    {-# LANGUAGE ScopedTypeVariables #-}
    {-# OPTIONS -cpp #-}
    -----------------------------------------------------------------------------
    -- |
    -- Module      :  Data.Sequence
    -- Copyright   :  (c) Ross Paterson 2005
    --                (c) Louis Wasserman 2009
    -- License     :  BSD-style
    -- Maintainer  :  libraries@haskell.org
    -- Stability   :  experimental
    -- Portability :  portable
    --
    -- General purpose finite sequences.
    -- Apart from being finite and having strict operations, sequences
    -- also differ from lists in supporting a wider variety of operations
    -- efficiently.
    --
    -- An amortized running time is given for each operation, with /n/ referring
    -- to the length of the sequence and /i/ being the integral index used by
    -- some operations.  These bounds hold even in a persistent (shared) setting.
    --
    -- The implementation uses 2-3 finger trees annotated with sizes,
    -- as described in section 4.2 of
    --
    --    * Ralf Hinze and Ross Paterson,
    --	\"Finger trees: a simple general-purpose data structure\",
    --	/Journal of Functional Programming/ 16:2 (2006) pp 197-217.
    --	<http://www.soi.city.ac.uk/~ross/papers/FingerTree.html>
    --
    -- /Note/: Many of these operations have the same names as similar
    -- operations on lists in the "Prelude".  The ambiguity may be resolved
    -- using either qualification or the @hiding@ clause.
    --
    -----------------------------------------------------------------------------

    module Data.Sequence (
    	Seq,
    	-- * Construction
    	empty,		-- :: Seq a
    	singleton,	-- :: a -> Seq a
    	(<|),		-- :: a -> Seq a -> Seq a
    	(|>),		-- :: Seq a -> a -> Seq a
    	(><),		-- :: Seq a -> Seq a -> Seq a
    	fromList,	-- :: [a] -> Seq a
    	-- ** Repetition
    	replicate,	-- :: Int -> a -> Seq a
    	replicateA,	-- :: Applicative f => Int -> f a -> f (Seq a)
    	replicateM,	-- :: Monad m => Int -> m a -> m (Seq a)
    	-- ** Iterative construction
    	iterateN,	-- :: Int -> (a -> a) -> a -> Seq a
    	unfoldr,	-- :: (b -> Maybe (a, b)) -> b -> Seq a
    	unfoldl,	-- :: (b -> Maybe (b, a)) -> b -> Seq a
    	-- * Deconstruction
    	-- | Additional functions for deconstructing sequences are available
    	-- via the 'Foldable' instance of 'Seq'.

    	-- ** Queries
    	null,		-- :: Seq a -> Bool
    	length,		-- :: Seq a -> Int
    	-- ** Views
    	ViewL(..),
    	viewl,		-- :: Seq a -> ViewL a
    	ViewR(..),
    	viewr,		-- :: Seq a -> ViewR a
    	-- * Scans
    	scanl,		-- :: (a -> b -> a) -> a -> Seq b -> Seq a
    	scanl1,		-- :: (a -> a -> a) -> Seq a -> Seq a
    	scanr,		-- :: (a -> b -> b) -> b -> Seq a -> Seq b
    	scanr1,		-- :: (a -> a -> a) -> Seq a -> Seq a
    	-- * Sublists
    	tails,		-- :: Seq a -> Seq (Seq a)
    	inits,		-- :: Seq a -> Seq (Seq a)
    	-- ** Sequential searches
    	takeWhileL,	-- :: (a -> Bool) -> Seq a -> Seq a
    	takeWhileR,	-- :: (a -> Bool) -> Seq a -> Seq a
    	dropWhileL,	-- :: (a -> Bool) -> Seq a -> Seq a
    	dropWhileR,	-- :: (a -> Bool) -> Seq a -> Seq a
    	spanl,		-- :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
    	spanr,		-- :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
    	breakl,		-- :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
    	breakr,		-- :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
    	partition,	-- :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
    	filter,		-- :: (a -> Bool) -> Seq a -> Seq a
    	-- * Sorting
    	sort,		-- :: Ord a => Seq a -> Seq a
    	sortBy,		-- :: (a -> a -> Ordering) -> Seq a -> Seq a
    	unstableSort,	-- :: Ord a => Seq a -> Seq a
    	unstableSortBy,	-- :: (a -> a -> Ordering) -> Seq a -> Seq a
    	-- * Indexing
    	index,		-- :: Seq a -> Int -> a
    	adjust,		-- :: (a -> a) -> Int -> Seq a -> Seq a
    	update,		-- :: Int -> a -> Seq a -> Seq a
    	take,		-- :: Int -> Seq a -> Seq a
    	drop,		-- :: Int -> Seq a -> Seq a
    	splitAt,	-- :: Int -> Seq a -> (Seq a, Seq a)
    	-- ** Indexing with predicates
    	-- | These functions perform sequential searches from the left
    	-- or right ends of the sequence, returning indices of matching
    	-- elements.
    	elemIndexL,	-- :: Eq a => a -> Seq a -> Maybe Int
    	elemIndicesL,	-- :: Eq a => a -> Seq a -> [Int]
    	elemIndexR,	-- :: Eq a => a -> Seq a -> Maybe Int
    	elemIndicesR,	-- :: Eq a => a -> Seq a -> [Int]
    	findIndexL,	-- :: (a -> Bool) -> Seq a -> Maybe Int
    	findIndicesL,	-- :: (a -> Bool) -> Seq a -> [Int]
    	findIndexR,	-- :: (a -> Bool) -> Seq a -> Maybe Int
    	findIndicesR,	-- :: (a -> Bool) -> Seq a -> [Int]
    	-- * Folds
    	-- | General folds are available via the 'Foldable' instance of 'Seq'.
    	foldlWithIndex,	-- :: (b -> Int -> a -> b) -> b -> Seq a -> b
    	foldrWithIndex, -- :: (Int -> a -> b -> b) -> b -> Seq a -> b
    	-- * Transformations
    	mapWithIndex,	-- :: (Int -> a -> b) -> Seq a -> Seq b
    	reverse,	-- :: Seq a -> Seq a
    	-- ** Zips
    	zip,		-- :: Seq a -> Seq b -> Seq (a, b)
    	zipWith, 	-- :: (a -> b -> c) -> Seq a -> Seq b -> Seq c
    	zip3,		-- :: Seq a -> Seq b -> Seq c -> Seq (a, b, c)
    	zipWith3,	-- :: (a -> b -> c -> d) -> Seq a -> Seq b -> Seq c -> Seq d
    	zip4,		-- :: Seq a -> Seq b -> Seq c -> Seq d -> Seq (a, b, c, d)
    	zipWith4,	-- :: (a -> b -> c -> d -> e) -> Seq a -> Seq b -> Seq c -> Seq d -> Seq e
    #if TESTING
    	valid,
    #endif
    	) where

    import Prelude hiding (
    	Functor(..),
    	null, length, take, drop, splitAt, foldl, foldl1, foldr, foldr1,
    	scanl, scanl1, scanr, scanr1, replicate, zip, zipWith, zip3, zipWith3,
    	takeWhile, dropWhile, iterate, reverse, filter, mapM, sum, all)
    import qualified Data.List (foldl', sortBy)
    import Control.Applicative (Applicative(..), (<$>), WrappedMonad(..), liftA, liftA2, liftA3)
    import Control.Monad (MonadPlus(..), ap)
    import Data.Monoid (Monoid(..))
    import Data.Functor (Functor(..))
    import Data.Foldable
    import Data.Traversable
    #ifndef __GLASGOW_HASKELL__
    import Data.Typeable (Typeable, typeOf, typeOfDefault)
    #endif
    import Data.Typeable (TyCon, Typeable1(..), mkTyCon, mkTyConApp )

    #ifdef __GLASGOW_HASKELL__
    import GHC.Exts (build)
    import Text.Read (Lexeme(Ident), lexP, parens, prec,
    	readPrec, readListPrec, readListPrecDefault)
    import Data.Data (Data(..), DataType, Constr, Fixity(..),
                                 mkConstr, mkDataType, constrIndex, gcast1)
    #endif

    #if TESTING
    import Control.Monad (liftM, liftM2, liftM3, liftM4)
    import qualified Data.List (zipWith)
    import Test.QuickCheck hiding ((><))
    #endif

    infixr 5 `consTree`
    infixl 5 `snocTree`

    infixr 5 ><
    infixr 5 <|, :<
    infixl 5 |>, :>

    class Sized a where
    	size :: a -> Int

    -- | General-purpose finite sequences.
    newtype Seq a = Seq (FingerTree (Elem a))

    instance Functor Seq where
    	fmap f (Seq xs) = Seq (fmap (fmap f) xs)
    #ifdef __GLASGOW_HASKELL__
    	x <$ s = replicate (length s) x
    #endif

    instance Foldable Seq where
    	foldr f z (Seq xs) = foldr (flip (foldr f)) z xs
    	foldl f z (Seq xs) = foldl (foldl f) z xs

    	foldr1 f (Seq xs) = getElem (foldr1 f' xs)
    	  where f' (Elem x) (Elem y) = Elem (f x y)

    	foldl1 f (Seq xs) = getElem (foldl1 f' xs)
    	  where f' (Elem x) (Elem y) = Elem (f x y)

    instance Traversable Seq where
    	traverse f (Seq xs) = Seq <$> traverse (traverse f) xs

    instance Monad Seq where
    	return = singleton
    	xs >>= f = foldl' add empty xs
    	  where add ys x = ys >< f x

    instance MonadPlus Seq where
    	mzero = empty
    	mplus = (><)

    instance Eq a => Eq (Seq a) where
    	xs == ys = length xs == length ys && toList xs == toList ys

    instance Ord a => Ord (Seq a) where
    	compare xs ys = compare (toList xs) (toList ys)

    #if TESTING
    instance Show a => Show (Seq a) where
    	showsPrec p (Seq x) = showsPrec p x
    #else
    instance Show a => Show (Seq a) where
    	showsPrec p xs = showParen (p > 10) $
    		showString "fromList " . shows (toList xs)
    #endif

    instance Read a => Read (Seq 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

    instance Monoid (Seq a) where
    	mempty = empty
    	mappend = (><)

    #include "Typeable.h"
    INSTANCE_TYPEABLE1(Seq,seqTc,"Seq")

    #if __GLASGOW_HASKELL__
    instance Data a => Data (Seq a) where
    	gfoldl f z s	= case viewl s of
    		EmptyL	-> z empty
    		x :< xs -> z (<|) `f` x `f` xs

    	gunfold k z c	= case constrIndex c of
    		1 -> z empty
    		2 -> k (k (z (<|)))
    		_ -> error "gunfold"

    	toConstr xs
    	  | null xs	= emptyConstr
    	  | otherwise	= consConstr

    	dataTypeOf _	= seqDataType

    	dataCast1 f	= gcast1 f

    emptyConstr, consConstr :: Constr
    emptyConstr = mkConstr seqDataType "empty" [] Prefix
    consConstr  = mkConstr seqDataType "<|" [] Infix

    seqDataType :: DataType
    seqDataType = mkDataType "Data.Sequence.Seq" [emptyConstr, consConstr]
    #endif

    -- Finger trees

    data FingerTree a
    	= Empty
    	| Single a
    	| Deep {-# UNPACK #-} !Int !(Digit a) (FingerTree (Node a)) !(Digit a)
    #if TESTING
    	deriving Show
    #endif

    instance Sized a => Sized (FingerTree a) where
    	{-# SPECIALIZE instance Sized (FingerTree (Elem a)) #-}
    	{-# SPECIALIZE instance Sized (FingerTree (Node a)) #-}
    	size Empty		= 0
    	size (Single x)		= size x
    	size (Deep v _ _ _)	= v

    instance Foldable FingerTree where
    	foldr _ z Empty = z
    	foldr f z (Single x) = x `f` z
    	foldr f z (Deep _ pr m sf) =
    		foldr f (foldr (flip (foldr f)) (foldr f z sf) m) pr

    	foldl _ z Empty = z
    	foldl f z (Single x) = z `f` x
    	foldl f z (Deep _ pr m sf) =
    		foldl f (foldl (foldl f) (foldl f z pr) m) sf

    	foldr1 _ Empty = error "foldr1: empty sequence"
    	foldr1 _ (Single x) = x
    	foldr1 f (Deep _ pr m sf) =
    		foldr f (foldr (flip (foldr f)) (foldr1 f sf) m) pr

    	foldl1 _ Empty = error "foldl1: empty sequence"
    	foldl1 _ (Single x) = x
    	foldl1 f (Deep _ pr m sf) =
    		foldl f (foldl (foldl f) (foldl1 f pr) m) sf

    instance Functor FingerTree where
    	fmap _ Empty = Empty
    	fmap f (Single x) = Single (f x)
    	fmap f (Deep v pr m sf) =
    		Deep v (fmap f pr) (fmap (fmap f) m) (fmap f sf)

    instance Traversable FingerTree where
    	traverse _ Empty = pure Empty
    	traverse f (Single x) = Single <$> f x
    	traverse f (Deep v pr m sf) =
    		Deep v <$> traverse f pr <*> traverse (traverse f) m <*>
    			traverse f sf

    {-# INLINE deep #-}
    {-# SPECIALIZE INLINE deep :: Digit (Elem a) -> FingerTree (Node (Elem a)) -> Digit (Elem a) -> FingerTree (Elem a) #-}
    {-# SPECIALIZE INLINE deep :: Digit (Node a) -> FingerTree (Node (Node a)) -> Digit (Node a) -> FingerTree (Node a) #-}
    deep		:: Sized a => Digit a -> FingerTree (Node a) -> Digit a -> FingerTree a
    deep pr m sf	=  Deep (size pr + size m + size sf) pr m sf

    {-# INLINE pullL #-}
    pullL :: Sized a => Int -> FingerTree (Node a) -> Digit a -> FingerTree a
    pullL s m sf = case viewLTree m of
    	Nothing2	-> digitToTree' s sf
    	Just2 pr m'	-> Deep s (nodeToDigit pr) m' sf

    {-# INLINE pullR #-}
    pullR :: Sized a => Int -> Digit a -> FingerTree (Node a) -> FingerTree a
    pullR s pr m = case viewRTree m of
    	Nothing2	-> digitToTree' s pr
    	Just2 m' sf	-> Deep s pr m' (nodeToDigit sf)

    {-# SPECIALIZE deepL :: Maybe (Digit (Elem a)) -> FingerTree (Node (Elem a)) -> Digit (Elem a) -> FingerTree (Elem a) #-}
    {-# SPECIALIZE deepL :: Maybe (Digit (Node a)) -> FingerTree (Node (Node a)) -> Digit (Node a) -> FingerTree (Node a) #-}
    deepL :: Sized a => Maybe (Digit a) -> FingerTree (Node a) -> Digit a -> FingerTree a
    deepL Nothing m sf	= pullL (size m + size sf) m sf
    deepL (Just pr) m sf	= deep pr m sf

    {-# SPECIALIZE deepR :: Digit (Elem a) -> FingerTree (Node (Elem a)) -> Maybe (Digit (Elem a)) -> FingerTree (Elem a) #-}
    {-# SPECIALIZE deepR :: Digit (Node a) -> FingerTree (Node (Node a)) -> Maybe (Digit (Node a)) -> FingerTree (Node a) #-}
    deepR :: Sized a => Digit a -> FingerTree (Node a) -> Maybe (Digit a) -> FingerTree a
    deepR pr m Nothing	= pullR (size m + size pr) pr m
    deepR pr m (Just sf)	= deep pr m sf

    -- Digits

    data Digit a
    	= One a
    	| Two a a
    	| Three a a a
    	| Four a a a a
    #if TESTING
    	deriving Show
    #endif

    instance Foldable Digit where
    	foldr f z (One a) = a `f` z
    	foldr f z (Two a b) = a `f` (b `f` z)
    	foldr f z (Three a b c) = a `f` (b `f` (c `f` z))
    	foldr f z (Four a b c d) = a `f` (b `f` (c `f` (d `f` z)))

    	foldl f z (One a) = z `f` a
    	foldl f z (Two a b) = (z `f` a) `f` b
    	foldl f z (Three a b c) = ((z `f` a) `f` b) `f` c
    	foldl f z (Four a b c d) = (((z `f` a) `f` b) `f` c) `f` d

    	foldr1 _ (One a) = a
    	foldr1 f (Two a b) = a `f` b
    	foldr1 f (Three a b c) = a `f` (b `f` c)
    	foldr1 f (Four a b c d) = a `f` (b `f` (c `f` d))

    	foldl1 _ (One a) = a
    	foldl1 f (Two a b) = a `f` b
    	foldl1 f (Three a b c) = (a `f` b) `f` c
    	foldl1 f (Four a b c d) = ((a `f` b) `f` c) `f` d

    instance Functor Digit where
    	fmap = fmapDefault

    instance Traversable Digit where
    	{-# INLINE traverse #-}
    	traverse f (One a) = One <$> f a
    	traverse f (Two a b) = Two <$> f a <*> f b
    	traverse f (Three a b c) = Three <$> f a <*> f b <*> f c
    	traverse f (Four a b c d) = Four <$> f a <*> f b <*> f c <*> f d

    instance Sized a => Sized (Digit a) where
    	{-# INLINE size #-}
    	size = foldl1 (+) . fmap size

    {-# SPECIALIZE digitToTree :: Digit (Elem a) -> FingerTree (Elem a) #-}
    {-# SPECIALIZE digitToTree :: Digit (Node a) -> FingerTree (Node a) #-}
    digitToTree	:: Sized a => Digit a -> FingerTree a
    digitToTree (One a) = Single a
    digitToTree (Two a b) = deep (One a) Empty (One b)
    digitToTree (Three a b c) = deep (Two a b) Empty (One c)
    digitToTree (Four a b c d) = deep (Two a b) Empty (Two c d)

    -- | Given the size of a digit and the digit itself, efficiently converts
    -- it to a FingerTree.
    digitToTree' :: Int -> Digit a -> FingerTree a
    digitToTree' n (Four a b c d) = Deep n (Two a b) Empty (Two c d)
    digitToTree' n (Three a b c) = Deep n (Two a b) Empty (One c)
    digitToTree' n (Two a b) = Deep n (One a) Empty (One b)
    digitToTree' n (One a) = n `seq` Single a

    -- Nodes

    data Node a
    	= Node2 {-# UNPACK #-} !Int a a
    	| Node3 {-# UNPACK #-} !Int a a a
    #if TESTING
    	deriving Show
    #endif

    instance Foldable Node where
    	foldr f z (Node2 _ a b) = a `f` (b `f` z)
    	foldr f z (Node3 _ a b c) = a `f` (b `f` (c `f` z))

    	foldl f z (Node2 _ a b) = (z `f` a) `f` b
    	foldl f z (Node3 _ a b c) = ((z `f` a) `f` b) `f` c

    instance Functor Node where
    	{-# INLINE fmap #-}
    	fmap = fmapDefault

    instance Traversable Node where
    	{-# INLINE traverse #-}
    	traverse f (Node2 v a b) = Node2 v <$> f a <*> f b
    	traverse f (Node3 v a b c) = Node3 v <$> f a <*> f b <*> f c

    instance Sized (Node a) where
    	size (Node2 v _ _)	= v
    	size (Node3 v _ _ _)	= v

    {-# INLINE node2 #-}
    {-# SPECIALIZE node2 :: Elem a -> Elem a -> Node (Elem a) #-}
    {-# SPECIALIZE node2 :: Node a -> Node a -> Node (Node a) #-}
    node2		:: Sized a => a -> a -> Node a
    node2 a b	=  Node2 (size a + size b) a b

    {-# INLINE node3 #-}
    {-# SPECIALIZE node3 :: Elem a -> Elem a -> Elem a -> Node (Elem a) #-}
    {-# SPECIALIZE node3 :: Node a -> Node a -> Node a -> Node (Node a) #-}
    node3		:: Sized a => a -> a -> a -> Node a
    node3 a b c	=  Node3 (size a + size b + size c) a b c

    nodeToDigit :: Node a -> Digit a
    nodeToDigit (Node2 _ a b) = Two a b
    nodeToDigit (Node3 _ a b c) = Three a b c

    -- Elements

    newtype Elem a  =  Elem { getElem :: a }

    instance Sized (Elem a) where
    	size _ = 1

    instance Functor Elem where
    	fmap f (Elem x) = Elem (f x)

    instance Foldable Elem where
    	foldr f z (Elem x) = f x z
    	foldl f z (Elem x) = f z x

    instance Traversable Elem where
    	traverse f (Elem x) = Elem <$> f x

    #ifdef TESTING
    instance (Show a) => Show (Elem a) where
    	showsPrec p (Elem x) = showsPrec p x
    #endif

    -------------------------------------------------------
    -- Applicative construction
    -------------------------------------------------------

    newtype Id a = Id {runId :: a}

    instance Functor Id where
    	fmap f (Id x) = Id (f x)

    instance Monad Id where
    	return = Id
    	m >>= k = k (runId m)

    instance Applicative Id where
    	pure = return
    	(<*>) = ap

    -- | This is essentially a clone of Control.Monad.State.Strict.
    newtype State s a = State {runState :: s -> (s, a)}

    instance Functor (State s) where
    	fmap = liftA

    instance Monad (State s) where
    	{-# INLINE return #-}
    	{-# INLINE (>>=) #-}
    	return x = State $ \ s -> (s, x)
    	m >>= k = State $ \ s -> case runState m s of
    		(s', x)	-> runState (k x) s'

    instance Applicative (State s) where
    	pure = return
    	(<*>) = ap

    execState :: State s a -> s -> a
    execState m x = snd (runState m x)

    -- | A helper method: a strict version of mapAccumL.
    mapAccumL' :: Traversable t => (a -> b -> (a, c)) -> a -> t b -> (a, t c)
    mapAccumL' f s t = runState (traverse (State . flip f) t) s

    -- | 'applicativeTree' takes an Applicative-wrapped construction of a
    -- piece of a FingerTree, assumed to always have the same size (which
    -- is put in the second argument), and replicates it as many times as
    -- specified.  This is a generalization of 'replicateA', which itself
    -- is a generalization of many Data.Sequence methods.
    {-# SPECIALIZE applicativeTree :: Int -> Int -> State s a -> State s (FingerTree a) #-}
    {-# SPECIALIZE applicativeTree :: Int -> Int -> Id a -> Id (FingerTree a) #-}
    -- Special note: the Id specialization automatically does node sharing,
    -- reducing memory usage of the resulting tree to /O(log n)/.
    applicativeTree :: forall f a. Applicative f => Int -> Int -> f a -> f (FingerTree a)
    applicativeTree n mSize m = mSize `seq` case n of
    	0 -> emptyTree
    	1 -> liftA Single m
    	2 -> deepA one emptyTree one
    	3 -> deepA two emptyTree one
    	4 -> deepA two emptyTree two
    	5 -> deepA three emptyTree two
    	6 -> deepA three emptyTree three
    	7 -> deepA four emptyTree three
    	8 -> deepA four emptyTree four
    	_ -> let (q, r) = n `quotRem` 3 in q `seq` case r of
    		0 -> deepA three (applicativeTree (q - 2) mSize' n3) three
    		1 -> deepA four (applicativeTree (q - 2) mSize' n3) three
    		_ -> deepA four (applicativeTree (q - 2) mSize' n3) four
      where
    	one = liftA One m
    	two = liftA2 Two m m
    	three = liftA3 Three m m m
    	four = liftA3 Four m m m <*> m
    	deepA = liftA3 (Deep (n * mSize))
    	mSize' = 3 * mSize
    	n3 = liftA3 (Node3 mSize') m m m

            emptyTree :: forall b. f (FingerTree b)
    	emptyTree = pure Empty

    ------------------------------------------------------------------------
    -- Construction
    ------------------------------------------------------------------------

    -- | /O(1)/. The empty sequence.
    empty		:: Seq a
    empty		=  Seq Empty

    -- | /O(1)/. A singleton sequence.
    singleton	:: a -> Seq a
    singleton x	=  Seq (Single (Elem x))

    -- | /O(log n)/. @replicate n x@ is a sequence consisting of @n@ copies of @x@.
    replicate	:: Int -> a -> Seq a
    replicate n x
      | n >= 0	= runId (replicateA n (Id x))
      | otherwise	= error "replicate takes a nonnegative integer argument"

    -- | 'replicateA' is an 'Applicative' version of 'replicate', and makes
    -- /O(log n)/ calls to '<*>' and 'pure'.
    --
    -- > replicateA n x = sequenceA (replicate n x)
    replicateA :: Applicative f => Int -> f a -> f (Seq a)
    replicateA n x
      | n >= 0	= Seq <$> applicativeTree n 1 (Elem <$> x)
      | otherwise	= error "replicateA takes a nonnegative integer argument"

    -- | 'replicateM' is a sequence counterpart of 'Control.Monad.replicateM'.
    --
    -- > replicateM n x = sequence (replicate n x)
    replicateM :: Monad m => Int -> m a -> m (Seq a)
    replicateM n x
      | n >= 0	= unwrapMonad (replicateA n (WrapMonad x))
      | otherwise	= error "replicateM takes a nonnegative integer argument"

    -- | /O(1)/. Add an element to the left end of a sequence.
    -- Mnemonic: a triangle with the single element at the pointy end.
    (<|)		:: a -> Seq a -> Seq a
    x <| Seq xs	=  Seq (Elem x `consTree` xs)

    {-# SPECIALIZE consTree :: Elem a -> FingerTree (Elem a) -> FingerTree (Elem a) #-}
    {-# SPECIALIZE consTree :: Node a -> FingerTree (Node a) -> FingerTree (Node a) #-}
    consTree	:: Sized a => a -> FingerTree a -> FingerTree a
    consTree a Empty	= Single a
    consTree a (Single b)	= deep (One a) Empty (One b)
    consTree a (Deep s (Four b c d e) m sf) = m `seq`
    	Deep (size a + s) (Two a b) (node3 c d e `consTree` m) sf
    consTree a (Deep s (Three b c d) m sf) =
    	Deep (size a + s) (Four a b c d) m sf
    consTree a (Deep s (Two b c) m sf) =
    	Deep (size a + s) (Three a b c) m sf
    consTree a (Deep s (One b) m sf) =
    	Deep (size a + s) (Two a b) m sf

    -- | /O(1)/. Add an element to the right end of a sequence.
    -- Mnemonic: a triangle with the single element at the pointy end.
    (|>)		:: Seq a -> a -> Seq a
    Seq xs |> x	=  Seq (xs `snocTree` Elem x)

    {-# SPECIALIZE snocTree :: FingerTree (Elem a) -> Elem a -> FingerTree (Elem a) #-}
    {-# SPECIALIZE snocTree :: FingerTree (Node a) -> Node a -> FingerTree (Node a) #-}
    snocTree	:: Sized a => FingerTree a -> a -> FingerTree a
    snocTree Empty a	=  Single a
    snocTree (Single a) b	=  deep (One a) Empty (One b)
    snocTree (Deep s pr m (Four a b c d)) e = m `seq`
    	Deep (s + size e) pr (m `snocTree` node3 a b c) (Two d e)
    snocTree (Deep s pr m (Three a b c)) d =
    	Deep (s + size d) pr m (Four a b c d)
    snocTree (Deep s pr m (Two a b)) c =
    	Deep (s + size c) pr m (Three a b c)
    snocTree (Deep s pr m (One a)) b =
    	Deep (s + size b) pr m (Two a b)

    -- | /O(log(min(n1,n2)))/. Concatenate two sequences.
    (><)		:: Seq a -> Seq a -> Seq a
    Seq xs >< Seq ys = Seq (appendTree0 xs ys)

    -- The appendTree/addDigits gunk below is machine generated

    appendTree0 :: FingerTree (Elem a) -> FingerTree (Elem a) -> FingerTree (Elem a)
    appendTree0 Empty xs =
    	xs
    appendTree0 xs Empty =
    	xs
    appendTree0 (Single x) xs =
    	x `consTree` xs
    appendTree0 xs (Single x) =
    	xs `snocTree` x
    appendTree0 (Deep s1 pr1 m1 sf1) (Deep s2 pr2 m2 sf2) =
    	Deep (s1 + s2) pr1 (addDigits0 m1 sf1 pr2 m2) sf2

    addDigits0 :: FingerTree (Node (Elem a)) -> Digit (Elem a) -> Digit (Elem a) -> FingerTree (Node (Elem a)) -> FingerTree (Node (Elem a))
    addDigits0 m1 (One a) (One b) m2 =
    	appendTree1 m1 (node2 a b) m2
    addDigits0 m1 (One a) (Two b c) m2 =
    	appendTree1 m1 (node3 a b c) m2
    addDigits0 m1 (One a) (Three b c d) m2 =
    	appendTree2 m1 (node2 a b) (node2 c d) m2
    addDigits0 m1 (One a) (Four b c d e) m2 =
    	appendTree2 m1 (node3 a b c) (node2 d e) m2
    addDigits0 m1 (Two a b) (One c) m2 =
    	appendTree1 m1 (node3 a b c) m2
    addDigits0 m1 (Two a b) (Two c d) m2 =
    	appendTree2 m1 (node2 a b) (node2 c d) m2
    addDigits0 m1 (Two a b) (Three c d e) m2 =
    	appendTree2 m1 (node3 a b c) (node2 d e) m2
    addDigits0 m1 (Two a b) (Four c d e f) m2 =
    	appendTree2 m1 (node3 a b c) (node3 d e f) m2
    addDigits0 m1 (Three a b c) (One d) m2 =
    	appendTree2 m1 (node2 a b) (node2 c d) m2
    addDigits0 m1 (Three a b c) (Two d e) m2 =
    	appendTree2 m1 (node3 a b c) (node2 d e) m2
    addDigits0 m1 (Three a b c) (Three d e f) m2 =
    	appendTree2 m1 (node3 a b c) (node3 d e f) m2
    addDigits0 m1 (Three a b c) (Four d e f g) m2 =
    	appendTree3 m1 (node3 a b c) (node2 d e) (node2 f g) m2
    addDigits0 m1 (Four a b c d) (One e) m2 =
    	appendTree2 m1 (node3 a b c) (node2 d e) m2
    addDigits0 m1 (Four a b c d) (Two e f) m2 =
    	appendTree2 m1 (node3 a b c) (node3 d e f) m2
    addDigits0 m1 (Four a b c d) (Three e f g) m2 =
    	appendTree3 m1 (node3 a b c) (node2 d e) (node2 f g) m2
    addDigits0 m1 (Four a b c d) (Four e f g h) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node2 g h) m2

    appendTree1 :: FingerTree (Node a) -> Node a -> FingerTree (Node a) -> FingerTree (Node a)
    appendTree1 Empty a xs =
    	a `consTree` xs
    appendTree1 xs a Empty =
    	xs `snocTree` a
    appendTree1 (Single x) a xs =
    	x `consTree` a `consTree` xs
    appendTree1 xs a (Single x) =
    	xs `snocTree` a `snocTree` x
    appendTree1 (Deep s1 pr1 m1 sf1) a (Deep s2 pr2 m2 sf2) =
    	Deep (s1 + size a + s2) pr1 (addDigits1 m1 sf1 a pr2 m2) sf2

    addDigits1 :: FingerTree (Node (Node a)) -> Digit (Node a) -> Node a -> Digit (Node a) -> FingerTree (Node (Node a)) -> FingerTree (Node (Node a))
    addDigits1 m1 (One a) b (One c) m2 =
    	appendTree1 m1 (node3 a b c) m2
    addDigits1 m1 (One a) b (Two c d) m2 =
    	appendTree2 m1 (node2 a b) (node2 c d) m2
    addDigits1 m1 (One a) b (Three c d e) m2 =
    	appendTree2 m1 (node3 a b c) (node2 d e) m2
    addDigits1 m1 (One a) b (Four c d e f) m2 =
    	appendTree2 m1 (node3 a b c) (node3 d e f) m2
    addDigits1 m1 (Two a b) c (One d) m2 =
    	appendTree2 m1 (node2 a b) (node2 c d) m2
    addDigits1 m1 (Two a b) c (Two d e) m2 =
    	appendTree2 m1 (node3 a b c) (node2 d e) m2
    addDigits1 m1 (Two a b) c (Three d e f) m2 =
    	appendTree2 m1 (node3 a b c) (node3 d e f) m2
    addDigits1 m1 (Two a b) c (Four d e f g) m2 =
    	appendTree3 m1 (node3 a b c) (node2 d e) (node2 f g) m2
    addDigits1 m1 (Three a b c) d (One e) m2 =
    	appendTree2 m1 (node3 a b c) (node2 d e) m2
    addDigits1 m1 (Three a b c) d (Two e f) m2 =
    	appendTree2 m1 (node3 a b c) (node3 d e f) m2
    addDigits1 m1 (Three a b c) d (Three e f g) m2 =
    	appendTree3 m1 (node3 a b c) (node2 d e) (node2 f g) m2
    addDigits1 m1 (Three a b c) d (Four e f g h) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node2 g h) m2
    addDigits1 m1 (Four a b c d) e (One f) m2 =
    	appendTree2 m1 (node3 a b c) (node3 d e f) m2
    addDigits1 m1 (Four a b c d) e (Two f g) m2 =
    	appendTree3 m1 (node3 a b c) (node2 d e) (node2 f g) m2
    addDigits1 m1 (Four a b c d) e (Three f g h) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node2 g h) m2
    addDigits1 m1 (Four a b c d) e (Four f g h i) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node3 g h i) m2

    appendTree2 :: FingerTree (Node a) -> Node a -> Node a -> FingerTree (Node a) -> FingerTree (Node a)
    appendTree2 Empty a b xs =
    	a `consTree` b `consTree` xs
    appendTree2 xs a b Empty =
    	xs `snocTree` a `snocTree` b
    appendTree2 (Single x) a b xs =
    	x `consTree` a `consTree` b `consTree` xs
    appendTree2 xs a b (Single x) =
    	xs `snocTree` a `snocTree` b `snocTree` x
    appendTree2 (Deep s1 pr1 m1 sf1) a b (Deep s2 pr2 m2 sf2) =
    	Deep (s1 + size a + size b + s2) pr1 (addDigits2 m1 sf1 a b pr2 m2) sf2

    addDigits2 :: FingerTree (Node (Node a)) -> Digit (Node a) -> Node a -> Node a -> Digit (Node a) -> FingerTree (Node (Node a)) -> FingerTree (Node (Node a))
    addDigits2 m1 (One a) b c (One d) m2 =
    	appendTree2 m1 (node2 a b) (node2 c d) m2
    addDigits2 m1 (One a) b c (Two d e) m2 =
    	appendTree2 m1 (node3 a b c) (node2 d e) m2
    addDigits2 m1 (One a) b c (Three d e f) m2 =
    	appendTree2 m1 (node3 a b c) (node3 d e f) m2
    addDigits2 m1 (One a) b c (Four d e f g) m2 =
    	appendTree3 m1 (node3 a b c) (node2 d e) (node2 f g) m2
    addDigits2 m1 (Two a b) c d (One e) m2 =
    	appendTree2 m1 (node3 a b c) (node2 d e) m2
    addDigits2 m1 (Two a b) c d (Two e f) m2 =
    	appendTree2 m1 (node3 a b c) (node3 d e f) m2
    addDigits2 m1 (Two a b) c d (Three e f g) m2 =
    	appendTree3 m1 (node3 a b c) (node2 d e) (node2 f g) m2
    addDigits2 m1 (Two a b) c d (Four e f g h) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node2 g h) m2
    addDigits2 m1 (Three a b c) d e (One f) m2 =
    	appendTree2 m1 (node3 a b c) (node3 d e f) m2
    addDigits2 m1 (Three a b c) d e (Two f g) m2 =
    	appendTree3 m1 (node3 a b c) (node2 d e) (node2 f g) m2
    addDigits2 m1 (Three a b c) d e (Three f g h) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node2 g h) m2
    addDigits2 m1 (Three a b c) d e (Four f g h i) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node3 g h i) m2
    addDigits2 m1 (Four a b c d) e f (One g) m2 =
    	appendTree3 m1 (node3 a b c) (node2 d e) (node2 f g) m2
    addDigits2 m1 (Four a b c d) e f (Two g h) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node2 g h) m2
    addDigits2 m1 (Four a b c d) e f (Three g h i) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node3 g h i) m2
    addDigits2 m1 (Four a b c d) e f (Four g h i j) m2 =
    	appendTree4 m1 (node3 a b c) (node3 d e f) (node2 g h) (node2 i j) m2

    appendTree3 :: FingerTree (Node a) -> Node a -> Node a -> Node a -> FingerTree (Node a) -> FingerTree (Node a)
    appendTree3 Empty a b c xs =
    	a `consTree` b `consTree` c `consTree` xs
    appendTree3 xs a b c Empty =
    	xs `snocTree` a `snocTree` b `snocTree` c
    appendTree3 (Single x) a b c xs =
    	x `consTree` a `consTree` b `consTree` c `consTree` xs
    appendTree3 xs a b c (Single x) =
    	xs `snocTree` a `snocTree` b `snocTree` c `snocTree` x
    appendTree3 (Deep s1 pr1 m1 sf1) a b c (Deep s2 pr2 m2 sf2) =
    	Deep (s1 + size a + size b + size c + s2) pr1 (addDigits3 m1 sf1 a b c pr2 m2) sf2

    addDigits3 :: FingerTree (Node (Node a)) -> Digit (Node a) -> Node a -> Node a -> Node a -> Digit (Node a) -> FingerTree (Node (Node a)) -> FingerTree (Node (Node a))
    addDigits3 m1 (One a) b c d (One e) m2 =
    	appendTree2 m1 (node3 a b c) (node2 d e) m2
    addDigits3 m1 (One a) b c d (Two e f) m2 =
    	appendTree2 m1 (node3 a b c) (node3 d e f) m2
    addDigits3 m1 (One a) b c d (Three e f g) m2 =
    	appendTree3 m1 (node3 a b c) (node2 d e) (node2 f g) m2
    addDigits3 m1 (One a) b c d (Four e f g h) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node2 g h) m2
    addDigits3 m1 (Two a b) c d e (One f) m2 =
    	appendTree2 m1 (node3 a b c) (node3 d e f) m2
    addDigits3 m1 (Two a b) c d e (Two f g) m2 =
    	appendTree3 m1 (node3 a b c) (node2 d e) (node2 f g) m2
    addDigits3 m1 (Two a b) c d e (Three f g h) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node2 g h) m2
    addDigits3 m1 (Two a b) c d e (Four f g h i) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node3 g h i) m2
    addDigits3 m1 (Three a b c) d e f (One g) m2 =
    	appendTree3 m1 (node3 a b c) (node2 d e) (node2 f g) m2
    addDigits3 m1 (Three a b c) d e f (Two g h) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node2 g h) m2
    addDigits3 m1 (Three a b c) d e f (Three g h i) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node3 g h i) m2
    addDigits3 m1 (Three a b c) d e f (Four g h i j) m2 =
    	appendTree4 m1 (node3 a b c) (node3 d e f) (node2 g h) (node2 i j) m2
    addDigits3 m1 (Four a b c d) e f g (One h) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node2 g h) m2
    addDigits3 m1 (Four a b c d) e f g (Two h i) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node3 g h i) m2
    addDigits3 m1 (Four a b c d) e f g (Three h i j) m2 =
    	appendTree4 m1 (node3 a b c) (node3 d e f) (node2 g h) (node2 i j) m2
    addDigits3 m1 (Four a b c d) e f g (Four h i j k) m2 =
    	appendTree4 m1 (node3 a b c) (node3 d e f) (node3 g h i) (node2 j k) m2

    appendTree4 :: FingerTree (Node a) -> Node a -> Node a -> Node a -> Node a -> FingerTree (Node a) -> FingerTree (Node a)
    appendTree4 Empty a b c d xs =
    	a `consTree` b `consTree` c `consTree` d `consTree` xs
    appendTree4 xs a b c d Empty =
    	xs `snocTree` a `snocTree` b `snocTree` c `snocTree` d
    appendTree4 (Single x) a b c d xs =
    	x `consTree` a `consTree` b `consTree` c `consTree` d `consTree` xs
    appendTree4 xs a b c d (Single x) =
    	xs `snocTree` a `snocTree` b `snocTree` c `snocTree` d `snocTree` x
    appendTree4 (Deep s1 pr1 m1 sf1) a b c d (Deep s2 pr2 m2 sf2) =
    	Deep (s1 + size a + size b + size c + size d + s2) pr1 (addDigits4 m1 sf1 a b c d pr2 m2) sf2

    addDigits4 :: FingerTree (Node (Node a)) -> Digit (Node a) -> Node a -> Node a -> Node a -> Node a -> Digit (Node a) -> FingerTree (Node (Node a)) -> FingerTree (Node (Node a))
    addDigits4 m1 (One a) b c d e (One f) m2 =
    	appendTree2 m1 (node3 a b c) (node3 d e f) m2
    addDigits4 m1 (One a) b c d e (Two f g) m2 =
    	appendTree3 m1 (node3 a b c) (node2 d e) (node2 f g) m2
    addDigits4 m1 (One a) b c d e (Three f g h) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node2 g h) m2
    addDigits4 m1 (One a) b c d e (Four f g h i) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node3 g h i) m2
    addDigits4 m1 (Two a b) c d e f (One g) m2 =
    	appendTree3 m1 (node3 a b c) (node2 d e) (node2 f g) m2
    addDigits4 m1 (Two a b) c d e f (Two g h) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node2 g h) m2
    addDigits4 m1 (Two a b) c d e f (Three g h i) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node3 g h i) m2
    addDigits4 m1 (Two a b) c d e f (Four g h i j) m2 =
    	appendTree4 m1 (node3 a b c) (node3 d e f) (node2 g h) (node2 i j) m2
    addDigits4 m1 (Three a b c) d e f g (One h) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node2 g h) m2
    addDigits4 m1 (Three a b c) d e f g (Two h i) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node3 g h i) m2
    addDigits4 m1 (Three a b c) d e f g (Three h i j) m2 =
    	appendTree4 m1 (node3 a b c) (node3 d e f) (node2 g h) (node2 i j) m2
    addDigits4 m1 (Three a b c) d e f g (Four h i j k) m2 =
    	appendTree4 m1 (node3 a b c) (node3 d e f) (node3 g h i) (node2 j k) m2
    addDigits4 m1 (Four a b c d) e f g h (One i) m2 =
    	appendTree3 m1 (node3 a b c) (node3 d e f) (node3 g h i) m2
    addDigits4 m1 (Four a b c d) e f g h (Two i j) m2 =
    	appendTree4 m1 (node3 a b c) (node3 d e f) (node2 g h) (node2 i j) m2
    addDigits4 m1 (Four a b c d) e f g h (Three i j k) m2 =
    	appendTree4 m1 (node3 a b c) (node3 d e f) (node3 g h i) (node2 j k) m2
    addDigits4 m1 (Four a b c d) e f g h (Four i j k l) m2 =
    	appendTree4 m1 (node3 a b c) (node3 d e f) (node3 g h i) (node3 j k l) m2

    -- | Builds a sequence from a seed value.  Takes time linear in the
    -- number of generated elements.  /WARNING:/ If the number of generated
    -- elements is infinite, this method will not terminate.
    unfoldr :: (b -> Maybe (a, b)) -> b -> Seq a
    unfoldr f = unfoldr' empty
      -- uses tail recursion rather than, for instance, the List implementation.
      where unfoldr' as b = maybe as (\ (a, b') -> unfoldr' (as |> a) b') (f b)

    -- | @'unfoldl' f x@ is equivalent to @'reverse' ('unfoldr' (swap . f) x)@.
    unfoldl :: (b -> Maybe (b, a)) -> b -> Seq a
    unfoldl f = unfoldl' empty
      where unfoldl' as b = maybe as (\ (b', a) -> unfoldl' (a <| as) b') (f b)

    -- | /O(n)/.  Constructs a sequence by repeated application of a function
    -- to a seed value.
    --
    -- > iterateN n f x = fromList (Prelude.take n (Prelude.iterate f x))
    iterateN :: Int -> (a -> a) -> a -> Seq a
    iterateN n f x
      | n >= 0	= replicateA n (State (\ y -> (f y, y))) `execState` x
      | otherwise	= error "iterateN takes a nonnegative integer argument"

    ------------------------------------------------------------------------
    -- Deconstruction
    ------------------------------------------------------------------------

    -- | /O(1)/. Is this the empty sequence?
    null		:: Seq a -> Bool
    null (Seq Empty) = True
    null _		=  False

    -- | /O(1)/. The number of elements in the sequence.
    length		:: Seq a -> Int
    length (Seq xs) =  size xs

    -- Views

    data Maybe2 a b = Nothing2 | Just2 a b

    -- | View of the left end of a sequence.
    data ViewL a
    	= EmptyL	-- ^ empty sequence
    	| a :< Seq a	-- ^ leftmost element and the rest of the sequence
    #ifndef __HADDOCK__
    # if __GLASGOW_HASKELL__
    	deriving (Eq, Ord, Show, Read, Data)
    # else
    	deriving (Eq, Ord, Show, Read)
    # endif
    #else
    instance Eq a => Eq (ViewL a)
    instance Ord a => Ord (ViewL a)
    instance Show a => Show (ViewL a)
    instance Read a => Read (ViewL a)
    instance Data a => Data (ViewL a)
    #endif

    INSTANCE_TYPEABLE1(ViewL,viewLTc,"ViewL")

    instance Functor ViewL where
    	fmap = fmapDefault

    instance Foldable ViewL where
    	foldr _ z EmptyL = z
    	foldr f z (x :< xs) = f x (foldr f z xs)

    	foldl _ z EmptyL = z
    	foldl f z (x :< xs) = foldl f (f z x) xs

    	foldl1 _ EmptyL = error "foldl1: empty view"
    	foldl1 f (x :< xs) = foldl f x xs

    instance Traversable ViewL where
    	traverse _ EmptyL	= pure EmptyL
    	traverse f (x :< xs)	= (:<) <$> f x <*> traverse f xs

    -- | /O(1)/. Analyse the left end of a sequence.
    viewl		::  Seq a -> ViewL a
    viewl (Seq xs)	=  case viewLTree xs of
    	Nothing2 -> EmptyL
    	Just2 (Elem x) xs' -> x :< Seq xs'

    {-# SPECIALIZE viewLTree :: FingerTree (Elem a) -> Maybe2 (Elem a) (FingerTree (Elem a)) #-}
    {-# SPECIALIZE viewLTree :: FingerTree (Node a) -> Maybe2 (Node a) (FingerTree (Node a)) #-}
    viewLTree	:: Sized a => FingerTree a -> Maybe2 a (FingerTree a)
    viewLTree Empty			= Nothing2
    viewLTree (Single a)		= Just2 a Empty
    viewLTree (Deep s (One a) m sf) = Just2 a (pullL (s - size a) m sf)
    viewLTree (Deep s (Two a b) m sf) =
    	Just2 a (Deep (s - size a) (One b) m sf)
    viewLTree (Deep s (Three a b c) m sf) =
    	Just2 a (Deep (s - size a) (Two b c) m sf)
    viewLTree (Deep s (Four a b c d) m sf) =
    	Just2 a (Deep (s - size a) (Three b c d) m sf)

    -- | View of the right end of a sequence.
    data ViewR a
    	= EmptyR	-- ^ empty sequence
    	| Seq a :> a	-- ^ the sequence minus the rightmost element,
    			-- and the rightmost element
    #ifndef __HADDOCK__
    # if __GLASGOW_HASKELL__
    	deriving (Eq, Ord, Show, Read, Data)
    # else
    	deriving (Eq, Ord, Show, Read)
    # endif
    #else
    instance Eq a => Eq (ViewR a)
    instance Ord a => Ord (ViewR a)
    instance Show a => Show (ViewR a)
    instance Read a => Read (ViewR a)
    instance Data a => Data (ViewR a)
    #endif

    INSTANCE_TYPEABLE1(ViewR,viewRTc,"ViewR")

    instance Functor ViewR where
    	fmap = fmapDefault

    instance Foldable ViewR where
    	foldr _ z EmptyR = z
    	foldr f z (xs :> x) = foldr f (f x z) xs

    	foldl _ z EmptyR = z
    	foldl f z (xs :> x) = foldl f z xs `f` x

    	foldr1 _ EmptyR = error "foldr1: empty view"
    	foldr1 f (xs :> x) = foldr f x xs

    instance Traversable ViewR where
    	traverse _ EmptyR	= pure EmptyR
    	traverse f (xs :> x)	= (:>) <$> traverse f xs <*> f x

    -- | /O(1)/. Analyse the right end of a sequence.
    viewr		::  Seq a -> ViewR a
    viewr (Seq xs)	=  case viewRTree xs of
    	Nothing2 -> EmptyR
    	Just2 xs' (Elem x) -> Seq xs' :> x

    {-# SPECIALIZE viewRTree :: FingerTree (Elem a) -> Maybe2 (FingerTree (Elem a)) (Elem a) #-}
    {-# SPECIALIZE viewRTree :: FingerTree (Node a) -> Maybe2 (FingerTree (Node a)) (Node a) #-}
    viewRTree	:: Sized a => FingerTree a -> Maybe2 (FingerTree a) a
    viewRTree Empty			= Nothing2
    viewRTree (Single z)		= Just2 Empty z
    viewRTree (Deep s pr m (One z)) = Just2 (pullR (s - size z) pr m) z
    viewRTree (Deep s pr m (Two y z)) =
    	Just2 (Deep (s - size z) pr m (One y)) z
    viewRTree (Deep s pr m (Three x y z)) =
    	Just2 (Deep (s - size z) pr m (Two x y)) z
    viewRTree (Deep s pr m (Four w x y z)) =
    	Just2 (Deep (s - size z) pr m (Three w x y)) z

    ------------------------------------------------------------------------
    -- Scans
    --
    -- These are not particularly complex applications of the Traversable
    -- functor, though making the correspondence with Data.List exact
    -- requires the use of (<|) and (|>).
    --
    -- Note that save for the single (<|) or (|>), we maintain the original
    -- structure of the Seq, not having to do any restructuring of our own.
    --
    -- wasserman.louis@gmail.com, 5/23/09
    ------------------------------------------------------------------------

    -- | 'scanl' is similar to 'foldl', but returns a sequence of reduced
    -- values from the left:
    --
    -- > scanl f z (fromList [x1, x2, ...]) = fromList [z, z `f` x1, (z `f` x1) `f` x2, ...]
    scanl :: (a -> b -> a) -> a -> Seq b -> Seq a
    scanl f z0 xs = z0 <| snd (mapAccumL (\ x z -> let x' = f x z in (x', x')) z0 xs)

    -- | 'scanl1' is a variant of 'scanl' that has no starting value argument:
    --
    -- > scanl1 f (fromList [x1, x2, ...]) = fromList [x1, x1 `f` x2, ...]
    scanl1 :: (a -> a -> a) -> Seq a -> Seq a
    scanl1 f xs = case viewl xs of
    	EmptyL		-> error "scanl1 takes a nonempty sequence as an argument"
    	x :< xs'	-> scanl f x xs'

    -- | 'scanr' is the right-to-left dual of 'scanl'.
    scanr :: (a -> b -> b) -> b -> Seq a -> Seq b
    scanr f z0 xs = snd (mapAccumR (\ z x -> let z' = f x z in (z', z')) z0 xs) |> z0

    -- | 'scanr1' is a variant of 'scanr' that has no starting value argument.
    scanr1 :: (a -> a -> a) -> Seq a -> Seq a
    scanr1 f xs = case viewr xs of
    	EmptyR		-> error "scanr1 takes a nonempty sequence as an argument"
    	xs' :> x	-> scanr f x xs'

    -- Indexing

    -- | /O(log(min(i,n-i)))/. The element at the specified position,
    -- counting from 0.  The argument should thus be a non-negative
    -- integer less than the size of the sequence.
    -- If the position is out of range, 'index' fails with an error.
    index		:: Seq a -> Int -> a
    index (Seq xs) i
      | 0 <= i && i < size xs = case lookupTree i xs of
    				Place _ (Elem x) -> x
      | otherwise	= error "index out of bounds"

    data Place a = Place {-# UNPACK #-} !Int a
    #if TESTING
    	deriving Show
    #endif

    {-# SPECIALIZE lookupTree :: Int -> FingerTree (Elem a) -> Place (Elem a) #-}
    {-# SPECIALIZE lookupTree :: Int -> FingerTree (Node a) -> Place (Node a) #-}
    lookupTree :: Sized a => Int -> FingerTree a -> Place a
    lookupTree _ Empty = error "lookupTree of empty tree"
    lookupTree i (Single x) = Place i x
    lookupTree i (Deep _ pr m sf)
      | i < spr	=  lookupDigit i pr
      | i < spm	=  case lookupTree (i - spr) m of
    			Place i' xs -> lookupNode i' xs
      | otherwise	=  lookupDigit (i - spm) sf
      where	spr	= size pr
    	spm	= spr + size m

    {-# SPECIALIZE lookupNode :: Int -> Node (Elem a) -> Place (Elem a) #-}
    {-# SPECIALIZE lookupNode :: Int -> Node (Node a) -> Place (Node a) #-}
    lookupNode :: Sized a => Int -> Node a -> Place a
    lookupNode i (Node2 _ a b)
      | i < sa	= Place i a
      | otherwise	= Place (i - sa) b
      where	sa	= size a
    lookupNode i (Node3 _ a b c)
      | i < sa	= Place i a
      | i < sab	= Place (i - sa) b
      | otherwise	= Place (i - sab) c
      where	sa	= size a
    	sab	= sa + size b

    {-# SPECIALIZE lookupDigit :: Int -> Digit (Elem a) -> Place (Elem a) #-}
    {-# SPECIALIZE lookupDigit :: Int -> Digit (Node a) -> Place (Node a) #-}
    lookupDigit :: Sized a => Int -> Digit a -> Place a
    lookupDigit i (One a) = Place i a
    lookupDigit i (Two a b)
      | i < sa	= Place i a
      | otherwise	= Place (i - sa) b
      where	sa	= size a
    lookupDigit i (Three a b c)
      | i < sa	= Place i a
      | i < sab	= Place (i - sa) b
      | otherwise	= Place (i - sab) c
      where	sa	= size a
    	sab	= sa + size b
    lookupDigit i (Four a b c d)
      | i < sa	= Place i a
      | i < sab	= Place (i - sa) b
      | i < sabc	= Place (i - sab) c
      | otherwise	= Place (i - sabc) d
      where	sa	= size a
    	sab	= sa + size b
    	sabc	= sab + size c

    -- | /O(log(min(i,n-i)))/. Replace the element at the specified position.
    -- If the position is out of range, the original sequence is returned.
    update		:: Int -> a -> Seq a -> Seq a
    update i x	= adjust (const x) i

    -- | /O(log(min(i,n-i)))/. Update the element at the specified position.
    -- If the position is out of range, the original sequence is returned.
    adjust		:: (a -> a) -> Int -> Seq a -> Seq a
    adjust f i (Seq xs)
      | 0 <= i && i < size xs = Seq (adjustTree (const (fmap f)) i xs)
      | otherwise	= Seq xs

    {-# SPECIALIZE adjustTree :: (Int -> Elem a -> Elem a) -> Int -> FingerTree (Elem a) -> FingerTree (Elem a) #-}
    {-# SPECIALIZE adjustTree :: (Int -> Node a -> Node a) -> Int -> FingerTree (Node a) -> FingerTree (Node a) #-}
    adjustTree	:: Sized a => (Int -> a -> a) ->
    			Int -> FingerTree a -> FingerTree a
    adjustTree _ _ Empty = error "adjustTree of empty tree"
    adjustTree f i (Single x) = Single (f i x)
    adjustTree f i (Deep s pr m sf)
      | i < spr	= Deep s (adjustDigit f i pr) m sf
      | i < spm	= Deep s pr (adjustTree (adjustNode f) (i - spr) m) sf
      | otherwise	= Deep s pr m (adjustDigit f (i - spm) sf)
      where	spr	= size pr
    	spm	= spr + size m

    {-# SPECIALIZE adjustNode :: (Int -> Elem a -> Elem a) -> Int -> Node (Elem a) -> Node (Elem a) #-}
    {-# SPECIALIZE adjustNode :: (Int -> Node a -> Node a) -> Int -> Node (Node a) -> Node (Node a) #-}
    adjustNode	:: Sized a => (Int -> a -> a) -> Int -> Node a -> Node a
    adjustNode f i (Node2 s a b)
      | i < sa	= Node2 s (f i a) b
      | otherwise	= Node2 s a (f (i - sa) b)
      where	sa	= size a
    adjustNode f i (Node3 s a b c)
      | i < sa	= Node3 s (f i a) b c
      | i < sab	= Node3 s a (f (i - sa) b) c
      | otherwise	= Node3 s a b (f (i - sab) c)
      where	sa	= size a
    	sab	= sa + size b

    {-# SPECIALIZE adjustDigit :: (Int -> Elem a -> Elem a) -> Int -> Digit (Elem a) -> Digit (Elem a) #-}
    {-# SPECIALIZE adjustDigit :: (Int -> Node a -> Node a) -> Int -> Digit (Node a) -> Digit (Node a) #-}
    adjustDigit	:: Sized a => (Int -> a -> a) -> Int -> Digit a -> Digit a
    adjustDigit f i (One a) = One (f i a)
    adjustDigit f i (Two a b)
      | i < sa	= Two (f i a) b
      | otherwise	= Two a (f (i - sa) b)
      where	sa	= size a
    adjustDigit f i (Three a b c)
      | i < sa	= Three (f i a) b c
      | i < sab	= Three a (f (i - sa) b) c
      | otherwise	= Three a b (f (i - sab) c)
      where	sa	= size a
    	sab	= sa + size b
    adjustDigit f i (Four a b c d)
      | i < sa	= Four (f i a) b c d
      | i < sab	= Four a (f (i - sa) b) c d
      | i < sabc	= Four a b (f (i - sab) c) d
      | otherwise	= Four a b c (f (i- sabc) d)
      where	sa	= size a
    	sab	= sa + size b
    	sabc	= sab + size c

    -- | A generalization of 'fmap', 'mapWithIndex' takes a mapping function
    -- that also depends on the element's index, and applies it to every
    -- element in the sequence.
    mapWithIndex :: (Int -> a -> b) -> Seq a -> Seq b
    mapWithIndex f xs = snd (mapAccumL' (\ i x -> (i + 1, f i x)) 0 xs)

    -- Splitting

    -- | /O(log(min(i,n-i)))/. The first @i@ elements of a sequence.
    -- If @i@ is negative, @'take' i s@ yields the empty sequence.
    -- If the sequence contains fewer than @i@ elements, the whole sequence
    -- is returned.
    take		:: Int -> Seq a -> Seq a
    take i		=  fst . splitAt i

    -- | /O(log(min(i,n-i)))/. Elements of a sequence after the first @i@.
    -- If @i@ is negative, @'drop' i s@ yields the whole sequence.
    -- If the sequence contains fewer than @i@ elements, the empty sequence
    -- is returned.
    drop		:: Int -> Seq a -> Seq a
    drop i		=  snd . splitAt i

    -- | /O(log(min(i,n-i)))/. Split a sequence at a given position.
    -- @'splitAt' i s = ('take' i s, 'drop' i s)@.
    splitAt			:: Int -> Seq a -> (Seq a, Seq a)
    splitAt i (Seq xs)	=  (Seq l, Seq r)
      where	(l, r)		=  split i xs

    split :: Int -> FingerTree (Elem a) ->
    	(FingerTree (Elem a), FingerTree (Elem a))
    split i Empty	= i `seq` (Empty, Empty)
    split i xs
      | size xs > i	= (l, consTree x r)
      | otherwise	= (xs, Empty)
      where Split l x r = splitTree i xs

    data Split t a = Split t a t
    #if TESTING
    	deriving Show
    #endif

    {-# SPECIALIZE splitTree :: Int -> FingerTree (Elem a) -> Split (FingerTree (Elem a)) (Elem a) #-}
    {-# SPECIALIZE splitTree :: Int -> FingerTree (Node a) -> Split (FingerTree (Node a)) (Node a) #-}
    splitTree :: Sized a => Int -> FingerTree a -> Split (FingerTree a) a
    splitTree _ Empty = error "splitTree of empty tree"
    splitTree i (Single x) = i `seq` Split Empty x Empty
    splitTree i (Deep _ pr m sf)
      | i < spr	= case splitDigit i pr of
    			Split l x r -> Split (maybe Empty digitToTree l) x (deepL r m sf)
      | i < spm	= case splitTree im m of
    			Split ml xs mr -> case splitNode (im - size ml) xs of
    			    Split l x r -> Split (deepR pr ml l) x (deepL r mr sf)
      | otherwise	= case splitDigit (i - spm) sf of
    			Split l x r -> Split (deepR pr m l) x (maybe Empty digitToTree r)
      where	spr	= size pr
    	spm	= spr + size m
    	im	= i - spr

    {-# SPECIALIZE splitNode :: Int -> Node (Elem a) -> Split (Maybe (Digit (Elem a))) (Elem a) #-}
    {-# SPECIALIZE splitNode :: Int -> Node (Node a) -> Split (Maybe (Digit (Node a))) (Node a) #-}
    splitNode :: Sized a => Int -> Node a -> Split (Maybe (Digit a)) a
    splitNode i (Node2 _ a b)
      | i < sa	= Split Nothing a (Just (One b))
      | otherwise	= Split (Just (One a)) b Nothing
      where	sa	= size a
    splitNode i (Node3 _ a b c)
      | i < sa	= Split Nothing a (Just (Two b c))
      | i < sab	= Split (Just (One a)) b (Just (One c))
      | otherwise	= Split (Just (Two a b)) c Nothing
      where	sa	= size a
    	sab	= sa + size b

    {-# SPECIALIZE splitDigit :: Int -> Digit (Elem a) -> Split (Maybe (Digit (Elem a))) (Elem a) #-}
    {-# SPECIALIZE splitDigit :: Int -> Digit (Node a) -> Split (Maybe (Digit (Node a))) (Node a) #-}
    splitDigit :: Sized a => Int -> Digit a -> Split (Maybe (Digit a)) a
    splitDigit i (One a) = i `seq` Split Nothing a Nothing
    splitDigit i (Two a b)
      | i < sa	= Split Nothing a (Just (One b))
      | otherwise	= Split (Just (One a)) b Nothing
      where	sa	= size a
    splitDigit i (Three a b c)
      | i < sa	= Split Nothing a (Just (Two b c))
      | i < sab	= Split (Just (One a)) b (Just (One c))
      | otherwise	= Split (Just (Two a b)) c Nothing
      where	sa	= size a
    	sab	= sa + size b
    splitDigit i (Four a b c d)
      | i < sa	= Split Nothing a (Just (Three b c d))
      | i < sab	= Split (Just (One a)) b (Just (Two c d))
      | i < sabc	= Split (Just (Two a b)) c (Just (One d))
      | otherwise	= Split (Just (Three a b c)) d Nothing
      where	sa	= size a
    	sab	= sa + size b
    	sabc	= sab + size c

    -- | /O(n)/.  Returns a sequence of all suffixes of this sequence,
    -- longest first.  For example,
    --
    -- > tails (fromList "abc") = fromList [fromList "abc", fromList "bc", fromList "c", fromList ""]
    --
    -- Evaluating the /i/th suffix takes /O(log(min(i, n-i)))/, but evaluating
    -- every suffix in the sequence takes /O(n)/ due to sharing.
    tails			:: Seq a -> Seq (Seq a)
    tails (Seq xs)		= Seq (tailsTree (Elem . Seq) xs) |> empty

    -- | /O(n)/.  Returns a sequence of all prefixes of this sequence,
    -- shortest first.  For example,
    --
    -- > inits (fromList "abc") = fromList [fromList "", fromList "a", fromList "ab", fromList "abc"]
    --
    -- Evaluating the /i/th prefix takes /O(log(min(i, n-i)))/, but evaluating
    -- every prefix in the sequence takes /O(n)/ due to sharing.
    inits			:: Seq a -> Seq (Seq a)
    inits (Seq xs) 		= empty <| Seq (initsTree (Elem . Seq) xs)

    -- This implementation of tails (and, analogously, inits) has the
    -- following algorithmic advantages:
    --	Evaluating each tail in the sequence takes linear total time,
    --	which is better than we could say for
    -- 		@fromList [drop n xs | n <- [0..length xs]]@.
    --	Evaluating any individual tail takes logarithmic time, which is
    --	better than we can say for either
    -- 		@scanr (<|) empty xs@ or @iterateN (length xs + 1) (\ xs -> let _ :< xs' = viewl xs in xs') xs@.
    --
    -- Moreover, if we actually look at every tail in the sequence, the
    -- following benchmarks demonstrate that this implementation is modestly
    -- faster than any of the above:
    --
    -- Times (ms)
    --               min      mean    +/-sd    median    max
    -- Seq.tails:   21.986   24.961   10.169   22.417   86.485
    -- scanr:       85.392   87.942    2.488   87.425  100.217
    -- iterateN:       29.952   31.245    1.574   30.412   37.268
    --
    -- The algorithm for tails (and, analogously, inits) is as follows:
    --
    -- A Node in the FingerTree of tails is constructed by evaluating the
    -- corresponding tail of the FingerTree of Nodes, considering the first
    -- Node in this tail, and constructing a Node in which each tail of this
    -- Node is made to be the prefix of the remaining tree.  This ends up
    -- working quite elegantly, as the remainder of the tail of the FingerTree
    -- of Nodes becomes the middle of a new tail, the suffix of the Node is
    -- the prefix, and the suffix of the original tree is retained.
    --
    -- In particular, evaluating the /i/th tail involves making as
    -- many partial evaluations as the Node depth of the /i/th element.
    -- In addition, when we evaluate the /i/th tail, and we also evaluate
    -- the /j/th tail, and /m/ Nodes are on the path to both /i/ and /j/,
    -- each of those /m/ evaluations are shared between the computation of
    -- the /i/th and /j/th tails.
    --
    -- wasserman.louis@gmail.com, 7/16/09

    tailsDigit :: Digit a -> Digit (Digit a)
    tailsDigit (One a) = One (One a)
    tailsDigit (Two a b) = Two (Two a b) (One b)
    tailsDigit (Three a b c) = Three (Three a b c) (Two b c) (One c)
    tailsDigit (Four a b c d) = Four (Four a b c d) (Three b c d) (Two c d) (One d)

    initsDigit :: Digit a -> Digit (Digit a)
    initsDigit (One a) = One (One a)
    initsDigit (Two a b) = Two (One a) (Two a b)
    initsDigit (Three a b c) = Three (One a) (Two a b) (Three a b c)
    initsDigit (Four a b c d) = Four (One a) (Two a b) (Three a b c) (Four a b c d)

    tailsNode :: Node a -> Node (Digit a)
    tailsNode (Node2 s a b) = Node2 s (Two a b) (One b)
    tailsNode (Node3 s a b c) = Node3 s (Three a b c) (Two b c) (One c)

    initsNode :: Node a -> Node (Digit a)
    initsNode (Node2 s a b) = Node2 s (One a) (Two a b)
    initsNode (Node3 s a b c) = Node3 s (One a) (Two a b) (Three a b c)

    {-# SPECIALIZE tailsTree :: (FingerTree (Elem a) -> Elem b) -> FingerTree (Elem a) -> FingerTree (Elem b) #-}
    {-# SPECIALIZE tailsTree :: (FingerTree (Node a) -> Node b) -> FingerTree (Node a) -> FingerTree (Node b) #-}
    -- | Given a function to apply to tails of a tree, applies that function
    -- to every tail of the specified tree.
    tailsTree :: (Sized a, Sized b) => (FingerTree a -> b) -> FingerTree a -> FingerTree b
    tailsTree _ Empty = Empty
    tailsTree f (Single x) = Single (f (Single x))
    tailsTree f (Deep n pr m sf) =
    	Deep n (fmap (\ pr' -> f (deep pr' m sf)) (tailsDigit pr))
    		(tailsTree f' m)
    		(fmap (f . digitToTree) (tailsDigit sf))
      where	f' ms = let Just2 node m' = viewLTree ms in
    		fmap (\ pr' -> f (deep pr' m' sf)) (tailsNode node)

    {-# SPECIALIZE initsTree :: (FingerTree (Elem a) -> Elem b) -> FingerTree (Elem a) -> FingerTree (Elem b) #-}
    {-# SPECIALIZE initsTree :: (FingerTree (Node a) -> Node b) -> FingerTree (Node a) -> FingerTree (Node b) #-}
    -- | Given a function to apply to inits of a tree, applies that function
    -- to every init of the specified tree.
    initsTree :: (Sized a, Sized b) => (FingerTree a -> b) -> FingerTree a -> FingerTree b
    initsTree _ Empty = Empty
    initsTree f (Single x) = Single (f (Single x))
    initsTree f (Deep n pr m sf) =
    	Deep n (fmap (f . digitToTree) (initsDigit pr))
    		(initsTree f' m)
    		(fmap (f . deep pr m) (initsDigit sf))
      where	f' ms =  let Just2 m' node = viewRTree ms in
    		 fmap (\ sf' -> f (deep pr m' sf')) (initsNode node)

    {-# INLINE foldlWithIndex #-}
    -- | 'foldlWithIndex' is a version of 'foldl' that also provides access
    -- to the index of each element.
    foldlWithIndex :: (b -> Int -> a -> b) -> b -> Seq a -> b
    foldlWithIndex f z xs = foldl (\ g x i -> i `seq` f (g (i - 1)) i x) (const z) xs (length xs - 1)

    {-# INLINE foldrWithIndex #-}
    -- | 'foldrWithIndex' is a version of 'foldr' that also provides access
    -- to the index of each element.
    foldrWithIndex :: (Int -> a -> b -> b) -> b -> Seq a -> b
    foldrWithIndex f z xs = foldr (\ x g i -> i `seq` f i x (g (i+1))) (const z) xs 0

    {-# INLINE listToMaybe' #-}
    -- 'listToMaybe\'' is a good consumer version of 'listToMaybe'.
    listToMaybe' :: [a] -> Maybe a
    listToMaybe' = foldr (\ x _ -> Just x) Nothing

    -- | /O(i)/ where /i/ is the prefix length.  'takeWhileL', applied
    -- to a predicate @p@ and a sequence @xs@, returns the longest prefix
    -- (possibly empty) of @xs@ of elements that satisfy @p@.
    takeWhileL :: (a -> Bool) -> Seq a -> Seq a
    takeWhileL p = fst . spanl p

    -- | /O(i)/ where /i/ is the suffix length.  'takeWhileR', applied
    -- to a predicate @p@ and a sequence @xs@, returns the longest suffix
    -- (possibly empty) of @xs@ of elements that satisfy @p@.
    --
    -- @'takeWhileR' p xs@ is equivalent to @'reverse' ('takeWhileL' p ('reverse' xs))@.
    takeWhileR :: (a -> Bool) -> Seq a -> Seq a
    takeWhileR p = fst . spanr p

    -- | /O(i)/ where /i/ is the prefix length.  @'dropWhileL' p xs@ returns
    -- the suffix remaining after @'takeWhileL' p xs@.
    dropWhileL :: (a -> Bool) -> Seq a -> Seq a
    dropWhileL p = snd . spanl p

    -- | /O(i)/ where /i/ is the suffix length.  @'dropWhileR' p xs@ returns
    -- the prefix remaining after @'takeWhileR' p xs@.
    --
    -- @'dropWhileR' p xs@ is equivalent to @'reverse' ('dropWhileL' p ('reverse' xs))@.
    dropWhileR :: (a -> Bool) -> Seq a -> Seq a
    dropWhileR p = snd . spanr p

    -- | /O(i)/ where /i/ is the prefix length.  'spanl', applied to
    -- a predicate @p@ and a sequence @xs@, returns a pair whose first
    -- element is the longest prefix (possibly empty) of @xs@ of elements that
    -- satisfy @p@ and the second element is the remainder of the sequence.
    spanl :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
    spanl p = breakl (not . p)

    -- | /O(i)/ where /i/ is the suffix length.  'spanr', applied to a
    -- predicate @p@ and a sequence @xs@, returns a pair whose /first/ element
    -- is the longest /suffix/ (possibly empty) of @xs@ of elements that
    -- satisfy @p@ and the second element is the remainder of the sequence.
    spanr :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
    spanr p = breakr (not . p)

    {-# INLINE breakl #-}
    -- | /O(i)/ where /i/ is the breakpoint index.  'breakl', applied to a
    -- predicate @p@ and a sequence @xs@, returns a pair whose first element
    -- is the longest prefix (possibly empty) of @xs@ of elements that
    -- /do not satisfy/ @p@ and the second element is the remainder of
    -- the sequence.
    --
    -- @'breakl' p@ is equivalent to @'spanl' (not . p)@.
    breakl :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
    breakl p xs = foldr (\ i _ -> splitAt i xs) (xs, empty) (findIndicesL p xs)

    {-# INLINE breakr #-}
    -- | @'breakr' p@ is equivalent to @'spanr' (not . p)@.
    breakr :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
    breakr p xs = foldr (\ i _ -> flipPair (splitAt (i + 1) xs)) (xs, empty) (findIndicesR p xs)
      where flipPair (x, y) = (y, x)

    -- | /O(n)/.  The 'partition' function takes a predicate @p@ and a
    -- sequence @xs@ and returns sequences of those elements which do and
    -- do not satisfy the predicate.
    partition :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
    partition p = foldl part (empty, empty)
      where part (xs, ys) x
    	  | p x		= (xs |> x, ys)
    	  | otherwise 	= (xs, ys |> x)

    -- | /O(n)/.  The 'filter' function takes a predicate @p@ and a sequence
    -- @xs@ and returns a sequence of those elements which satisfy the
    -- predicate.
    filter :: (a -> Bool) -> Seq a -> Seq a
    filter p = foldl (\ xs x -> if p x then xs |> x else xs) empty

    -- Indexing sequences

    -- | 'elemIndexL' finds the leftmost index of the specified element,
    -- if it is present, and otherwise 'Nothing'.
    elemIndexL :: Eq a => a -> Seq a -> Maybe Int
    elemIndexL x = findIndexL (x ==)

    -- | 'elemIndexR' finds the rightmost index of the specified element,
    -- if it is present, and otherwise 'Nothing'.
    elemIndexR :: Eq a => a -> Seq a -> Maybe Int
    elemIndexR x = findIndexR (x ==)

    -- | 'elemIndicesL' finds the indices of the specified element, from
    -- left to right (i.e. in ascending order).
    elemIndicesL :: Eq a => a -> Seq a -> [Int]
    elemIndicesL x = findIndicesL (x ==)

    -- | 'elemIndicesR' finds the indices of the specified element, from
    -- right to left (i.e. in descending order).
    elemIndicesR :: Eq a => a -> Seq a -> [Int]
    elemIndicesR x = findIndicesR (x ==)

    -- | @'findIndexL' p xs@ finds the index of the leftmost element that
    -- satisfies @p@, if any exist.
    findIndexL :: (a -> Bool) -> Seq a -> Maybe Int
    findIndexL p = listToMaybe' . findIndicesL p

    -- | @'findIndexR' p xs@ finds the index of the rightmost element that
    -- satisfies @p@, if any exist.
    findIndexR :: (a -> Bool) -> Seq a -> Maybe Int
    findIndexR p = listToMaybe' . findIndicesR p

    {-# INLINE findIndicesL #-}
    -- | @'findIndicesL' p@ finds all indices of elements that satisfy @p@,
    -- in ascending order.
    findIndicesL :: (a -> Bool) -> Seq a -> [Int]
    #if __GLASGOW_HASKELL__
    findIndicesL p xs = build (\ c n -> let g i x z = if p x then c i z else z in
    				foldrWithIndex g n xs)
    #else
    findIndicesL p xs = foldrWithIndex g [] xs
        where g i x is = if p x then i:is else is
    #endif

    {-# INLINE findIndicesR #-}
    -- | @'findIndicesR' p@ finds all indices of elements that satisfy @p@,
    -- in descending order.
    findIndicesR :: (a -> Bool) -> Seq a -> [Int]
    #if __GLASGOW_HASKELL__
    findIndicesR p xs = build (\ c n -> let g z i x = if p x then c i z else z in
    				foldlWithIndex g n xs)
    #else
    findIndicesR p xs = foldlWithIndex g [] xs
        where g is i x = if p x then i:is else is
    #endif

    ------------------------------------------------------------------------
    -- Lists
    ------------------------------------------------------------------------

    -- | /O(n)/. Create a sequence from a finite list of elements.
    -- There is a function 'toList' in the opposite direction for all
    -- instances of the 'Foldable' class, including 'Seq'.
    fromList  	:: [a] -> Seq a
    fromList  	=  Data.List.foldl' (|>) empty

    ------------------------------------------------------------------------
    -- Reverse
    ------------------------------------------------------------------------

    -- | /O(n)/. The reverse of a sequence.
    reverse :: Seq a -> Seq a
    reverse (Seq xs) = Seq (reverseTree id xs)

    reverseTree :: (a -> a) -> FingerTree a -> FingerTree a
    reverseTree _ Empty = Empty
    reverseTree f (Single x) = Single (f x)
    reverseTree f (Deep s pr m sf) =
    	Deep s (reverseDigit f sf)
    		(reverseTree (reverseNode f) m)
    		(reverseDigit f pr)

    {-# INLINE reverseDigit #-}
    reverseDigit :: (a -> a) -> Digit a -> Digit a
    reverseDigit f (One a) = One (f a)
    reverseDigit f (Two a b) = Two (f b) (f a)
    reverseDigit f (Three a b c) = Three (f c) (f b) (f a)
    reverseDigit f (Four a b c d) = Four (f d) (f c) (f b) (f a)

    reverseNode :: (a -> a) -> Node a -> Node a
    reverseNode f (Node2 s a b) = Node2 s (f b) (f a)
    reverseNode f (Node3 s a b c) = Node3 s (f c) (f b) (f a)

    ------------------------------------------------------------------------
    -- Zipping
    ------------------------------------------------------------------------

    -- | /O(min(n1,n2))/.  'zip' takes two sequences and returns a sequence
    -- of corresponding pairs.  If one input is short, excess elements are
    -- discarded from the right end of the longer sequence.
    zip :: Seq a -> Seq b -> Seq (a, b)
    zip = zipWith (,)

    -- | /O(min(n1,n2))/.  'zipWith' generalizes 'zip' by zipping with the
    -- function given as the first argument, instead of a tupling function.
    -- For example, @zipWith (+)@ is applied to two sequences to take the
    -- sequence of corresponding sums.
    zipWith :: (a -> b -> c) -> Seq a -> Seq b -> Seq c
    zipWith f xs ys
      | length xs <= length ys	= zipWith' f xs ys
      | otherwise			= zipWith' (flip f) ys xs

    -- like 'zipWith', but assumes length xs <= length ys
    zipWith' :: (a -> b -> c) -> Seq a -> Seq b -> Seq c
    zipWith' f xs ys = snd (mapAccumL k ys xs)
      where
        k kys x = case viewl kys of
                   (z :< zs) -> (zs, f x z)
                   EmptyL    -> error "zipWith': unexpected EmptyL"

    -- | /O(min(n1,n2,n3))/.  'zip3' takes three sequences and returns a
    -- sequence of triples, analogous to 'zip'.
    zip3 :: Seq a -> Seq b -> Seq c -> Seq (a,b,c)
    zip3 = zipWith3 (,,)

    -- | /O(min(n1,n2,n3))/.  'zipWith3' takes a function which combines
    -- three elements, as well as three sequences and returns a sequence of
    -- their point-wise combinations, analogous to 'zipWith'.
    zipWith3 :: (a -> b -> c -> d) -> Seq a -> Seq b -> Seq c -> Seq d
    zipWith3 f s1 s2 s3 = zipWith ($) (zipWith f s1 s2) s3

    -- | /O(min(n1,n2,n3,n4))/.  'zip4' takes four sequences and returns a
    -- sequence of quadruples, analogous to 'zip'.
    zip4 :: Seq a -> Seq b -> Seq c -> Seq d -> Seq (a,b,c,d)
    zip4 = zipWith4 (,,,)

    -- | /O(min(n1,n2,n3,n4))/.  'zipWith4' takes a function which combines
    -- four elements, as well as four sequences and returns a sequence of
    -- their point-wise combinations, analogous to 'zipWith'.
    zipWith4 :: (a -> b -> c -> d -> e) -> Seq a -> Seq b -> Seq c -> Seq d -> Seq e
    zipWith4 f s1 s2 s3 s4 = zipWith ($) (zipWith ($) (zipWith f s1 s2) s3) s4

    ------------------------------------------------------------------------
    -- Sorting
    --
    -- sort and sortBy are implemented by simple deforestations of
    -- 	\ xs -> fromList2 (length xs) . Data.List.sortBy cmp . toList
    -- which does not get deforested automatically, it would appear.
    --
    -- Unstable sorting is performed by a heap sort implementation based on
    -- pairing heaps.  Because the internal structure of sequences is quite
    -- varied, it is difficult to get blocks of elements of roughly the same
    -- length, which would improve merge sort performance.  Pairing heaps,
    -- on the other hand, are relatively resistant to the effects of merging
    -- heaps of wildly different sizes, as guaranteed by its amortized
    -- constant-time merge operation.  Moreover, extensive use of SpecConstr
    -- transformations can be done on pairing heaps, especially when we're
    -- only constructing them to immediately be unrolled.
    --
    -- On purely random sequences of length 50000, with no RTS options,
    -- I get the following statistics, in which heapsort is about 42.5%
    -- faster:  (all comparisons done with -O2)
    --
    -- Times (ms)            min      mean    +/-sd    median    max
    -- to/from list:       103.802  108.572    7.487  106.436  143.339
    -- unstable heapsort:   60.686   62.968    4.275   61.187   79.151
    --
    -- Heapsort, it would seem, is less of a memory hog than Data.List.sortBy.
    -- The gap is narrowed when more memory is available, but heapsort still
    -- wins, 15% faster, with +RTS -H128m:
    --
    -- Times (ms)            min    mean    +/-sd  median    max
    -- to/from list:       42.692  45.074   2.596  44.600  56.601
    -- unstable heapsort:  37.100  38.344   3.043  37.715  55.526
    --
    -- In addition, on strictly increasing sequences the gap is even wider
    -- than normal; heapsort is 68.5% faster with no RTS options:
    -- Times (ms)            min    mean    +/-sd  median    max
    -- to/from list:       52.236  53.574   1.987  53.034  62.098
    -- unstable heapsort:  16.433  16.919   0.931  16.681  21.622
    --
    -- This may be attributed to the elegant nature of the pairing heap.
    --
    -- wasserman.louis@gmail.com, 7/20/09
    ------------------------------------------------------------------------

    -- | /O(n log n)/.  'sort' sorts the specified 'Seq' by the natural
    -- ordering of its elements.  The sort is stable.
    -- If stability is not required, 'unstableSort' can be considerably
    -- faster, and in particular uses less memory.
    sort :: Ord a => Seq a -> Seq a
    sort = sortBy compare

    -- | /O(n log n)/.  'sortBy' sorts the specified 'Seq' according to the
    -- specified comparator.  The sort is stable.
    -- If stability is not required, 'unstableSortBy' can be considerably
    -- faster, and in particular uses less memory.
    sortBy :: (a -> a -> Ordering) -> Seq a -> Seq a
    sortBy cmp xs = fromList2 (length xs) (Data.List.sortBy cmp (toList xs))

    -- | /O(n log n)/.  'unstableSort' sorts the specified 'Seq' by
    -- the natural ordering of its elements, but the sort is not stable.
    -- This algorithm is frequently faster and uses less memory than 'sort',
    -- and performs extremely well -- frequently twice as fast as 'sort' --
    -- when the sequence is already nearly sorted.
    unstableSort :: Ord a => Seq a -> Seq a
    unstableSort = unstableSortBy compare

    -- | /O(n log n)/.  A generalization of 'unstableSort', 'unstableSortBy'
    -- takes an arbitrary comparator and sorts the specified sequence.
    -- The sort is not stable.  This algorithm is frequently faster and
    -- uses less memory than 'sortBy', and performs extremely well --
    -- frequently twice as fast as 'sortBy' -- when the sequence is already
    -- nearly sorted.
    unstableSortBy :: (a -> a -> Ordering) -> Seq a -> Seq a
    unstableSortBy cmp (Seq xs) =
    	fromList2 (size xs) $ maybe [] (unrollPQ cmp) $
    		toPQ cmp (\ (Elem x) -> PQueue x Nil) xs

    -- | fromList2, given a list and its length, constructs a completely
    -- balanced Seq whose elements are that list using the applicativeTree
    -- generalization.
    fromList2 :: Int -> [a] -> Seq a
    fromList2 n = execState (replicateA n (State ht))
      where
        ht (x:xs) = (xs, x)
        ht []     = error "fromList2: short list"

    -- | A 'PQueue' is a simple pairing heap.
    data PQueue e = PQueue e (PQL e)
    data PQL e = Nil | {-# UNPACK #-} !(PQueue e) :& PQL e

    infixr 8 :&

    #if TESTING

    instance Functor PQueue where
    	fmap f (PQueue x ts) = PQueue (f x) (fmap f ts)

    instance Functor PQL where
    	fmap f (q :& qs) = fmap f q :& fmap f qs
    	fmap _ Nil = Nil

    instance Show e => Show (PQueue e) where
    	show = unlines . draw . fmap show

    -- borrowed wholesale from Data.Tree, as Data.Tree actually depends
    -- on Data.Sequence
    draw :: PQueue String -> [String]
    draw (PQueue x ts0) = x : drawSubTrees ts0
      where drawSubTrees Nil = []
    	drawSubTrees (t :& Nil) =
    		"|" : shift "`- " "   " (draw t)
    	drawSubTrees (t :& ts) =
    		"|" : shift "+- " "|  " (draw t) ++ drawSubTrees ts

    	shift first other = Data.List.zipWith (++) (first : repeat other)
    #endif

    -- | 'unrollPQ', given a comparator function, unrolls a 'PQueue' into
    -- a sorted list.
    unrollPQ :: (e -> e -> Ordering) -> PQueue e -> [e]
    unrollPQ cmp = unrollPQ'
      where
    	{-# INLINE unrollPQ' #-}
    	unrollPQ' (PQueue x ts) = x:mergePQs0 ts
    	(<>) = mergePQ cmp
    	mergePQs0 Nil = []
    	mergePQs0 (t :& Nil) = unrollPQ' t
    	mergePQs0 (t1 :& t2 :& ts) = mergePQs (t1 <> t2) ts
    	mergePQs t ts = t `seq` case ts of
    		Nil		-> unrollPQ' t
    		t1 :& Nil	-> unrollPQ' (t <> t1)
    		t1 :& t2 :& ts'	-> mergePQs (t <> (t1 <> t2)) ts'

    -- | 'toPQ', given an ordering function and a mechanism for queueifying
    -- elements, converts a 'FingerTree' to a 'PQueue'.
    toPQ :: (e -> e -> Ordering) -> (a -> PQueue e) -> FingerTree a -> Maybe (PQueue e)
    toPQ _ _ Empty = Nothing
    toPQ _ f (Single x) = Just (f x)
    toPQ cmp f (Deep _ pr m sf) = Just (maybe (pr' <> sf') ((pr' <> sf') <>) (toPQ cmp fNode m))
      where
    	fDigit digit = case fmap f digit of
    		One a		-> a
    		Two a b		-> a <> b
    		Three a b c	-> a <> b <> c
    		Four a b c d	-> (a <> b) <> (c <> d)
    	(<>) = mergePQ cmp
    	fNode = fDigit . nodeToDigit
    	pr' = fDigit pr
    	sf' = fDigit sf

    -- | 'mergePQ' merges two 'PQueue's.
    mergePQ :: (a -> a -> Ordering) -> PQueue a -> PQueue a -> PQueue a
    mergePQ cmp q1@(PQueue x1 ts1) q2@(PQueue x2 ts2)
      | cmp x1 x2 == GT	= PQueue x2 (q1 :& ts2)
      | otherwise		= PQueue x1 (q2 :& ts1)

    #if TESTING

    ------------------------------------------------------------------------
    -- QuickCheck
    ------------------------------------------------------------------------

    instance Arbitrary a => Arbitrary (Seq a) where
    	arbitrary = liftM Seq arbitrary
    	shrink (Seq x) = map Seq (shrink x)

    instance Arbitrary a => Arbitrary (Elem a) where
    	arbitrary = liftM Elem arbitrary

    instance (Arbitrary a, Sized a) => Arbitrary (FingerTree a) where
    	arbitrary = sized arb
    	  where arb :: (Arbitrary a, Sized a) => Int -> Gen (FingerTree a)
    		arb 0 = return Empty
    		arb 1 = liftM Single arbitrary
    		arb n = liftM3 deep arbitrary (arb (n `div` 2)) arbitrary

    	shrink (Deep _ (One a) Empty (One b)) = [Single a, Single b]
    	shrink (Deep _ pr m sf) =
    		[deep pr' m sf | pr' <- shrink pr] ++
    		[deep pr m' sf | m' <- shrink m] ++
    		[deep pr m sf' | sf' <- shrink sf]
    	shrink (Single x) = map Single (shrink x)
    	shrink Empty = []

    instance (Arbitrary a, Sized a) => Arbitrary (Node a) where
    	arbitrary = oneof [
    		liftM2 node2 arbitrary arbitrary,
    		liftM3 node3 arbitrary arbitrary arbitrary]

    	shrink (Node2 _ a b) =
    		[node2 a' b | a' <- shrink a] ++
    		[node2 a b' | b' <- shrink b]
    	shrink (Node3 _ a b c) =
    		[node2 a b, node2 a c, node2 b c] ++
    		[node3 a' b c | a' <- shrink a] ++
    		[node3 a b' c | b' <- shrink b] ++
    		[node3 a b c' | c' <- shrink c]

    instance Arbitrary a => Arbitrary (Digit a) where
    	arbitrary = oneof [
    		liftM One arbitrary,
    		liftM2 Two arbitrary arbitrary,
    		liftM3 Three arbitrary arbitrary arbitrary,
    		liftM4 Four arbitrary arbitrary arbitrary arbitrary]

    	shrink (One a) = map One (shrink a)
    	shrink (Two a b) = [One a, One b]
    	shrink (Three a b c) = [Two a b, Two a c, Two b c]
    	shrink (Four a b c d) = [Three a b c, Three a b d, Three a c d, Three b c d]

    ------------------------------------------------------------------------
    -- Valid trees
    ------------------------------------------------------------------------

    class Valid a where
    	valid :: a -> Bool

    instance Valid (Elem a) where
    	valid _ = True

    instance Valid (Seq a) where
    	valid (Seq xs) = valid xs

    instance (Sized a, Valid a) => Valid (FingerTree a) where
    	valid Empty = True
    	valid (Single x) = valid x
    	valid (Deep s pr m sf) =
    		s == size pr + size m + size sf && valid pr && valid m && valid sf

    instance (Sized a, Valid a) => Valid (Node a) where
    	valid node = size node == sum (fmap size node) && all valid node

    instance Valid a => Valid (Digit a) where
    	valid = all valid

    #endif