper is errorneous:
      - it can be proved that for delta=2 and delta>=5 there does
        not exist any ratio that would work
      - delta=4.5 and ratio=2 does not work

      That leaves two reasonable variants, delta=3 and delta=4,
      both with ratio=2.

      - A lower [delta] leads to a more 'perfectly' balanced tree.
      - A higher [delta] performs less rebalancing.

      In the benchmarks, delta=3 is faster on insert operations,
      and delta=4 has slightly better deletes. As the insert speedup
      is larger, we currently use delta=3.

    --------------------------------------------------------------------}
    delta,ratio :: Int
    delta = 3
    ratio = 2

    -- The balance function is equivalent to the following:
    --
    --   balance :: a -> Set a -> Set a -> Set a
    --   balance x l r
    --     | sizeL + sizeR <= 1   = Bin sizeX x l r
    --     | sizeR > delta*sizeL  = rotateL x l r
    --     | sizeL > delta*sizeR  = rotateR x l r
    --     | otherwise            = Bin sizeX x l r
    --     where
    --       sizeL = size l
    --       sizeR = size r
    --       sizeX = sizeL + sizeR + 1
    --
    --   rotateL :: a -> Set a -> Set a -> Set a
    --   rotateL x l r@(Bin _ _ ly ry) | size ly < ratio*size ry = singleL x l r
    --                                 | otherwise               = doubleL x l r
    --   rotateR :: a -> Set a -> Set a -> Set a
    --   rotateR x l@(Bin _ _ ly ry) r | size ry < ratio*size ly = singleR x l r
    --                                 | otherwise               = doubleR x l r
    --
    --   singleL, singleR :: a -> Set a -> Set a -> Set a
    --   singleL x1 t1 (Bin _ x2 t2 t3)  = bin x2 (bin x1 t1 t2) t3
    --   singleR x1 (Bin _ x2 t1 t2) t3  = bin x2 t1 (bin x1 t2 t3)
    --
    --   doubleL, doubleR :: a -> Set a -> Set a -> Set a
    --   doubleL x1 t1 (Bin _ x2 (Bin _ x3 t2 t3) t4) = bin x3 (bin x1 t1 t2) (bin x2 t3 t4)
    --   doubleR x1 (Bin _ x2 t1 (Bin _ x3 t2 t3)) t4 = bin x3 (bin x2 t1 t2) (bin x1 t3 t4)
    --
    -- It is only written in such a way that every node is pattern-matched only once.
    --
    -- Only balanceL and balanceR are needed at the moment, so balance is not here anymore.
    -- In case it is needed, it can be found in Data.Map.

    -- Functions balanceL and balanceR are specialised versions of balance.
    -- balanceL only checks whether the left subtree is too big,
    -- balanceR only checks whether the right subtree is too big.

    -- balanceL is called when left subtree might have been inserted to or when
    -- right subtree might have been deleted from.
    balanceL :: a -> Set a -> Set a -> Set a
    balanceL x l r = case r of
      Tip -> case l of
               Tip -> Bin 1 x Tip Tip
               (Bin _ _ Tip Tip) -> Bin 2 x l Tip
               (Bin _ lx Tip (Bin _ lrx _ _)) -> Bin 3 lrx (Bin 1 lx Tip Tip) (Bin 1 x Tip Tip)
               (Bin _ lx ll@(Bin _ _ _ _) Tip) -> Bin 3 lx ll (Bin 1 x Tip Tip)
               (Bin ls lx ll@(Bin lls _ _ _) lr@(Bin lrs lrx lrl lrr))
                 | lrs < ratio*lls -> Bin (1+ls) lx ll (Bin (1+lrs) x lr Tip)
                 | otherwise -> Bin (1+ls) lrx (Bin (1+lls+size lrl) lx ll lrl) (Bin (1+size lrr) x lrr Tip)

      (Bin rs _ _ _) -> case l of
               Tip -> Bin (1+rs) x Tip r

               (Bin ls lx ll lr)
                  | ls > delta*rs  -> case (ll, lr) of
                       (Bin lls _ _ _, Bin lrs lrx lrl lrr)
                         | lrs < ratio*lls -> Bin (1+ls+rs) lx ll (Bin (1+rs+lrs) x lr r)
                         | otherwise -> Bin (1+ls+rs) lrx (Bin (1+lls+size lrl) lx ll lrl) (Bin (1+rs+size lrr) x lrr r)
                       (_, _) -> error "Failure in Data.Map.balanceL"
                  | otherwise -> Bin (1+ls+rs) x l r
    {-# NOINLINE balanceL #-}

    -- balanceR is called when right subtree might have been inserted to or when
    -- left subtree might have been deleted from.
    balanceR :: a -> Set a -> Set a -> Set a
    balanceR x l r = case l of
      Tip -> case r of
               Tip -> Bin 1 x Tip Tip
               (Bin _ _ Tip Tip) -> Bin 2 x Tip r
               (Bin _ rx Tip rr@(Bin _ _ _ _)) -> Bin 3 rx (Bin 1 x Tip Tip) rr
               (Bin _ rx (Bin _ rlx _ _) Tip) -> Bin 3 rlx (Bin 1 x Tip Tip) (Bin 1 rx Tip Tip)
               (Bin rs rx rl@(Bin rls rlx rll rlr) rr@(Bin rrs _ _ _))
                 | rls < ratio*rrs -> Bin (1+rs) rx (Bin (1+rls) x Tip rl) rr
                 | otherwise -> Bin (1+rs) rlx (Bin (1+size rll) x Tip rll) (Bin (1+rrs+size rlr) rx rlr rr)

      (Bin ls _ _ _) -> case r of
               Tip -> Bin (1+ls) x l Tip

               (Bin rs rx rl rr)
                  | rs > delta*ls  -> case (rl, rr) of
                       (Bin rls rlx rll rlr, Bin rrs _ _ _)
                         | rls < ratio*rrs -> Bin (1+ls+rs) rx (Bin (1+ls+rls) x l rl) rr
                         | otherwise -> Bin (1+ls+rs) rlx (Bin (1+ls+size rll) x l rll) (Bin (1+rrs+size rlr) rx rlr rr)
                       (_, _) -> error "Failure in Data.Map.balanceR"
                  | otherwise -> Bin (1+ls+rs) x l r
    {-# NOINLINE balanceR #-}

    {--------------------------------------------------------------------
      The bin constructor maintains the size of the tree
    --------------------------------------------------------------------}
    bin :: a -> Set a -> Set a -> Set a
    bin x l r
      = Bin (size l + size r + 1) x l r
    {-# INLINE bin #-}

    {--------------------------------------------------------------------
      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 subset less than all
    -- elements in the second, and so on).
    --
    -- Examples:
    --
    -- > splitRoot (fromList [1..6]) ==
    -- >   [fromList [1,2,3],fromList [4],fromList [5,6]]
    --
    -- > splitRoot empty == []
    --
    --  Note that the current implementation does not return more than three subsets,
    --  but you should not depend on this behaviour because it can change in the
    --  future without notice.
    splitRoot :: Set a -> [Set a]
    splitRoot orig =
      case orig of
        Tip           -> []
        Bin _ v l r -> [l, singleton v, r]
    {-# INLINE splitRoot #-}

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

    > Set> putStrLn $ showTreeWith True False $ fromDistinctAscList [1..5]
    > 4
    > +--2
    > |  +--1
    > |  +--3
    > +--5
    >
    > Set> putStrLn $ showTreeWith True True $ fromDistinctAscList [1..5]
    > 4
    > |
    > +--2
    > |  |
    > |  +--1
    > |  |
    > |  +--3
    > |
    > +--5
    >
    > Set> putStrLn $ showTreeWith False True $ fromDistinctAscList [1..5]
    > +--5
    > |
    > 4
    > |
    > |  +--3
    > |  |
    > +--2
    >    |
    >    +--1

    -}
    showTreeWith :: Show a => Bool -> Bool -> Set a -> String
    showTreeWith hang wide t
      | hang      = (showsTreeHang wide [] t) ""
      | otherwise = (showsTree wide [] [] t) ""

    showsTree :: Show a => Bool -> [String] -> [String] -> Set a -> ShowS
    showsTree wide lbars rbars t
      = case t of
          Tip -> showsBars lbars . showString "|\n"
          Bin _ x Tip Tip
              -> showsBars lbars . shows x . showString "\n"
          Bin _ x l r
              -> showsTree wide (withBar rbars) (withEmpty rbars) r .
                 showWide wide rbars .
                 showsBars lbars . shows x . showString "\n" .
                 showWide wide lbars .
                 showsTree wide (withEmpty lbars) (withBar lbars) l

    showsTreeHang :: Show a => Bool -> [String] -> Set a -> ShowS
    showsTreeHang wide bars t
      = case t of
          Tip -> showsBars bars . showString "|\n"
          Bin _ x Tip Tip
              -> showsBars bars . shows x . showString "\n"
          Bin _ x l r
              -> showsBars bars . shows x . showString "\n" .
                 showWide wide bars .
                 showsTreeHang wide (withBar bars) l .
                 showWide wide bars .
                 showsTreeHang wide (withEmpty bars) r

    showWide :: Bool -> [String] -> String -> String
    showWide wide bars
      | wide      = showString (concat (reverse bars)) . showString "|\n"
      | otherwise = id

    showsBars :: [String] -> ShowS
    showsBars bars
      = case bars of
          [] -> id
          _  -> showString (concat (reverse (tail bars))) . showString node

    node :: String
    node           = "+--"

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

    {--------------------------------------------------------------------
      Assertions
    --------------------------------------------------------------------}
    -- | /O(n)/. Test if the internal set structure is valid.
    valid :: Ord a => Set a -> Bool
    valid t
      = balanced t && ordered t && validsize t

    ordered :: Ord a => Set a -> Bool
    ordered t
      = bounded (const True) (const True) t
      where
        bounded lo hi t'
          = case t' of
              Tip         -> True
              Bin _ x l r -> (lo x) && (hi x) && bounded lo (<x) l && bounded (>x) hi r

    balanced :: Set a -> Bool
    balanced t
      = case t of
          Tip         -> True
          Bin _ _ l r -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) &&
                         balanced l && balanced r

    validsize :: Set a -> Bool
    validsize t
      = (realsize t == Just (size t))
      where
        realsize t'
          = case t' of
              Tip          -> Just 0
              Bin sz _ l r -> case (realsize l,realsize r) of
                                (Just n,Just m)  | n+m+1 == sz  -> Just sz
                                _                -> Nothing