Data/Sequence.hs

    {-# LANGUAGE CPP #-}
    #if __GLASGOW_HASKELL__
    {-# LANGUAGE DeriveDataTypeable, StandaloneDeriving #-}
    #endif
    #if __GLASGOW_HASKELL__ >= 703
    {-# LANGUAGE Trustworthy #-}
    #endif
    -----------------------------------------------------------------------------
    -- |
    -- 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 (
    #if !defined(TESTING)
        Seq,
    #else
        Seq(..), Elem(..), FingerTree(..), Node(..), Digit(..),
    #endif
        -- * 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
        Sized(..),
        deep,
        node2,
        node3,
    #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
    import Control.Applicative (Applicative(..), (<$>), Alternative,
                                WrappedMonad(..), liftA, liftA2, liftA3)
    import qualified Control.Applicative as Applicative (Alternative(..))
    import Control.DeepSeq (NFData(rnf))
    import Control.Monad (MonadPlus(..), ap)
    import Data.Monoid (Monoid(..))
    import Data.Functor (Functor(..))
    import Data.Foldable
    import Data.Traversable
    import Data.Typeable

    #ifdef __GLASGOW_HASKELL__
    import GHC.Exts (build)
    import Text.Read (Lexeme(Ident), lexP, parens, prec,
        readPrec, readListPrec, readListPrecDefault)
    import Data.Data
    #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 NFData a => NFData (Seq a) where
        rnf (Seq xs) = rnf xs

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

    instance Applicative Seq where
        pure = singleton
        fs <*> xs = foldl' add empty fs
          where add ys f = ys >< fmap f xs

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

    instance Alternative Seq where
        empty = empty
        (<|>) = (><)

    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

    instance NFData a => NFData (FingerTree a) where
        rnf (Empty) = ()
        rnf (Single x) = rnf x
        rnf (Deep _ pr m sf) = rnf pr `seq` rnf m `seq` rnf sf

    {-# INLINE deep #-}
    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
        {-# INLINE fmap #-}
        fmap f (One a) = One (f a)
        fmap f (Two a b) = Two (f a) (f b)
        fmap f (Three a b c) = Three (f a) (f b) (f c)
        fmap f (Four a b c d) = Four (f a) (f b) (f c) (f d)

    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 NFData a => NFData (Digit a) where
        rnf (One a) = rnf a
        rnf (Two a b) = rnf a `seq` rnf b
        rnf (Three a b c) = rnf a `seq` rnf b `seq` rnf c
        rnf (Four a b c d) = rnf a `seq` rnf b `seq` rnf c `seq` rnf 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 f (Node2 v a b) = Node2 v (f a) (f b)
        fmap f (Node3 v a b c) = Node3 v (f a) (f b) (f c)

    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 NFData a => NFData (Node a) where
        rnf (Node2 _ a b) = rnf a `seq` rnf b
        rnf (Node3 _ a b c) = rnf a `seq` rnf b `seq` rnf c

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

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

    {-# INLINE node3 #-}
    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 }
    #if TESTING
        deriving Show
    #endif

    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

    instance NFData a => NFData (Elem a) where
        rnf (Elem x) = rnf x

    -------------------------------------------------------
    -- 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 :: Applicative f => Int -> Int -> f a -> f (FingerTree a)
    applicativeTree n mSize m = mSize `seq` case n of
        0 -> pure Empty
        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 = 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' ('fmap' 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
    #if __GLASGOW_HASKELL__
        deriving (Eq, Ord, Show, Read, Data)
    #else
        deriving (Eq, Ord, Show, Read)
    #endif

    INSTANCE_TYPEABLE1(ViewL,viewLTc,"ViewL")

    instance Functor ViewL where
        {-# INLINE fmap #-}
        fmap _ EmptyL       = EmptyL
        fmap f (x :< xs)    = f x :< fmap f xs

    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
    #if __GLASGOW_HASKELL__
        deriving (Eq, Ord, Show, Read, Data)
    #else
        deriving (Eq, Ord, Show, Read)
    #endif

    INSTANCE_TYPEABLE1(ViewR,viewRTc,"ViewR")

    instance Functor ViewR where
        {-# INLINE fmap #-}
        fmap _ EmptyR       = EmptyR
        fmap f (xs :> x)    = fmap f xs :> f x

    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)