{-# language BangPatterns #-}
{-# language PatternGuards #-}
{-# language TypeFamilies #-}
{-# language LambdaCase #-}
{-# language FlexibleContexts #-}
{-# language UndecidableInstances #-}
{-# language MagicHash #-}
-----------------------------------------------------------------------------
-- |
-- Module      :  Set.Internal
-- Copyright   :  (c) Daan Leijen 2002, (c) Edward Kmett 2017-2018
-- License     :  BSD-style
-- Maintainer  :  libraries@haskell.org
-- Portability :  portable
--
-- An efficient implementation of unpacked sets using backpack,
-- based on Data.Set.Internal from containers.
--
-- 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).
--
-- This isn't required here, because we get to know the Ord Key dictionary

-- [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 Set.Internal (
  -- * Set type
    Set(..)

  -- * 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
  , foldMap
  , foldr
  , foldl
  -- ** Strict folds
  , foldr'
  , foldl'

  -- * 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 Control.DeepSeq (NFData(rnf))
import Data.Bits (shiftL, shiftR)
import Data.Data
import Data.Default.Class
import qualified Data.List as List
import Data.Semigroup (Semigroup((<>), stimes), stimesIdempotentMonoid)
import GHC.Exts (build, lazy, isTrue#, reallyUnsafePtrEquality#)
import qualified GHC.Exts as GHCExts
import Prelude hiding (filter,foldMap,foldl,foldr,null,map,take,drop,splitAt)
import Text.Read

import Key

-- | The same as a regular Haskell pair, but
--
-- @
-- (x :*: _|_) = (_|_ :*: y) = _|_
-- @
data StrictPair a b = !a :*: !b

infixr 1 :*:

-- | Convert a strict pair to a standard pair.
toPair :: StrictPair a b -> (a, b)
toPair (x :*: y) = (x, y)
{-# inline toPair #-}

ptrEq :: a -> a -> Bool
ptrEq x y = isTrue# (reallyUnsafePtrEquality# x y)
{-# inline ptrEq #-}

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

-- | /O(m*log(n\/m+1)), m <= n/. See 'difference'.
(\\) :: Set -> Set -> Set
(\\) = difference
{-# inline (\\) #-}

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

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

instance Default Set where
  def = Tip

type Size = Int

instance Monoid Set where
  mempty  = empty
  mconcat = unions
  mappend = (<>)

instance Semigroup Set where
  (<>) = union
  stimes  = stimesIdempotentMonoid

foldMap :: Monoid m => (Key -> m) -> Set -> m
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 #-}

instance Data Key => Data Set 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]

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

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

-- | /O(log n)/. Is the element in the set?
member :: Key -> Set -> Bool
member !_ Tip = False
member x (Bin _ y l r) = case compare x y of
  LT -> member x l
  GT -> member x r
  EQ -> True

-- | /O(log n)/. Is the element not in the set?
notMember :: Key -> Set -> Bool
notMember a t = not $ member a t

-- | /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 :: Key -> Set -> Maybe Key
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

-- | /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 :: Key -> Set -> Maybe Key
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

-- | /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 :: Key -> Set -> Maybe Key
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

-- | /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 :: Key -> Set -> Maybe Key
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

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

-- | /O(1)/. Create a singleton set.
singleton :: Key -> Set
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 :: Key -> Set -> Set
insert x0 = go x0 x0 where
  go :: Key -> Key -> Set -> Set
  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

-- 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 :: Key -> Set -> Set
insertR x0 = go x0 x0 where
  go :: Key -> Key -> Set -> Set
  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

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

-- See Note: Type of local 'go' function
delete :: Key -> Set -> Set
delete = go where
  go :: Key -> Set -> Set
  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

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

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

isSubsetOfX :: Set -> Set -> 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

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

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

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

-- | /O(log n)/. The maximal element of a set.
findMax :: Set -> Key
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 -> Set
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 -> Set
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 :: [Set] -> Set
unions = List.foldl' union empty

-- | /O(m*log(n\/m + 1)), m <= n/. The union of two sets, preferring the first set when
-- equal elements are encountered.
union :: Set -> Set -> Set
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

{--------------------------------------------------------------------
  Difference
--------------------------------------------------------------------}
-- | /O(m*log(n\/m + 1)), m <= n/. Difference of two sets.
difference :: Set -> Set -> Set
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

{--------------------------------------------------------------------
  Intersection
--------------------------------------------------------------------}
-- | /O(m*log(n\/m + 1)), m <= n/. The intersection of two sets.
-- Keyents 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 :: Set -> Set -> Set
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

{--------------------------------------------------------------------
  Filter and partition
--------------------------------------------------------------------}
-- | /O(n)/. Filter all elements that satisfy the predicate.
filter :: (Key -> Bool) -> Set -> Set
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 :: (Key -> Bool) -> Set -> (Set,Set)
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 :: (Key -> Key) -> Set -> Set
map f = fromList . List.map f . toList

-- | /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 :: (Key -> Key) -> Set -> Set
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, such that @'foldr' f z == 'Prelude.foldr' f z . 'toAscList'@.
--
-- For example,
--
-- > toAscList set = foldr (:) [] set
foldr :: (Key -> b -> b) -> b -> Set -> 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' :: (Key -> b -> b) -> b -> Set -> 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 -> Key -> a) -> a -> Set -> 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 -> Key -> a) -> a -> Set -> 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 -> [Key]
elems = toAscList

{--------------------------------------------------------------------
  Lists
--------------------------------------------------------------------}
instance GHCExts.IsList Set where
  type Item Set = Key
  fromList = fromList
  toList   = toList

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

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

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

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

foldlFB :: (a -> Key -> a) -> a -> Set -> 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 #-}

-- | /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 :: [Key] -> Set
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 = List.foldl' 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)

{--------------------------------------------------------------------
  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 :: [Key] -> Set
fromAscList xs = fromDistinctAscList (combineEq xs)

-- | /O(n)/. Build a set from a descending list in linear time.
-- /The precondition (input list is descending) is not checked./
fromDescList :: [Key] -> Set
fromDescList xs = fromDistinctDescList (combineEq xs)

-- [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 :: [Key] -> [Key]
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 :: [Key] -> Set
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 :: [Key] -> Set
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 Set where
  t1 == t2  = (size t1 == size t2) && (toAscList t1 == toAscList t2)

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

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

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

{--------------------------------------------------------------------
  Read
--------------------------------------------------------------------}
instance Read Key => Read Set where
  readPrec = parens $ prec 10 $ do
    Ident "fromList" <- lexP
    xs <- readPrec
    return (fromList xs)

  readListPrec = readListPrecDefault

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

instance NFData Key => NFData Set 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 :: Key -> Set -> (Set,Set)
split x t = toPair $ splitS x t

splitS :: Key -> Set -> StrictPair Set Set
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)

-- | /O(log n)/. Performs a 'split' but also returns whether the pivot
-- element was found in the original set.
splitMember :: Key -> Set -> (Set,Bool,Set)
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)

{--------------------------------------------------------------------
  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 :: Key -> Set -> Int
findIndex = go 0 where
  go :: Int -> Key -> Set -> 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

-- | /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 :: Key -> Set -> Maybe Int
lookupIndex = go 0 where
  go :: Int -> Key -> Set -> 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

-- | /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 -> Key
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 -> Set
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 -> Set
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 -> Set
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 -> (Set, Set)
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 :: (Key -> Bool) -> Set -> Set
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 :: (Key -> Bool) -> Set -> Set
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 :: (Key -> Bool) -> Set -> (Set, Set)
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 :: Key -> Set -> Set -> Set
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 :: Key -> Set -> Set
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 -> Set -> Set
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 -> Set -> Set
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 -> (Key, Set)
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 -> (Key, Set)
deleteFindMax t
  | Just r <- maxView t = r
  | otherwise = (error "Set.deleteFindMax: can not return the maximal element of an empty set", Tip)

minViewSure :: Key -> Set -> Set -> StrictPair Key Set
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 -> Maybe (Key, Set)
minView Tip = Nothing
minView (Bin _ x l r) = Just $! toPair $ minViewSure x l r

maxViewSure :: Key -> Set -> Set -> StrictPair Key Set
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 -> Maybe (Key, Set)
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 :: Key -> Set -> Set -> Set
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 :: Key -> Set -> Set -> Set
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 :: Key -> Set -> Set -> Set
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 -> [Set]
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 Key => Set -> 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 Key => Bool -> Bool -> Set -> String
showTreeWith hang wide t
  | hang      = (showsTreeHang wide [] t) ""
  | otherwise = (showsTree wide [] [] t) ""

showsTree :: Show Key => Bool -> [String] -> [String] -> Set -> 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 Key => Bool -> [String] -> Set -> 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 :: Set -> Bool
valid t = balanced t && ordered t && validsize t

ordered :: Set -> 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 -> 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 -> 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