content: {-@ LIQUID "--notermination" @-}
  {-@ LIQUID "--pruneunsorted" @-}
  
  {-# 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 (
  
      liquidCanaryFusion,
  
      -- * 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, loopWrapperLE, 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.Ptr
  import Foreign.ForeignPtr
  import Foreign.Storable         (Storable(..))
  
  import Data.Word                (Word8, Word64)
  import System.IO.Unsafe         (unsafePerformIO)
  
  -- LIQUID
  import Prelude hiding (undefined)
  -- import Language.Haskell.Liquid.Prelude  (liquidAssume, liquidAssert)
  
  {-@ qualif PlusOnePos(v: int): 0 <= (v + 1)              @-}
  {-@ qualif LePlusOne(v: int, x: int): v <= (x + 1)       @-}
  {-@ qualif LeDiff(v: int, x: int, y:int): v <= (x - y)   @-}
  {-@ qualif PlenEq(v: Ptr a, x: int): x <= (plen v)       @-}
  {-@ qualif BlenEq(v: int, x:ByteString): v = (bLength x) @-}
  {-@ qualif PSnd(v: a, x:b): v = (psnd x)                 @-}
  
  {-@ data PairS a b <p :: x0:a -> b -> Bool> = (:*:) (x::a) (y::b<p x>)  @-}
  
  {-@ measure pfst :: (PairS a b) -> a
        pfst ((:*:) x y) = x
    @-}
  
  {-@ measure psnd :: (PairS a b) -> b
        psnd ((:*:) x y) = y
    @-}
  
  {-@ measure isJustS    :: (MaybeS a) -> Bool
        isJustS (JustS x)  = true
        isJustS NothingS = false
    @-}
  
  {-@ qualif PlusOne(v:int, x:int): v = x + 1 @-}
  
  {-@ type MaybeSJ a   = {v: MaybeS a | (isJustS v)}                 @-}
  
  {-@ type AccEFLJ acc = acc -> Word8 -> (PairS acc (MaybeSJ Word8)) @-}
  {-@ type NoAccEFLJ   =        Word8 ->             (MaybeSJ Word8) @-}
  
  {- liquidCanaryFusion :: x:Int -> {v: Int | v > x} @-}
  liquidCanaryFusion     :: Int -> Int
  liquidCanaryFusion x   = x - 1
  
  
  -- -----------------------------------------------------------------------------
  --
  -- Useful macros, until we have bang patterns
  --
  
  {-@ lazy undef @-}
  undef :: a -> b
  undef x = undef x
  
  #define STRICT1(f) f a | a `seq` False = undef ()
  #define STRICT2(f) f a b | a `seq` b `seq` False = undef ()
  #define STRICT3(f) f a b c | a `seq` b `seq` c `seq` False = undef ()
  #define STRICT4(f) f a b c d | a `seq` b `seq` c `seq` d `seq` False = undef ()
  #define STRICT5(f) f a b c d e | a `seq` b `seq` c `seq` d `seq` e `seq` False = undef ()
  
  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) -> AccEFLJ Word8 @-}
  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)) -> AccEFLJ acc @-}
  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) -> AccEFLJ Int @-}
  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 :: p:(PairS acc arr) -> {v:arr | v = (psnd p)} @-}
  loopArr :: (PairS acc arr) -> arr
  loopArr (_ :*: arr) = arr
  #if defined(__GLASGOW_HASKELL__)
  {-# INLINE [1] loopArr #-}
  #endif
  
  {-@ loopAcc :: p:(PairS acc arr) -> {v:acc | v = (pfst p)} @-}
  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 :: p:(PairS acc arr) -> ({v:acc | v = (pfst p)}, {v:arr | v = (psnd p)}) @-}
  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 :: AccEFLJ acc -> acc -> b:ByteString -> (PairS acc (ByteStringSZ b)) @-}
  
  loopU f start (PS z s i) = unsafePerformIO $ withForeignPtr z $ \a -> do
      (ps, acc) <- createAndTrimEQ i $ \p -> do
        (acc' :*: i') <- go (a `plusPtr` s) p start
        return (0 :: Int, i', acc')
      return (acc :*: ps)
  
    where
      go p ma = trans i 0 0
          where
              STRICT4(trans)
              {- LIQUID WITNESS -}
              trans (d :: Int) 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 (d-1) (a_off+1) ma_off' acc'
  
  -- a_off = i - d
  {-@ qualif Decr(v:Int, x: Int, y:Int): v = x - y @-}
  
  #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 :: (AccEFLJ acc) -> acc -> b:L.ByteString -> (PairS acc (L.LByteStringSZ b)) @-}
  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 :: AccEFLJ acc -> acc -> b:ByteString -> (PairS acc (ByteStringSZ b)) @-}
  loopUp :: AccEFL acc -> acc -> ByteString -> PairS acc ByteString
  loopUp f a arr = loopWrapper (doUpLoop f a) arr
  {-# INLINE loopUp #-}
  
  {-@ loopDown :: AccEFLJ acc -> acc -> b:ByteString -> (PairS acc (ByteStringSZ b)) @-}
  loopDown :: AccEFL acc -> acc -> ByteString -> PairS acc ByteString
  loopDown f a arr = loopWrapper (doDownLoop f a) arr
  {-# INLINE loopDown #-}
  
  {-@ loopNoAcc :: NoAccEFLJ -> b:ByteString -> (PairS NoAcc (ByteStringSZ b)) @-}
  loopNoAcc :: NoAccEFL -> ByteString -> PairS NoAcc ByteString
  loopNoAcc f arr = loopWrapper (doNoAccLoop f NoAcc) arr
  {-# INLINE loopNoAcc #-}
  
  {-@ loopMap :: MapEFL -> b:ByteString -> (PairS NoAcc (ByteStringSZ b)) @-}
  loopMap :: MapEFL -> ByteString -> PairS NoAcc ByteString
  loopMap f arr = loopWrapper (doMapLoop f NoAcc) arr
  {-# INLINE loopMap #-}
  
  {-@ loopFilter :: FilterEFL -> b:ByteString -> (PairS NoAcc (ByteStringLE b)) @-}
  loopFilter :: FilterEFL -> ByteString -> PairS NoAcc ByteString
  loopFilter f arr = loopWrapperLE (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 TripleSLE a N = PairS <{\z v -> v <= (N - (psnd z))}> (PairS <{\x y -> true}> a Nat) {v:Nat | v <= N} @-}
  {-@ type TripleS   a N = PairS <{\z v -> v <= (N - (psnd z))}> (PairS <{\x y -> true}> a Nat) {v:Nat | v  = N} @-}
  
  
  {-@ type ImperativeLoopLE acc =  s:(PtrV Word8)
                              -> d:(PtrV Word8)
                              -> n:{v: Nat | ((v <= (plen d)) && (v <= (plen s))) }
                              -> IO (TripleSLE acc n)
    @-}
  
  {-@ type ImperativeLoop   acc =  s:(PtrV Word8)
                                -> d:(PtrV Word8)
                                -> n:{v: Nat | ((v <= (plen d)) && (v <= (plen s))) }
                                -> IO (TripleS acc n)
    @-}
  
  
  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
  
  {-@ loopWrapperLE :: ImperativeLoopLE acc -> b:ByteString -> PairS acc (ByteStringLE b) @-}
  loopWrapperLE :: ImperativeLoop acc -> ByteString -> PairS acc ByteString
  loopWrapperLE 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)
  
  -- LIQUID DUPLICATECODE
  {-@ loopWrapper :: ImperativeLoop acc -> b:ByteString -> PairS acc (ByteStringSZ b) @-}
  loopWrapper :: ImperativeLoop acc -> ByteString -> PairS acc ByteString
  loopWrapper body (PS srcFPtr srcOffset srcLen) = unsafePerformIO $
      withForeignPtr srcFPtr $ \srcPtr -> do
      (ps, acc) <- createAndTrimEQ srcLen $ \destPtr -> do
          (acc :*: destOffset :*: destLen) <- body (srcPtr `plusPtr` srcOffset) destPtr srcLen
          return $ (destOffset, id destLen, acc)
      return (acc :*: ps)
  
  
  
  
  {-@ doUpLoop :: AccEFLJ acc -> acc -> ImperativeLoop acc @-}
  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 :: Int) {- LIQUID CAST -} :*: 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 :: AccEFLJ acc -> acc -> ImperativeLoop acc @-}
  doDownLoop :: AccEFL acc -> acc -> ImperativeLoop acc
  doDownLoop f acc0 src dest len = loop (len-1) (len-1) acc0
    where STRICT3(loop)
          loop src_offDOWN dest_offDOWN acc
              | src_offDOWN < 0 = return (acc :*: dest_offDOWN + 1 :*: len - (dest_offDOWN + 1))
              | otherwise   = do
                  x <- peekByteOff src src_offDOWN
                  case f acc x of
                    (acc' :*: NothingS) -> loop (src_offDOWN - 1) dest_offDOWN acc'
                    (acc' :*: JustS x') -> pokeByteOff dest dest_offDOWN x'
                                        >> loop (src_offDOWN - 1) (dest_offDOWN - 1) acc'
  
  {-@ doNoAccLoop :: NoAccEFLJ -> noAcc -> ImperativeLoop noAcc @-}
  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 :: Int) {- LIQUID CAST -} :*: 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 :: MapEFL -> noAcc -> ImperativeLoop noAcc
  doMapLoop f noAcc src dest len = loop 0
    where STRICT1(loop)
          loop n
              | n >= len = return (noAcc :*: (0 :: Int) {- LIQUID CAST -} :*: 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 -> ImperativeLoopLE noAcc @-}
  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 :: Int) {- LIQUID CAST -} :*: 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
  
  -- LIQUID
  -- run two loops in sequence,
  -- think of it as: loop1 >> loop2
  
  {-@ sequenceLoops :: ImperativeLoop acc1 -> ImperativeLoop acc2 -> ImperativeLoop (PairS acc1 acc2) @-}
  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)
  