content: {-# LANGUAGE CPP #-}
  #if __GLASGOW_HASKELL__
  {-# LANGUAGE DeriveDataTypeable, StandaloneDeriving #-}
  #endif
  #if !defined(TESTING) && __GLASGOW_HASKELL__ >= 703
  {-# LANGUAGE Trustworthy #-}
  #endif
  -----------------------------------------------------------------------------
  -- |
  -- Module      :  Data.Set.Base
  -- Copyright   :  (c) Daan Leijen 2002
  -- License     :  BSD-style
  -- Maintainer  :  libraries@haskell.org
  -- Stability   :  provisional
  -- Portability :  portable
  --
  -- 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.
  --
  -- 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.
  -----------------------------------------------------------------------------
  
  -- [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.
  --
  -- For example, change 'member' so that its local 'go' function is not passing
  -- argument x and then look at the resulting code for hedgeInt.
  
  
  -- [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.Base (
              -- * 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
              , partition
              , split
              , splitMember
  
              -- * Map
              , map
              , mapMonotonic
  
              -- * 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
              , valid
  
              -- Internals (for testing)
              , bin
              , balanced
              , join
              , merge
              ) where
  
  import Prelude hiding (filter,foldl,foldr,null,map)
  import qualified Data.List as List
  import Data.Monoid (Monoid(..))
  import qualified Data.Foldable as Foldable
  import Data.Typeable
  import Control.DeepSeq (NFData(rnf))
  
  #if __GLASGOW_HASKELL__
  import GHC.Exts ( build )
  import Text.Read
  import Data.Data
  #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_1_OF_3(fn) fn arg _ _ | arg `seq` False = undefined
  
  {--------------------------------------------------------------------
    Operators
  --------------------------------------------------------------------}
  infixl 9 \\ --
  
  -- | /O(n+m)/. See 'difference'.
  (\\) :: Ord a => Set a -> Set a -> Set a
  m1 \\ m2 = difference m1 m2
  #if __GLASGOW_HASKELL__ >= 700
  {-# 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
  
  instance Ord a => Monoid (Set a) where
      mempty  = empty
      mappend = union
      mconcat = unions
  
  instance Foldable.Foldable Set where
      fold Tip = mempty
      fold (Bin _ k l r) = Foldable.fold l `mappend` k `mappend` Foldable.fold r
      foldr = foldr
      foldl = foldl
      foldMap _ Tip = mempty
      foldMap f (Bin _ k l r) = Foldable.foldMap f l `mappend` f k `mappend` Foldable.foldMap f r
  
  #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 a, Ord a) => Data (Set a) where
    gfoldl f z set = z fromList `f` (toList set)
    toConstr _     = error "toConstr"
    gunfold _ _    = error "gunfold"
    dataTypeOf _   = mkNoRepType "Data.Set.Set"
    dataCast1 f    = gcast1 f
  
  #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
      STRICT_1_OF_2(go)
      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__ >= 700
  {-# 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__ >= 700
  {-# 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
      STRICT_1_OF_2(goNothing)
      goNothing _ Tip = Nothing
      goNothing x (Bin _ y l r) | x <= y = goNothing x l
                                | otherwise = goJust x y r
  
      STRICT_1_OF_3(goJust)
      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__ >= 700
  {-# 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
      STRICT_1_OF_2(goNothing)
      goNothing _ Tip = Nothing
      goNothing x (Bin _ y l r) | x < y = goJust x y l
                                | otherwise = goNothing x r
  
      STRICT_1_OF_3(goJust)
      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__ >= 700
  {-# 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
      STRICT_1_OF_2(goNothing)
      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
  
      STRICT_1_OF_3(goJust)
      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__ >= 700
  {-# 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
      STRICT_1_OF_2(goNothing)
      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
  
      STRICT_1_OF_3(goJust)
      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__ >= 700
  {-# 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
  insert :: Ord a => a -> Set a -> Set a
  insert = go
    where
      go :: Ord a => a -> Set a -> Set a
      STRICT_1_OF_2(go)
      go x Tip = singleton x
      go x (Bin sz y l r) = case compare x y of
          LT -> balanceL y (go x l) r
          GT -> balanceR y l (go x r)
          EQ -> Bin sz x l r
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE insert #-}
  #else
  {-# INLINE insert #-}
  #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
  insertR :: Ord a => a -> Set a -> Set a
  insertR = go
    where
      go :: Ord a => a -> Set a -> Set a
      STRICT_1_OF_2(go)
      go x Tip = singleton x
      go x t@(Bin _ y l r) = case compare x y of
          LT -> balanceL y (go x l) r
          GT -> balanceR y l (go x r)
          EQ -> t
  #if __GLASGOW_HASKELL__ >= 700
  {-# 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
      STRICT_1_OF_2(go)
      go _ Tip = Tip
      go x (Bin _ y l r) = case compare x y of
          LT -> balanceR y (go x l) r
          GT -> balanceL y l (go x r)
          EQ -> glue l r
  #if __GLASGOW_HASKELL__ >= 700
  {-# 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__ >= 700
  {-# 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__ >= 700
  {-# 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__ >= 700
  {-# INLINABLE isSubsetOfX #-}
  #endif
  
  
  {--------------------------------------------------------------------
    Minimal, Maximal
  --------------------------------------------------------------------}
  -- | /O(log n)/. The minimal element of a set.
  findMin :: Set a -> a
  findMin (Bin _ x Tip _) = x
  findMin (Bin _ _ l _)   = findMin l
  findMin Tip             = error "Set.findMin: empty set has no minimal element"
  
  -- | /O(log n)/. The maximal element of a set.
  findMax :: Set a -> a
  findMax (Bin _ x _ Tip)  = x
  findMax (Bin _ _ _ r)    = findMax r
  findMax Tip              = error "Set.findMax: empty set has no maximal element"
  
  -- | /O(log n)/. Delete the minimal element.
  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.
  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__ >= 700
  {-# INLINABLE unions #-}
  #endif
  
  -- | /O(n+m)/. The union of two sets, preferring the first set when
  -- equal elements are encountered.
  -- The implementation uses the efficient /hedge-union/ algorithm.
  -- Hedge-union is more efficient on (bigset `union` smallset).
  union :: Ord a => Set a -> Set a -> Set a
  union Tip t2  = t2
  union t1 Tip  = t1
  union t1 t2 = hedgeUnion NothingS NothingS t1 t2
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE union #-}
  #endif
  
  hedgeUnion :: Ord a => MaybeS a -> MaybeS a -> Set a -> Set a -> Set a
  hedgeUnion _   _   t1  Tip = t1
  hedgeUnion blo bhi Tip (Bin _ x l r) = join x (filterGt blo l) (filterLt bhi r)
  hedgeUnion _   _   t1  (Bin _ x Tip Tip) = insertR x t1   -- According to benchmarks, this special case increases
                                                            -- performance up to 30%. It does not help in difference or intersection.
  hedgeUnion blo bhi (Bin _ x l r) t2 = join x (hedgeUnion blo bmi l (trim blo bmi t2))
                                               (hedgeUnion bmi bhi r (trim bmi bhi t2))
    where bmi = JustS x
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE hedgeUnion #-}
  #endif
  
  {--------------------------------------------------------------------
    Difference
  --------------------------------------------------------------------}
  -- | /O(n+m)/. Difference of two sets.
  -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
  difference :: Ord a => Set a -> Set a -> Set a
  difference Tip _   = Tip
  difference t1 Tip  = t1
  difference t1 t2   = hedgeDiff NothingS NothingS t1 t2
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE difference #-}
  #endif
  
  hedgeDiff :: Ord a => MaybeS a -> MaybeS a -> Set a -> Set a -> Set a
  hedgeDiff _   _   Tip           _ = Tip
  hedgeDiff blo bhi (Bin _ x l r) Tip = join x (filterGt blo l) (filterLt bhi r)
  hedgeDiff blo bhi t (Bin _ x l r) = merge (hedgeDiff blo bmi (trim blo bmi t) l)
                                            (hedgeDiff bmi bhi (trim bmi bhi t) r)
    where bmi = JustS x
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE hedgeDiff #-}
  #endif
  
  {--------------------------------------------------------------------
    Intersection
  --------------------------------------------------------------------}
  -- | /O(n+m)/. 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 t2 = hedgeInt NothingS NothingS t1 t2
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE intersection #-}
  #endif
  
  hedgeInt :: Ord a => MaybeS a -> MaybeS a -> Set a -> Set a -> Set a
  hedgeInt _ _ _   Tip = Tip
  hedgeInt _ _ Tip _   = Tip
  hedgeInt blo bhi (Bin _ x l r) t2 = let l' = hedgeInt blo bmi l (trim blo bmi t2)
                                          r' = hedgeInt bmi bhi r (trim bmi bhi t2)
                                      in if x `member` t2 then join x l' r' else merge l' r'
    where bmi = JustS x
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE hedgeInt #-}
  #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 (Bin _ x l r)
      | p x       = join x (filter p l) (filter p r)
      | otherwise = merge (filter p l) (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 _ Tip = (Tip, Tip)
  partition p (Bin _ x l r) = case (partition p l, partition p r) of
    ((l1, l2), (r1, r2))
      | p x       -> (join x l1 r1, merge l2 r2)
      | otherwise -> (merge l1 r1, join 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 a, Ord b) => (a->b) -> Set a -> Set b
  map f = fromList . List.map f . toList
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE map #-}
  #endif
  
  -- | /O(n)/. The
  --
  -- @'mapMonotonic' f s == 'map' f s@, but works only when @f@ is monotonic.
  -- /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
      STRICT_1_OF_2(go)
      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
      STRICT_1_OF_2(go)
      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
  --------------------------------------------------------------------}
  -- | /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.
  fromList :: Ord a => [a] -> Set a
  fromList = foldlStrict ins empty
    where
      ins t x = insert x t
  #if __GLASGOW_HASKELL__ >= 700
  {-# 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)
    where
    -- [combineEq xs] combines equal elements with [const] in an ordered list [xs]
    combineEq xs'
      = case xs' of
          []     -> []
          [x]    -> [x]
          (x:xx) -> combineEq' x xx
  
    combineEq' z [] = [z]
    combineEq' z (x:xs')
      | z==x      =   combineEq' z xs'
      | otherwise = z:combineEq' x xs'
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE fromAscList #-}
  #endif
  
  
  -- | /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./
  fromDistinctAscList :: [a] -> Set a
  fromDistinctAscList xs
    = create const (length xs) xs
    where
      -- 1) use continutations so that we use heap space instead of stack space.
      -- 2) special case for n==5 to create bushier trees.
      create c 0 xs' = c Tip xs'
      create c 5 xs' = case xs' of
                         (x1:x2:x3:x4:x5:xx)
                              -> c (bin x4 (bin x2 (singleton x1) (singleton x3)) (singleton x5)) xx
                         _ -> error "fromDistinctAscList create 5"
      create c n xs' = seq nr $ create (createR nr c) nl xs'
        where nl = n `div` 2
              nr = n - nl - 1
  
      createR n c l (x:ys) = create (createB l x c) n ys
      createR _ _ _ []     = error "fromDistinctAscList createR []"
      createB l x c r zs   = c (bin x 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)
  
  {--------------------------------------------------------------------
    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
  --------------------------------------------------------------------}
  
  #include "Typeable.h"
  INSTANCE_TYPEABLE1(Set,setTc,"Set")
  
  {--------------------------------------------------------------------
    NFData
  --------------------------------------------------------------------}
  
  instance NFData a => NFData (Set a) where
      rnf Tip           = ()
      rnf (Bin _ y l r) = rnf y `seq` rnf l `seq` rnf r
  
  {--------------------------------------------------------------------
    Utility functions that return sub-ranges of the original
    tree. Some functions take a `Maybe value` as an argument to
    allow comparisons against infinite values. These are called `blow`
    (Nothing is -\infty) and `bhigh` (here Nothing is +\infty).
    We use MaybeS value, which is a Maybe strict in the Just case.
  
    [trim blow bhigh t]   A tree that is either empty or where [x > blow]
                          and [x < bhigh] for the value [x] of the root.
    [filterGt blow t]     A tree where for all values [k]. [k > blow]
    [filterLt bhigh t]    A tree where for all values [k]. [k < bhigh]
  
    [split k t]           Returns two trees [l] and [r] where all values
                          in [l] are <[k] and all keys in [r] are >[k].
    [splitMember k t]     Just like [split] but also returns whether [k]
                          was found in the tree.
  --------------------------------------------------------------------}
  
  data MaybeS a = NothingS | JustS !a
  
  {--------------------------------------------------------------------
    [trim blo bhi t] trims away all subtrees that surely contain no
    values between the range [blo] to [bhi]. The returned tree is either
    empty or the key of the root is between @blo@ and @bhi@.
  --------------------------------------------------------------------}
  trim :: Ord a => MaybeS a -> MaybeS a -> Set a -> Set a
  trim NothingS   NothingS   t = t
  trim (JustS lx) NothingS   t = greater lx t where greater lo (Bin _ x _ r) | x <= lo = greater lo r
                                                    greater _  t' = t'
  trim NothingS   (JustS hx) t = lesser hx t  where lesser  hi (Bin _ x l _) | x >= hi = lesser  hi l
                                                    lesser  _  t' = t'
  trim (JustS lx) (JustS hx) t = middle lx hx t  where middle lo hi (Bin _ x _ r) | x <= lo = middle lo hi r
                                                       middle lo hi (Bin _ x l _) | x >= hi = middle lo hi l
                                                       middle _  _  t' = t'
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE trim #-}
  #endif
  
  {--------------------------------------------------------------------
    [filterGt b t] filter all values >[b] from tree [t]
    [filterLt b t] filter all values <[b] from tree [t]
  --------------------------------------------------------------------}
  filterGt :: Ord a => MaybeS a -> Set a -> Set a
  filterGt NothingS t = t
  filterGt (JustS b) t = filter' b t
    where filter' _   Tip = Tip
          filter' b' (Bin _ x l r) =
            case compare b' x of LT -> join x (filter' b' l) r
                                 EQ -> r
                                 GT -> filter' b' r
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE filterGt #-}
  #endif
  
  filterLt :: Ord a => MaybeS a -> Set a -> Set a
  filterLt NothingS t = t
  filterLt (JustS b) t = filter' b t
    where filter' _   Tip = Tip
          filter' b' (Bin _ x l r) =
            case compare x b' of LT -> join x l (filter' b' r)
                                 EQ -> l
                                 GT -> filter' b' l
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE filterLt #-}
  #endif
  
  {--------------------------------------------------------------------
    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 _ Tip = (Tip,Tip)
  split x (Bin _ y l r)
    = case compare x y of
        LT -> let (lt,gt) = split x l in (lt,join y gt r)
        GT -> let (lt,gt) = split x r in (join y l lt,gt)
        EQ -> (l,r)
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE split #-}
  #endif
  
  -- | /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 in (lt, found, join y gt r)
         GT -> let (lt, found, gt) = splitMember x r in (join y l lt, found, gt)
         EQ -> (l, True, r)
  #if __GLASGOW_HASKELL__ >= 700
  {-# INLINABLE splitMember #-}
  #endif
  
  {--------------------------------------------------------------------
    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.
      [join 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.
  
    Note: in contrast to Adam's paper, we use (<=) comparisons instead
    of (<) comparisons in [join], [merge] and [balance].
    Quickcheck (on [difference]) showed that this was necessary in order
    to maintain the invariants. It is quite unsatisfactory that I haven't
    been able to find out why this is actually the case! Fortunately, it
    doesn't hurt to be a bit more conservative.
  --------------------------------------------------------------------}
  
  {--------------------------------------------------------------------
    Join
  --------------------------------------------------------------------}
  join :: a -> Set a -> Set a -> Set a
  join x Tip r  = insertMin x r
  join x l Tip  = insertMax x l
  join x l@(Bin sizeL y ly ry) r@(Bin sizeR z lz rz)
    | delta*sizeL < sizeR  = balanceL z (join x l lz) rz
    | delta*sizeR < sizeL  = balanceR y ly (join 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 r
    | size l > size r = let (m,l') = deleteFindMax l in balanceR m l' r
    | otherwise       = let (m,r') = deleteFindMin r 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
    = case t of
        Bin _ x Tip r -> (x,r)
        Bin _ x l r   -> let (xm,l') = deleteFindMin l in (xm,balanceR x l' r)
        Tip           -> (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
    = case t of
        Bin _ x l Tip -> (x,l)
        Bin _ x l r   -> let (xm,r') = deleteFindMax r in (xm,balanceL x l r')
        Tip           -> (error "Set.deleteFindMax: can not return the maximal element of an empty set", Tip)
  
  -- | /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 x = Just (deleteFindMin x)
  
  -- | /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 x = Just (deleteFindMax x)
  
  {--------------------------------------------------------------------
    [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
  --------------------------------------------------------------------}
  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 #-}
  
  {--------------------------------------------------------------------
    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
  