    {-# OPTIONS_GHC -cpp -fglasgow-exts -fno-warn-orphans #-}
    -- |
    -- Module      : Data.ByteString.Fusion
    -- License     : BSD-style
    -- Maintainer  : dons@cse.unsw.edu.au
    -- Stability   : experimental
    -- Portability : portable
    --
    -- Functional array fusion for ByteStrings.
    --
    -- Originally based on code from the Data Parallel Haskell project,
    --      <http://www.cse.unsw.edu.au/~chak/project/dph>
    --

    -- #hide
    module Data.ByteString.Fusion (

        -- * Fusion utilities
        loopU, loopL, fuseEFL,
        NoAcc(NoAcc), loopArr, loopAcc, loopSndAcc, unSP,
        mapEFL, filterEFL, foldEFL, foldEFL', scanEFL, mapAccumEFL, mapIndexEFL,

        -- ** Alternative Fusion stuff
        -- | This replaces 'loopU' with 'loopUp'
        -- and adds several further special cases of loops.
        loopUp, loopDown, loopNoAcc, loopMap, loopFilter,
        loopWrapper, sequenceLoops,
        doUpLoop, doDownLoop, doNoAccLoop, doMapLoop, doFilterLoop,

        -- | These are the special fusion cases for combining each loop form perfectly.
        fuseAccAccEFL, fuseAccNoAccEFL, fuseNoAccAccEFL, fuseNoAccNoAccEFL,
        fuseMapAccEFL, fuseAccMapEFL, fuseMapNoAccEFL, fuseNoAccMapEFL,
        fuseMapMapEFL, fuseAccFilterEFL, fuseFilterAccEFL, fuseNoAccFilterEFL,
        fuseFilterNoAccEFL, fuseFilterFilterEFL, fuseMapFilterEFL, fuseFilterMapEFL,

        -- * Strict pairs and sums
        PairS(..), MaybeS(..)

      ) where

    import Data.ByteString.Internal
    import qualified Data.ByteString.Lazy.Internal as L

    import Foreign.ForeignPtr
    import Foreign.Ptr
    import Foreign.Storable         (Storable(..))

    import Data.Word                (Word8)
    import System.IO.Unsafe         (unsafePerformIO)

    -- -----------------------------------------------------------------------------
    --
    -- Useful macros, until we have bang patterns
    --

    #define STRICT1(f) f a | a `seq` False = undefined
    #define STRICT2(f) f a b | a `seq` b `seq` False = undefined
    #define STRICT3(f) f a b c | a `seq` b `seq` c `seq` False = undefined
    #define STRICT4(f) f a b c d | a `seq` b `seq` c `seq` d `seq` False = undefined
    #define STRICT5(f) f a b c d e | a `seq` b `seq` c `seq` d `seq` e `seq` False = undefined

    infixl 2 :*:

    -- |Strict pair
    data PairS a b = !a :*: !b deriving (Eq,Ord,Show)

    -- |Strict Maybe
    data MaybeS a = NothingS | JustS !a deriving (Eq,Ord,Show)

    -- |Data type for accumulators which can be ignored. The rewrite rules rely on
    -- the fact that no bottoms of this type are ever constructed; hence, we can
    -- assume @(_ :: NoAcc) `seq` x = x@.
    --
    data NoAcc = NoAcc

    -- |Type of loop functions
    type AccEFL acc = acc -> Word8 -> (PairS acc (MaybeS Word8))
    type NoAccEFL   =        Word8 ->             MaybeS Word8
    type MapEFL     =        Word8 ->                    Word8
    type FilterEFL  =        Word8 ->             Bool

    infixr 9 `fuseEFL`

    -- |Fuse to flat loop functions
    fuseEFL :: AccEFL acc1 -> AccEFL acc2 -> AccEFL (PairS acc1 acc2)
    fuseEFL f g (acc1 :*: acc2) e1 =
        case f acc1 e1 of
            acc1' :*: NothingS -> (acc1' :*: acc2) :*: NothingS
            acc1' :*: JustS e2 ->
                case g acc2 e2 of
                    acc2' :*: res -> (acc1' :*: acc2') :*: res
    #if defined(__GLASGOW_HASKELL__)
    {-# INLINE [1] fuseEFL #-}
    #endif

    -- | Special forms of loop arguments
    --
    -- * These are common special cases for the three function arguments of gen
    --   and loop; we give them special names to make it easier to trigger RULES
    --   applying in the special cases represented by these arguments.  The
    --   "INLINE [1]" makes sure that these functions are only inlined in the last
    --   two simplifier phases.
    --
    -- * In the case where the accumulator is not needed, it is better to always
    --   explicitly return a value `()', rather than just copy the input to the
    --   output, as the former gives GHC better local information.
    --

    -- | Element function expressing a mapping only
    #if !defined(LOOPNOACC_FUSION)
    mapEFL :: (Word8 -> Word8) -> AccEFL NoAcc
    mapEFL f = \_ e -> (NoAcc :*: (JustS $ f e))
    #else
    mapEFL :: (Word8 -> Word8) -> NoAccEFL
    mapEFL f = \e -> JustS (f e)
    #endif
    #if defined(__GLASGOW_HASKELL__)
    {-# INLINE [1] mapEFL #-}
    #endif

    -- | Element function implementing a filter function only
    #if !defined(LOOPNOACC_FUSION)
    filterEFL :: (Word8 -> Bool) -> AccEFL NoAcc
    filterEFL p = \_ e -> if p e then (NoAcc :*: JustS e) else (NoAcc :*: NothingS)
    #else
    filterEFL :: (Word8 -> Bool) -> NoAccEFL
    filterEFL p = \e -> if p e then JustS e else NothingS
    #endif

    #if defined(__GLASGOW_HASKELL__)
    {-# INLINE [1] filterEFL #-}
    #endif

    -- |Element function expressing a reduction only
    foldEFL :: (acc -> Word8 -> acc) -> AccEFL acc
    foldEFL f = \a e -> (f a e :*: NothingS)
    #if defined(__GLASGOW_HASKELL__)
    {-# INLINE [1] foldEFL #-}
    #endif

    -- | A strict foldEFL.
    foldEFL' :: (acc -> Word8 -> acc) -> AccEFL acc
    foldEFL' f = \a e -> let a' = f a e in a' `seq` (a' :*: NothingS)
    #if defined(__GLASGOW_HASKELL__)
    {-# INLINE [1] foldEFL' #-}
    #endif

    -- | Element function expressing a prefix reduction only
    --
    scanEFL :: (Word8 -> Word8 -> Word8) -> AccEFL Word8
    scanEFL f = \a e -> (f a e :*: JustS a)
    #if defined(__GLASGOW_HASKELL__)
    {-# INLINE [1] scanEFL #-}
    #endif

    -- | Element function implementing a map and fold
    --
    mapAccumEFL :: (acc -> Word8 -> (acc, Word8)) -> AccEFL acc
    mapAccumEFL f = \a e -> case f a e of (a', e') -> (a' :*: JustS e')
    #if defined(__GLASGOW_HASKELL__)
    {-# INLINE [1] mapAccumEFL #-}
    #endif

    -- | Element function implementing a map with index
    --
    mapIndexEFL :: (Int -> Word8 -> Word8) -> AccEFL Int
    mapIndexEFL f = \i e -> let i' = i+1 in i' `seq` (i' :*: JustS (f i e))
    #if defined(__GLASGOW_HASKELL__)
    {-# INLINE [1] mapIndexEFL #-}
    #endif

    -- | Projection functions that are fusion friendly (as in, we determine when
    -- they are inlined)
    loopArr :: (PairS acc arr) -> arr
    loopArr (_ :*: arr) = arr
    #if defined(__GLASGOW_HASKELL__)
    {-# INLINE [1] loopArr #-}
    #endif

    loopAcc :: (PairS acc arr) -> acc
    loopAcc (acc :*: _) = acc
    #if defined(__GLASGOW_HASKELL__)
    {-# INLINE [1] loopAcc #-}
    #endif

    loopSndAcc :: (PairS (PairS acc1 acc2) arr) -> (PairS acc2 arr)
    loopSndAcc ((_ :*: acc) :*: arr) = (acc :*: arr)
    #if defined(__GLASGOW_HASKELL__)
    {-# INLINE [1] loopSndAcc #-}
    #endif

    unSP :: (PairS acc arr) -> (acc, arr)
    unSP (acc :*: arr) = (acc, arr)
    #if defined(__GLASGOW_HASKELL__)
    {-# INLINE [1] unSP #-}
    #endif

    ------------------------------------------------------------------------
    --
    -- Loop combinator and fusion rules for flat arrays
    -- |Iteration over over ByteStrings

    -- | Iteration over over ByteStrings
    loopU :: AccEFL acc                 -- ^ mapping & folding, once per elem
          -> acc                        -- ^ initial acc value
          -> ByteString                 -- ^ input ByteString
          -> (PairS acc ByteString)

    loopU f start (PS z s i) = unsafePerformIO $ withForeignPtr z $ \a -> do
        (ps, acc) <- createAndTrim' i $ \p -> do
          (acc' :*: i') <- go (a `plusPtr` s) p start
          return (0, i', acc')
        return (acc :*: ps)

      where
        go p ma = trans 0 0
            where
                STRICT3(trans)
                trans a_off ma_off acc
                    | a_off >= i = return (acc :*: ma_off)
                    | otherwise  = do
                        x <- peekByteOff p a_off
                        let (acc' :*: oe) = f acc x
                        ma_off' <- case oe of
                            NothingS -> return ma_off
                            JustS e  -> do pokeByteOff ma ma_off e
                                           return $ ma_off + 1
                        trans (a_off+1) ma_off' acc'

    #if defined(__GLASGOW_HASKELL__)
    {-# INLINE [1] loopU #-}
    #endif

    {-# RULES

    "FPS loop/loop fusion!" forall em1 em2 start1 start2 arr.
      loopU em2 start2 (loopArr (loopU em1 start1 arr)) =
        loopSndAcc (loopU (em1 `fuseEFL` em2) (start1 :*: start2) arr)

      #-}

    --
    -- Functional list/array fusion for lazy ByteStrings.
    --
    loopL :: AccEFL acc          -- ^ mapping & folding, once per elem
          -> acc                 -- ^ initial acc value
          -> L.ByteString        -- ^ input ByteString
          -> PairS acc L.ByteString
    loopL f = loop
      where loop s L.Empty     = (s :*: L.Empty)
            loop s (L.Chunk x xs)
              | l == 0    = (s'' :*: ys)
              | otherwise = (s'' :*: L.Chunk y ys)
              where (s'  :*: y@(PS _ _ l)) = loopU f s x -- avoid circular dep on S.null
                    (s'' :*: ys)           = loop s' xs

    #if defined(__GLASGOW_HASKELL__)
    {-# INLINE [1] loopL #-}
    #endif

    {-# RULES

    "FPS lazy loop/loop fusion!" forall em1 em2 start1 start2 arr.
      loopL em2 start2 (loopArr (loopL em1 start1 arr)) =
        loopSndAcc (loopL (em1 `fuseEFL` em2) (start1 :*: start2) arr)

      #-}

    {-

    Alternate experimental formulation of loopU which partitions it into
    an allocating wrapper and an imperitive array-mutating loop.

    The point in doing this split is that we might be able to fuse multiple
    loops into a single wrapper. This would save reallocating another buffer.
    It should also give better cache locality by reusing the buffer.

    Note that this stuff needs ghc-6.5 from May 26 or later for the RULES to
    really work reliably.

    -}

    loopUp :: AccEFL acc -> acc -> ByteString -> PairS acc ByteString
    loopUp f a arr = loopWrapper (doUpLoop f a) arr
    {-# INLINE loopUp #-}

    loopDown :: AccEFL acc -> acc -> ByteString -> PairS acc ByteString
    loopDown f a arr = loopWrapper (doDownLoop f a) arr
    {-# INLINE loopDown #-}

    loopNoAcc :: NoAccEFL -> ByteString -> PairS NoAcc ByteString
    loopNoAcc f arr = loopWrapper (doNoAccLoop f NoAcc) arr
    {-# INLINE loopNoAcc #-}

    loopMap :: MapEFL -> ByteString -> PairS NoAcc ByteString
    loopMap f arr = loopWrapper (doMapLoop f NoAcc) arr
    {-# INLINE loopMap #-}

    loopFilter :: FilterEFL -> ByteString -> PairS NoAcc ByteString
    loopFilter f arr = loopWrapper (doFilterLoop f NoAcc) arr
    {-# INLINE loopFilter #-}

    -- The type of imperitive loops that fill in a destination array by
    -- reading a source array. They may not fill in the whole of the dest
    -- array if the loop is behaving as a filter, this is why we return
    -- the length that was filled in. The loop may also accumulate some
    -- value as it loops over the source array.
    --
    type ImperativeLoop acc =
        Ptr Word8          -- pointer to the start of the source byte array
     -> Ptr Word8          -- pointer to ther start of the destination byte array
     -> Int                -- length of the source byte array
     -> IO (PairS (PairS acc Int) Int) -- result and offset, length of dest that was filled

    loopWrapper :: ImperativeLoop acc -> ByteString -> PairS acc ByteString
    loopWrapper body (PS srcFPtr srcOffset srcLen) = unsafePerformIO $
        withForeignPtr srcFPtr $ \srcPtr -> do
        (ps, acc) <- createAndTrim' srcLen $ \destPtr -> do
            (acc :*: destOffset :*: destLen) <-
              body (srcPtr `plusPtr` srcOffset) destPtr srcLen
            return (destOffset, destLen, acc)
        return (acc :*: ps)

    doUpLoop :: AccEFL acc -> acc -> ImperativeLoop acc
    doUpLoop f acc0 src dest len = loop 0 0 acc0
      where STRICT3(loop)
            loop src_off dest_off acc
                | src_off >= len = return (acc :*: 0 :*: dest_off)
                | otherwise      = do
                    x <- peekByteOff src src_off
                    case f acc x of
                      (acc' :*: NothingS) -> loop (src_off+1) dest_off acc'
                      (acc' :*: JustS x') -> pokeByteOff dest dest_off x'
                                          >> loop (src_off+1) (dest_off+1) acc'

    doDownLoop :: AccEFL acc -> acc -> ImperativeLoop acc
    doDownLoop f acc0 src dest len = loop (len-1) (len-1) acc0
      where STRICT3(loop)
            loop src_off dest_off acc
                | src_off < 0 = return (acc :*: dest_off + 1 :*: len - (dest_off + 1))
                | otherwise   = do
                    x <- peekByteOff src src_off
                    case f acc x of
                      (acc' :*: NothingS) -> loop (src_off-1) dest_off acc'
                      (acc' :*: JustS x') -> pokeByteOff dest dest_off x'
                                          >> loop (src_off-1) (dest_off-1) acc'

    doNoAccLoop :: NoAccEFL -> noAcc -> ImperativeLoop noAcc
    doNoAccLoop f noAcc src dest len = loop 0 0
      where STRICT2(loop)
            loop src_off dest_off
                | src_off >= len = return (noAcc :*: 0 :*: dest_off)
                | otherwise      = do
                    x <- peekByteOff src src_off
                    case f x of
                      NothingS -> loop (src_off+1) dest_off
                      JustS x' -> pokeByteOff dest dest_off x'
                               >> loop (src_off+1) (dest_off+1)

    doMapLoop :: MapEFL -> noAcc -> ImperativeLoop noAcc
    doMapLoop f noAcc src dest len = loop 0
      where STRICT1(loop)
            loop n
                | n >= len = return (noAcc :*: 0 :*: len)
                | otherwise      = do
                    x <- peekByteOff src n
                    pokeByteOff dest n (f x)
                    loop (n+1) -- offset always the same, only pass 1 arg

    doFilterLoop :: FilterEFL -> noAcc -> ImperativeLoop noAcc
    doFilterLoop f noAcc src dest len = loop 0 0
      where STRICT2(loop)
            loop src_off dest_off
                | src_off >= len = return (noAcc :*: 0 :*: dest_off)
                | otherwise      = do
                    x <- peekByteOff src src_off
                    if f x
                      then pokeByteOff dest dest_off x
                        >> loop (src_off+1) (dest_off+1)
                      else loop (src_off+1) dest_off

    -- run two loops in sequence,
    -- think of it as: loop1 >> loop2
    sequenceLoops :: ImperativeLoop acc1
                  -> ImperativeLoop acc2
                  -> ImperativeLoop (PairS acc1 acc2)
    sequenceLoops loop1 loop2 src dest len0 = do
      (acc1 :*: off1 :*: len1) <- loop1 src dest len0
      (acc2 :*: off2 :*: len2) <-
        let src'  = dest `plusPtr` off1
            dest' = src' -- note that we are using dest == src
                         -- for the second loop as we are
                         -- mutating the dest array in-place!
         in loop2 src' dest' len1
      return ((acc1  :*: acc2) :*: off1 + off2 :*: len2)

      -- TODO: prove that this is associative! (I think it is)
      -- since we can't be sure how the RULES will combine loops.

    #if defined(__GLASGOW_HASKELL__)

    {-# INLINE [1] doUpLoop             #-}
    {-# INLINE [1] doDownLoop           #-}
    {-# INLINE [1] doNoAccLoop          #-}
    {-# INLINE [1] doMapLoop            #-}
    {-# INLINE [1] doFilterLoop         #-}

    {-# INLINE [1] loopWrapper          #-}
    {-# INLINE [1] sequenceLoops        #-}

    {-# INLINE [1] fuseAccAccEFL        #-}
    {-# INLINE [1] fuseAccNoAccEFL      #-}
    {-# INLINE [1] fuseNoAccAccEFL      #-}
    {-# INLINE [1] fuseNoAccNoAccEFL    #-}
    {-# INLINE [1] fuseMapAccEFL        #-}
    {-# INLINE [1] fuseAccMapEFL        #-}
    {-# INLINE [1] fuseMapNoAccEFL      #-}
    {-# INLINE [1] fuseNoAccMapEFL      #-}
    {-# INLINE [1] fuseMapMapEFL        #-}
    {-# INLINE [1] fuseAccFilterEFL     #-}
    {-# INLINE [1] fuseFilterAccEFL     #-}
    {-# INLINE [1] fuseNoAccFilterEFL   #-}
    {-# INLINE [1] fuseFilterNoAccEFL   #-}
    {-# INLINE [1] fuseFilterFilterEFL  #-}
    {-# INLINE [1] fuseMapFilterEFL     #-}
    {-# INLINE [1] fuseFilterMapEFL     #-}

    #endif

    {-# RULES

    "FPS loopArr/loopSndAcc" forall x.
      loopArr (loopSndAcc x) = loopArr x

    "FPS seq/NoAcc" forall (u::NoAcc) e.
      u `seq` e = e

    "FPS loop/loop wrapper elimination" forall loop1 loop2 arr.
      loopWrapper loop2 (loopArr (loopWrapper loop1 arr)) =
        loopSndAcc (loopWrapper (sequenceLoops loop1 loop2) arr)

    --
    -- n.b in the following, when reading n/m fusion, recall sequenceLoops
    -- is monadic, so its really n >> m fusion (i.e. m.n), not n . m fusion.
    --

    "FPS up/up loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doUpLoop f1 acc1) (doUpLoop f2 acc2) =
        doUpLoop (f1 `fuseAccAccEFL` f2) (acc1 :*: acc2)

    "FPS map/map loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doMapLoop f1 acc1) (doMapLoop f2 acc2) =
        doMapLoop (f1 `fuseMapMapEFL` f2) (acc1 :*: acc2)

    "FPS filter/filter loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doFilterLoop f1 acc1) (doFilterLoop f2 acc2) =
        doFilterLoop (f1 `fuseFilterFilterEFL` f2) (acc1 :*: acc2)

    "FPS map/filter loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doMapLoop f1 acc1) (doFilterLoop f2 acc2) =
        doNoAccLoop (f1 `fuseMapFilterEFL` f2) (acc1 :*: acc2)

    "FPS filter/map loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doFilterLoop f1 acc1) (doMapLoop f2 acc2) =
        doNoAccLoop (f1 `fuseFilterMapEFL` f2) (acc1 :*: acc2)

    "FPS map/up loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doMapLoop f1 acc1) (doUpLoop f2 acc2) =
        doUpLoop (f1 `fuseMapAccEFL` f2) (acc1 :*: acc2)

    "FPS up/map loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doUpLoop f1 acc1) (doMapLoop f2 acc2) =
        doUpLoop (f1 `fuseAccMapEFL` f2) (acc1 :*: acc2)

    "FPS filter/up loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doFilterLoop f1 acc1) (doUpLoop f2 acc2) =
        doUpLoop (f1 `fuseFilterAccEFL` f2) (acc1 :*: acc2)

    "FPS up/filter loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doUpLoop f1 acc1) (doFilterLoop f2 acc2) =
        doUpLoop (f1 `fuseAccFilterEFL` f2) (acc1 :*: acc2)

    "FPS down/down loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doDownLoop f1 acc1) (doDownLoop f2 acc2) =
        doDownLoop (f1 `fuseAccAccEFL` f2) (acc1 :*: acc2)

    "FPS map/down fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doMapLoop f1 acc1) (doDownLoop f2 acc2) =
        doDownLoop (f1 `fuseMapAccEFL` f2) (acc1 :*: acc2)

    "FPS down/map loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doDownLoop f1 acc1) (doMapLoop f2 acc2) =
        doDownLoop (f1 `fuseAccMapEFL` f2) (acc1 :*: acc2)

    "FPS filter/down fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doFilterLoop f1 acc1) (doDownLoop f2 acc2) =
        doDownLoop (f1 `fuseFilterAccEFL` f2) (acc1 :*: acc2)

    "FPS down/filter loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doDownLoop f1 acc1) (doFilterLoop f2 acc2) =
        doDownLoop (f1 `fuseAccFilterEFL` f2) (acc1 :*: acc2)

    "FPS noAcc/noAcc loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doNoAccLoop f1 acc1) (doNoAccLoop f2 acc2) =
        doNoAccLoop (f1 `fuseNoAccNoAccEFL` f2) (acc1 :*: acc2)

    "FPS noAcc/up loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doNoAccLoop f1 acc1) (doUpLoop f2 acc2) =
        doUpLoop (f1 `fuseNoAccAccEFL` f2) (acc1 :*: acc2)

    "FPS up/noAcc loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doUpLoop f1 acc1) (doNoAccLoop f2 acc2) =
        doUpLoop (f1 `fuseAccNoAccEFL` f2) (acc1 :*: acc2)

    "FPS map/noAcc loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doMapLoop f1 acc1) (doNoAccLoop f2 acc2) =
        doNoAccLoop (f1 `fuseMapNoAccEFL` f2) (acc1 :*: acc2)

    "FPS noAcc/map loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doNoAccLoop f1 acc1) (doMapLoop f2 acc2) =
        doNoAccLoop (f1 `fuseNoAccMapEFL` f2) (acc1 :*: acc2)

    "FPS filter/noAcc loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doFilterLoop f1 acc1) (doNoAccLoop f2 acc2) =
        doNoAccLoop (f1 `fuseFilterNoAccEFL` f2) (acc1 :*: acc2)

    "FPS noAcc/filter loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doNoAccLoop f1 acc1) (doFilterLoop f2 acc2) =
        doNoAccLoop (f1 `fuseNoAccFilterEFL` f2) (acc1 :*: acc2)

    "FPS noAcc/down loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doNoAccLoop f1 acc1) (doDownLoop f2 acc2) =
        doDownLoop (f1 `fuseNoAccAccEFL` f2) (acc1 :*: acc2)

    "FPS down/noAcc loop fusion" forall f1 f2 acc1 acc2.
      sequenceLoops (doDownLoop f1 acc1) (doNoAccLoop f2 acc2) =
        doDownLoop (f1 `fuseAccNoAccEFL` f2) (acc1 :*: acc2)

      #-}

    {-

    up      = up loop
    down    = down loop
    map     = map special case
    filter  = filter special case
    noAcc   = noAcc undirectional loop (unused)

    heirarchy:
      up     down
       ^     ^
        \   /
        noAcc
         ^ ^
        /   \
     map     filter

    each is a special case of the things above

    so we get rules that combine things on the same level
    and rules that combine things on different levels
    to get something on the higher level

    so all the cases:
    up/up         --> up     fuseAccAccEFL
    down/down     --> down   fuseAccAccEFL
    noAcc/noAcc   --> noAcc  fuseNoAccNoAccEFL

    noAcc/up      --> up     fuseNoAccAccEFL
    up/noAcc      --> up     fuseAccNoAccEFL
    noAcc/down    --> down   fuseNoAccAccEFL
    down/noAcc    --> down   fuseAccNoAccEFL

    and if we do the map, filter special cases then it adds a load more:

    map/map       --> map    fuseMapMapEFL
    filter/filter --> filter fuseFilterFilterEFL

    map/filter    --> noAcc  fuseMapFilterEFL
    filter/map    --> noAcc  fuseFilterMapEFL

    map/noAcc     --> noAcc  fuseMapNoAccEFL
    noAcc/map     --> noAcc  fuseNoAccMapEFL

    map/up        --> up     fuseMapAccEFL
    up/map        --> up     fuseAccMapEFL

    map/down      --> down   fuseMapAccEFL
    down/map      --> down   fuseAccMapEFL

    filter/noAcc  --> noAcc  fuseNoAccFilterEFL
    noAcc/filter  --> noAcc  fuseFilterNoAccEFL

    filter/up     --> up     fuseFilterAccEFL
    up/filter     --> up     fuseAccFilterEFL

    filter/down   --> down   fuseFilterAccEFL
    down/filter   --> down   fuseAccFilterEFL
    -}

    fuseAccAccEFL :: AccEFL acc1 -> AccEFL acc2 -> AccEFL (PairS acc1 acc2)
    fuseAccAccEFL f g (acc1 :*: acc2) e1 =
        case f acc1 e1 of
            acc1' :*: NothingS -> (acc1' :*: acc2) :*: NothingS
            acc1' :*: JustS e2 ->
                case g acc2 e2 of
                    acc2' :*: res -> (acc1' :*: acc2') :*: res

    fuseAccNoAccEFL :: AccEFL acc -> NoAccEFL -> AccEFL (PairS acc noAcc)
    fuseAccNoAccEFL f g (acc :*: noAcc) e1 =
        case f acc e1 of
            acc' :*: NothingS -> (acc' :*: noAcc) :*: NothingS
            acc' :*: JustS e2 -> (acc' :*: noAcc) :*: g e2

    fuseNoAccAccEFL :: NoAccEFL -> AccEFL acc -> AccEFL (PairS noAcc acc)
    fuseNoAccAccEFL f g (noAcc :*: acc) e1 =
        case f e1 of
            NothingS -> (noAcc :*: acc) :*: NothingS
            JustS e2 ->
                case g acc e2 of
                    acc' :*: res -> (noAcc :*: acc') :*: res

    fuseNoAccNoAccEFL :: NoAccEFL -> NoAccEFL -> NoAccEFL
    fuseNoAccNoAccEFL f g e1 =
        case f e1 of
            NothingS -> NothingS
            JustS e2 -> g e2

    fuseMapAccEFL :: MapEFL -> AccEFL acc -> AccEFL (PairS noAcc acc)
    fuseMapAccEFL f g (noAcc :*: acc) e1 =
        case g acc (f e1) of
            (acc' :*: res) -> (noAcc :*: acc') :*: res

    fuseAccMapEFL :: AccEFL acc -> MapEFL -> AccEFL (PairS acc noAcc)
    fuseAccMapEFL f g (acc :*: noAcc) e1 =
        case f acc e1 of
            (acc' :*: NothingS) -> (acc' :*: noAcc) :*: NothingS
            (acc' :*: JustS e2) -> (acc' :*: noAcc) :*: JustS (g e2)

    fuseMapMapEFL :: MapEFL -> MapEFL -> MapEFL
    fuseMapMapEFL   f g e1 = g (f e1)     -- n.b. perfect fusion

    fuseMapNoAccEFL :: MapEFL -> NoAccEFL -> NoAccEFL
    fuseMapNoAccEFL f g e1 = g (f e1)

    fuseNoAccMapEFL :: NoAccEFL -> MapEFL -> NoAccEFL
    fuseNoAccMapEFL f g e1 =
        case f e1 of
            NothingS -> NothingS
            JustS e2 -> JustS (g e2)

    fuseAccFilterEFL :: AccEFL acc -> FilterEFL -> AccEFL (PairS acc noAcc)
    fuseAccFilterEFL f g (acc :*: noAcc) e1 =
        case f acc e1 of
            acc' :*: NothingS -> (acc' :*: noAcc) :*: NothingS
            acc' :*: JustS e2 ->
                case g e2 of
                    False -> (acc' :*: noAcc) :*: NothingS
                    True  -> (acc' :*: noAcc) :*: JustS e2

    fuseFilterAccEFL :: FilterEFL -> AccEFL acc -> AccEFL (PairS noAcc acc)
    fuseFilterAccEFL f g (noAcc :*: acc) e1 =
        case f e1 of
            False -> (noAcc :*: acc) :*: NothingS
            True  ->
                case g acc e1 of
                    acc' :*: res -> (noAcc :*: acc') :*: res

    fuseNoAccFilterEFL :: NoAccEFL -> FilterEFL -> NoAccEFL
    fuseNoAccFilterEFL f g e1 =
        case f e1 of
            NothingS -> NothingS
            JustS e2 ->
                case g e2 of
                    False -> NothingS
                    True  -> JustS e2

    fuseFilterNoAccEFL :: FilterEFL -> NoAccEFL -> NoAccEFL
    fuseFilterNoAccEFL f g e1 =
        case f e1 of
            False -> NothingS
            True  -> g e1

    fuseFilterFilterEFL :: FilterEFL -> FilterEFL -> FilterEFL
    fuseFilterFilterEFL f g e1 = f e1 && g e1

    fuseMapFilterEFL :: MapEFL -> FilterEFL -> NoAccEFL
    fuseMapFilterEFL f g e1 =
        case f e1 of
            e2 -> case g e2 of
                False -> NothingS
                True  -> JustS e2

    fuseFilterMapEFL :: FilterEFL -> MapEFL -> NoAccEFL
    fuseFilterMapEFL f g e1 =
        case f e1 of
            False -> NothingS
            True  -> JustS (g e1)