module Data.List.HT.Private where import Data.List as List (find, transpose, unfoldr, isPrefixOf, findIndices, foldl', ) import Data.Maybe as Maybe (fromMaybe, catMaybes, ) import Data.Maybe.HT (toMaybe, ) import Control.Monad (guard, ) import Data.Tuple.HT (mapPair, mapFst, mapSnd, forcePair, ) import qualified Data.List.Key.Private as Key import qualified Data.List.Match.Private as Match import Prelude hiding (unzip, break, span, ) -- * Improved standard functions {- | This function is lazier than the one suggested in the Haskell 98 report. It is @inits undefined = [] : undefined@, in contrast to @Data.List.inits undefined = undefined@. -} inits :: [a] -> [[a]] inits xt = [] : case xt of [] -> [] x:xs -> map (x:) (inits xs) {- | Suggested implementation in the Haskell 98 report. It is not as lazy as possible. -} inits98 :: [a] -> [[a]] inits98 [] = [[]] inits98 (x:xs) = [[]] ++ map (x:) (inits98 xs) inits98' :: [a] -> [[a]] inits98' = foldr (\x prefixes -> [] : map (x:) prefixes) [[]] {- | This function is lazier than the one suggested in the Haskell 98 report. It is @tails undefined = ([] : undefined) : undefined@, in contrast to @Data.List.tails undefined = undefined@. -} tails :: [a] -> [[a]] tails xt = uncurry (:) $ case xt of [] -> ([],[]) xxs@(_:xs) -> (xxs, tails xs) tails' :: [a] -> [[a]] tails' = fst . breakAfter null . iterate tail tails98 :: [a] -> [[a]] tails98 [] = [[]] tails98 xxs@(_:xs) = xxs : tails98 xs {- | This function compares adjacent elements of a list. If two adjacent elements satisfy a relation then they are put into the same sublist. Example: > groupBy (<) "abcdebcdef" == ["abcde","bcdef"] In contrast to that 'Data.List.groupBy' compares the head of each sublist with each candidate for this sublist. This yields > List.groupBy (<) "abcdebcdef" == ["abcdebcdef"] The second @'b'@ is compared with the leading @'a'@. Thus it is put into the same sublist as @'a'@. The sublists are never empty. Thus the more precise result type would be @[(a,[a])]@. -} groupBy :: (a -> a -> Bool) -> [a] -> [[a]] groupBy = Key.groupBy group :: (Eq a) => [a] -> [[a]] group = groupBy (==) {- | Like standard 'unzip' but more lazy. It is @Data.List.unzip undefined == undefined@, but @unzip undefined == (undefined, undefined)@. -} unzip :: [(a,b)] -> ([a],[b]) unzip = forcePair . foldr (\ (x,y) ~(xs,ys) -> (x:xs,y:ys)) ([],[]) {- | 'Data.List.partition' of GHC 6.2.1 fails on infinite lists. But this one does not. -} {- The lazy pattern match @(y,z)@ is necessary since otherwise it fails on infinite lists. -} partition :: (a -> Bool) -> [a] -> ([a], [a]) partition p = forcePair . foldr (\x ~(y,z) -> if p x then (x : y, z) else (y, x : z)) ([],[]) {- | It is @Data.List.span f undefined = undefined@, whereas @span f undefined = (undefined, undefined)@. -} span, break :: (a -> Bool) -> [a] -> ([a],[a]) span p = let recourse xt = forcePair $ fromMaybe ([],xt) $ do (x,xs) <- viewL xt guard $ p x return $ mapFst (x:) $ recourse xs in recourse break p = span (not . p) -- * Split {- | Split the list at the occurrences of a separator into sub-lists. Remove the separators. This is a generalization of 'words'. -} chop :: (a -> Bool) -> [a] -> [[a]] chop p = uncurry (:) . foldr (\ x ~(y,ys) -> if p x then ([],y:ys) else ((x:y),ys) ) ([],[]) chop' :: (a -> Bool) -> [a] -> [[a]] chop' p = let recourse = uncurry (:) . mapSnd (switchL [] (const recourse)) . break p in recourse chopAtRun :: (Eq a) => (a -> Bool) -> [a] -> [[a]] chopAtRun p = let recourse [] = [[]] recourse y = let (z,zs) = break p (dropWhile p y) in z : recourse zs in recourse {- | Like 'break', but splits after the matching element. -} breakAfter :: (a -> Bool) -> [a] -> ([a], [a]) breakAfter p = let recourse [] = ([],[]) recourse (x:xs) = mapFst (x:) $ if p x then ([],xs) else recourse xs in forcePair . recourse {- | Split the list after each occurence of a terminator. Keep the terminator. There is always a list for the part after the last terminator. It may be empty. -} segmentAfter :: (a -> Bool) -> [a] -> [[a]] segmentAfter p = -- foldr (\ x ~yt@(y:ys) -> if p x then [x]:yt else (x:y):ys) [[]] uncurry (:) . foldr (\x ~(y,ys) -> mapFst (x:) $ if p x then ([],y:ys) else (y,ys)) ([],[]) propSegmentAfterConcat :: Eq a => (a -> Bool) -> [a] -> Bool propSegmentAfterConcat p xs = concat (segmentAfter p xs) == xs propSegmentAfterNumSeps :: (a -> Bool) -> [a] -> Bool propSegmentAfterNumSeps p xs = length (filter p xs) == length (tail (segmentAfter p xs)) propSegmentAfterLasts :: (a -> Bool) -> [a] -> Bool propSegmentAfterLasts p = all (p . last) . init . segmentAfter p propSegmentAfterInits :: (a -> Bool) -> [a] -> Bool propSegmentAfterInits p = all (all (not . p) . init) . init . segmentAfter p {- This test captures both infinitely many groups and infinitely big groups. -} propSegmentAfterInfinite :: (a -> Bool) -> a -> [a] -> Bool propSegmentAfterInfinite p x = flip seq True . (!!100) . concat . segmentAfter p . cycle . (x:) {- | Split the list before each occurence of a leading character. Keep these characters. There is always a list for the part before the first leading character. It may be empty. -} segmentBefore :: (a -> Bool) -> [a] -> [[a]] segmentBefore p = -- foldr (\ x ~(y:ys) -> (if p x then ([]:) else id) ((x:y):ys)) [[]] uncurry (:) . foldr (\ x ~(y,ys) -> let xs = x:y in if p x then ([],xs:ys) else (xs,ys)) ([],[]) propSegmentBeforeConcat :: Eq a => (a -> Bool) -> [a] -> Bool propSegmentBeforeConcat p xs = concat (segmentBefore p xs) == xs propSegmentBeforeNumSeps :: (a -> Bool) -> [a] -> Bool propSegmentBeforeNumSeps p xs = length (filter p xs) == length (tail (segmentBefore p xs)) propSegmentBeforeHeads :: (a -> Bool) -> [a] -> Bool propSegmentBeforeHeads p = all (p . head) . tail . segmentBefore p propSegmentBeforeTails :: (a -> Bool) -> [a] -> Bool propSegmentBeforeTails p = all (all (not . p) . tail) . tail . segmentBefore p propSegmentBeforeInfinite :: (a -> Bool) -> a -> [a] -> Bool propSegmentBeforeInfinite p x = flip seq True . (!!100) . concat . segmentBefore p . cycle . (x:) -- cf. Matroid.hs {- | @removeEach xs@ represents a list of sublists of @xs@, where each element of @xs@ is removed and the removed element is separated. It seems to be much simpler to achieve with @zip xs (map (flip List.delete xs) xs)@, but the implementation of 'removeEach' does not need the 'Eq' instance and thus can also be used for lists of functions. -} removeEach :: [a] -> [(a, [a])] removeEach = map (\(ys, pivot, zs) -> (pivot,ys++zs)) . splitEverywhere splitEverywhere :: [a] -> [([a], a, [a])] splitEverywhere xs = map (\(y, z:zs) -> (y,z,zs)) (init (zip (inits xs) (tails xs))) -- * inspect ends of a list {-# DEPRECATED splitLast "use viewR instead" #-} {- | It holds @splitLast xs == (init xs, last xs)@, but 'splitLast' is more efficient if the last element is accessed after the initial ones, because it avoids memoizing list. -} splitLast :: [a] -> ([a], a) splitLast [] = error "splitLast: empty list" splitLast [x] = ([], x) splitLast (x:xs) = let (xs', lastx) = splitLast xs in (x:xs', lastx) propSplitLast :: Eq a => [a] -> Bool propSplitLast xs = splitLast xs == (init xs, last xs) {- | Should be prefered to 'head' and 'tail'. -} {-# INLINE viewL #-} viewL :: [a] -> Maybe (a, [a]) viewL (x:xs) = Just (x,xs) viewL [] = Nothing {- | Should be prefered to 'init' and 'last'. -} viewR :: [a] -> Maybe ([a], a) viewR = foldr (\x -> Just . forcePair . maybe ([],x) (mapFst (x:))) Nothing propViewR :: Eq a => [a] -> Bool propViewR xs = maybe True ((init xs, last xs) == ) (viewR xs) {- | Should be prefered to 'head' and 'tail'. -} {-# INLINE switchL #-} switchL :: b -> (a -> [a] -> b) -> [a] -> b switchL n _ [] = n switchL _ j (x:xs) = j x xs switchL' :: b -> (a -> [a] -> b) -> [a] -> b switchL' n j = maybe n (uncurry j) . viewL {- | Should be prefered to 'init' and 'last'. -} {-# INLINE switchR #-} switchR :: b -> ([a] -> a -> b) -> [a] -> b switchR n j = maybe n (uncurry j) . viewR propSwitchR :: Eq a => [a] -> Bool propSwitchR xs = switchR True (\ixs lxs -> ixs == init xs && lxs == last xs) xs -- * List processing starting at the end {- | Remove the longest suffix of elements satisfying p. In contrast to @reverse . dropWhile p . reverse@ this works for infinite lists, too. -} dropWhileRev :: (a -> Bool) -> [a] -> [a] dropWhileRev p = foldr (\x xs -> if p x && null xs then [] else x:xs) [] dropWhileRev' :: (a -> Bool) -> [a] -> [a] dropWhileRev' p = concat . init . segmentAfter (not . p) {- | Alternative version of @reverse . takeWhile p . reverse@. -} takeWhileRev :: (a -> Bool) -> [a] -> [a] takeWhileRev p = last . segmentAfter (not . p) {- | Doesn't seem to be superior to the naive implementation. -} takeWhileRev' :: (a -> Bool) -> [a] -> [a] takeWhileRev' p = (\xs -> if fst (head xs) then map snd xs else []) . last . Key.aux groupBy (==) p {- | However it is more inefficient, because of repeatedly appending single elements. :-( -} takeWhileRev'' :: (a -> Bool) -> [a] -> [a] takeWhileRev'' p = foldl (\xs x -> if p x then xs++[x] else []) [] -- * List processing with Maybe and Either {- | @maybePrefixOf xs ys@ is @Just zs@ if @xs@ is a prefix of @ys@, where @zs@ is @ys@ without the prefix @xs@. Otherwise it is @Nothing@. -} maybePrefixOf :: Eq a => [a] -> [a] -> Maybe [a] maybePrefixOf (x:xs) (y:ys) = guard (x==y) >> maybePrefixOf xs ys maybePrefixOf [] ys = Just ys maybePrefixOf _ [] = Nothing {- | Partition a list into elements which evaluate to @Just@ or @Nothing@ by @f@. It holds @mapMaybe f == fst . partitionMaybe f@ and @partition p == partitionMaybe (\ x -> toMaybe (p x) x)@. -} partitionMaybe :: (a -> Maybe b) -> [a] -> ([b], [a]) partitionMaybe f = forcePair . foldr (\x -> maybe (mapSnd (x:)) (\y -> mapFst (y:)) (f x)) ([],[]) {- | This is the cousin of 'takeWhile' analogously to 'catMaybes' being the cousin of 'filter'. Example: Keep the heads of sublists until an empty list occurs. > takeWhileJust $ map (fmap fst . viewL) xs -} takeWhileJust :: [Maybe a] -> [a] takeWhileJust = foldr (\x acc -> maybe [] (:acc) x) [] unzipEithers :: [Either a b] -> ([a], [b]) unzipEithers = forcePair . foldr (either (\x -> mapFst (x:)) (\y -> mapSnd (y:))) ([],[]) -- * Sieve and slice {-| keep every k-th value from the list -} sieve, sieve', sieve'', sieve''' :: Int -> [a] -> [a] sieve k = unfoldr (\xs -> toMaybe (not (null xs)) (head xs, drop k xs)) sieve' k = map head . sliceVertical k sieve'' k x = map (x!!) [0,k..(length x-1)] sieve''' k = map head . takeWhile (not . null) . iterate (drop k) propSieve :: Eq a => Int -> [a] -> Bool propSieve n x = sieve n x == sieve' n x && sieve n x == sieve'' n x {- sliceHorizontal is faster than sliceHorizontal' but consumes slightly more memory (although it needs no swapping) -} sliceHorizontal, sliceHorizontal', sliceHorizontal'', sliceHorizontal''' :: Int -> [a] -> [[a]] sliceHorizontal n = map (sieve n) . take n . iterate (drop 1) sliceHorizontal' n = foldr (\x ys -> let y = last ys in Match.take ys ((x:y):ys)) (replicate n []) sliceHorizontal'' n = reverse . foldr (\x ~(y:ys) -> ys ++ [x:y]) (replicate n []) sliceHorizontal''' n = take n . transpose . takeWhile (not . null) . iterate (drop n) propSliceHorizontal :: Eq a => Int -> [a] -> Bool propSliceHorizontal n x = sliceHorizontal n x == sliceHorizontal' n x && sliceHorizontal n x == sliceHorizontal'' n x && sliceHorizontal n x == sliceHorizontal''' n x sliceVertical, sliceVertical' :: Int -> [a] -> [[a]] sliceVertical n = map (take n) . takeWhile (not . null) . iterate (drop n) {- takeWhile must be performed before (map take) in order to handle (n==0) correctly -} sliceVertical' n = unfoldr (\x -> toMaybe (not (null x)) (splitAt n x)) propSliceVertical :: Eq a => Int -> [a] -> Bool propSliceVertical n x = take 100000 (sliceVertical n x) == take 100000 (sliceVertical' n x) propSlice :: Eq a => Int -> [a] -> Bool propSlice n x = -- problems: sliceHorizontal 4 [] == [[],[],[],[]] sliceHorizontal n x == transpose (sliceVertical n x) && sliceVertical n x == transpose (sliceHorizontal n x) -- * Search&replace search :: (Eq a) => [a] -> [a] -> [Int] search sub str = findIndices (isPrefixOf sub) (tails str) markSublists :: (Eq a) => [a] -> [a] -> [Maybe [a]] markSublists sub ys = let ~(hd', rest') = foldr (\c ~(hd, rest) -> let xs = c:hd in case maybePrefixOf sub xs of Just suffix -> ([], Nothing : Just suffix : rest) Nothing -> (xs, rest)) ([],[]) ys in Just hd' : rest' replace :: (Eq a) => [a] -> [a] -> [a] -> [a] replace src dst xs = concatMap (fromMaybe dst) (markSublists src xs) propReplaceId :: (Eq a) => [a] -> [a] -> Bool propReplaceId xs ys = replace xs xs ys == ys propReplaceCycle :: (Eq a) => [a] -> [a] -> Bool propReplaceCycle xs ys = replace xs ys (cycle xs) == cycle ys {- | This is slightly wrong, because it re-replaces things. That's also the reason for inefficiency: The replacing can go on only when subsequent replacements are finished. Thus this functiob fails on infinite lists. -} replace' :: (Eq a) => [a] -> [a] -> [a] -> [a] replace' src dst = foldr (\x xs -> let y=x:xs in if isPrefixOf src y then dst ++ drop (length src) y else y) [] multiReplace :: Eq a => [([a], [a])] -> [a] -> [a] multiReplace dict = let recourse [] = [] recourse str@(s:ss) = maybe (s : recourse ss) (\(src, dst) -> dst ++ recourse (Match.drop src str)) (find (flip isPrefixOf str . fst) dict) in recourse propMultiReplaceSingle :: Eq a => [a] -> [a] -> [a] -> Bool propMultiReplaceSingle src dst x = replace src dst x == multiReplace [(src,dst)] x -- * Lists of lists {- | Transform > [[00,01,02,...], [[00], > [10,11,12,...], --> [10,01], > [20,21,22,...], [20,11,02], > ...] ...] With @concat . shear@ you can perform a Cantor diagonalization, that is an enumeration of all elements of the sub-lists where each element is reachable within a finite number of steps. It is also useful for polynomial multiplication (convolution). -} shear :: [[a]] -> [[a]] shear = map catMaybes . shearTranspose . transposeFill transposeFill :: [[a]] -> [[Maybe a]] transposeFill = unfoldr (\xs -> toMaybe (not (null xs)) (mapSnd (dropWhileRev null) $ unzipCons xs)) unzipCons :: [[a]] -> ([Maybe a], [[a]]) unzipCons = unzip . map ((\my -> (fmap fst my, maybe [] snd my)) . viewL) {- | It's somehow inverse to zipCons, but the difficult part is, that a trailing empty list on the right side is suppressed. -} unzipConsSkew :: [[a]] -> ([Maybe a], [[a]]) unzipConsSkew = let aux [] [] = ([],[]) -- one empty list at the end will be removed aux xs ys = mapSnd (xs:) $ prep ys prep = forcePair . switchL ([],[]) (\y ys -> let my = viewL y in mapFst (fmap fst my :) $ aux (maybe [] snd my) ys) in prep shear' :: [[a]] -> [[a]] shear' xs@(_:_) = let (y:ys,zs) = unzip (map (splitAt 1) xs) zipConc (a:as) (b:bs) = (a++b) : zipConc as bs zipConc [] bs = bs zipConc as [] = as in y : zipConc ys (shear' (dropWhileRev null zs)) {- Dropping trailing empty lists is necessary, otherwise finite lists are filled with empty lists. -} shear' [] = [] {- | Transform > [[00,01,02,...], [[00], > [10,11,12,...], --> [01,10], > [20,21,22,...], [02,11,20], > ...] ...] It's like 'shear' but the order of elements in the sub list is reversed. Its implementation seems to be more efficient than that of 'shear'. If the order does not matter, better choose 'shearTranspose'. -} shearTranspose :: [[a]] -> [[a]] shearTranspose = foldr zipConsSkew [] zipConsSkew :: [a] -> [[a]] -> [[a]] zipConsSkew xt yss = uncurry (:) $ case xt of x:xs -> ([x], zipCons xs yss) [] -> ([], yss) {- | zipCons is like @zipWith (:)@ but it keeps lists which are too long This version works also for @zipCons something undefined@. -} zipCons :: [a] -> [[a]] -> [[a]] zipCons (x:xs) yt = let (y,ys) = switchL ([],[]) (,) yt in (x:y) : zipCons xs ys zipCons [] ys = ys -- | zipCons' is like @zipWith (:)@ but it keeps lists which are too long zipCons' :: [a] -> [[a]] -> [[a]] zipCons' (x:xs) (y:ys) = (x:y) : zipCons' xs ys zipCons' [] ys = ys zipCons' xs [] = map (:[]) xs {- | Operate on each combination of elements of the first and the second list. In contrast to the list instance of 'Monad.liftM2' in holds the results in a list of lists. It holds @concat (outerProduct f xs ys) == liftM2 f xs ys@ -} outerProduct :: (a -> b -> c) -> [a] -> [b] -> [[c]] outerProduct f xs ys = map (flip map ys . f) xs -- * Miscellaneous {- | Take while first predicate holds, then continue taking while second predicate holds, and so on. -} takeWhileMulti :: [a -> Bool] -> [a] -> [a] takeWhileMulti [] _ = [] takeWhileMulti _ [] = [] takeWhileMulti aps@(p:ps) axs@(x:xs) = if p x then x : takeWhileMulti aps xs else takeWhileMulti ps axs takeWhileMulti' :: [a -> Bool] -> [a] -> [a] takeWhileMulti' ps xs = concatMap fst (tail (scanl (flip span . snd) (undefined,xs) ps)) propTakeWhileMulti :: (Eq a) => [a -> Bool] -> [a] -> Bool propTakeWhileMulti ps xs = takeWhileMulti ps xs == takeWhileMulti' ps xs {- Debug.QuickCheck.quickCheck (propTakeWhileMulti [(<0), (>0), odd, even, ((0::Int)==)]) -} {- | This is a combination of 'foldl'' and 'foldr' in the sense of 'propFoldl'r'. It is however more efficient because it avoids storing the whole input list as a result of sharing. -} foldl'r, foldl'rStrict, foldl'rNaive :: (b -> a -> b) -> b -> (c -> d -> d) -> d -> [(a,c)] -> (b,d) foldl'r f b0 g d0 = -- (\(k,d1) -> (k b0, d1)) . mapFst ($b0) . foldr (\(a,c) ~(k,d) -> (\b -> k $! f b a, g c d)) (id,d0) foldl'rStrict f b0 g d0 = mapFst ($b0) . foldr (\(a,c) ~(k,d) -> ((,) $! (\b -> k $! f b a)) $! g c d) (id,d0) foldl'rNaive f b g d xs = mapPair (foldl' f b, foldr g d) $ unzip xs propFoldl'r :: (Eq b, Eq d) => (b -> a -> b) -> b -> (c -> d -> d) -> d -> [(a,c)] -> Bool propFoldl'r f b g d xs = foldl'r f b g d xs == foldl'rNaive f b g d xs {- The results in GHCi surprise: *List.HT> mapSnd last $ foldl'rNaive (+) (0::Integer) (:) "" $ replicate 1000000 (1,'a') (1000000,'a') (0.44 secs, 141032856 bytes) *List.HT> mapSnd last $ foldl'r (+) (0::Integer) (:) "" $ replicate 1000000 (1,'a') (1000000,'a') (2.64 secs, 237424948 bytes) -} {- Debug.QuickCheck.quickCheck (\b d -> propFoldl'r (+) (b::Int) (++) (d::[Int])) -} {- | rotate left -} rotate, rotate', rotate'' :: Int -> [a] -> [a] rotate n x = Match.take x (drop (mod n (length x)) (cycle x)) {- | more efficient implementation of rotate' -} rotate' n x = uncurry (flip (++)) (splitAt (mod n (length x)) x) rotate'' n x = Match.take x (drop n (cycle x)) propRotate :: Eq a => Int -> [a] -> Bool propRotate n x = rotate n x == rotate' n x && rotate n x == rotate'' n x {- Debug.QuickCheck.quickCheck (\n x -> n>=0 Debug.QuickCheck.==> List.HT.propRotate n ((0::Int):x)) -} {-| Given two lists that are ordered (i.e. @p x y@ holds for subsequent @x@ and @y@) 'mergeBy' them into a list that is ordered, again. -} mergeBy :: (a -> a -> Bool) -> [a] -> [a] -> [a] mergeBy = Key.mergeBy allEqual :: Eq a => [a] -> Bool allEqual = and . mapAdjacent (==) isAscending :: (Ord a) => [a] -> Bool isAscending = and . isAscendingLazy isAscendingLazy :: (Ord a) => [a] -> [Bool] isAscendingLazy = mapAdjacent (<=) {- | This function combines every pair of neighbour elements in a list with a certain function. -} mapAdjacent :: (a -> a -> b) -> [a] -> [b] mapAdjacent f xs = zipWith f xs (tail xs) {- | Enumerate without Enum context. For Enum equivalent to enumFrom. -} range :: Num a => Int -> [a] range n = take n (iterate (+1) 0) {-# INLINE padLeft #-} padLeft :: a -> Int -> [a] -> [a] padLeft c n xs = replicate (n - length xs) c ++ xs {-# INLINE padRight #-} padRight, padRight1 :: a -> Int -> [a] -> [a] padRight c n xs = take n $ xs ++ repeat c padRight1 c n xs = xs ++ replicate (n - length xs) c {- | For an associative operation @op@ this computes @iterateAssociative op a = iterate (op a) a@ but it is even faster than @map (powerAssociative op a a) [0..]@ since it shares temporary results. The idea is: From the list @map (powerAssociative op a a) [0,(2*n)..]@ we compute the list @map (powerAssociative op a a) [0,n..]@, and iterate that until @n==1@. -} iterateAssociative :: (a -> a -> a) -> a -> [a] iterateAssociative op a = foldr (\pow xs -> pow : concatMap (\x -> [x, op x pow]) xs) undefined (iterate (\x -> op x x) a) {- | This is equal to 'iterateAssociative'. The idea is the following: The list we search is the fixpoint of the function: "Square all elements of the list, then spread it and fill the holes with successive numbers of their left neighbour." This also preserves log n applications per value. However it has a space leak, because for the value with index @n@ all elements starting at @div n 2@ must be kept. -} iterateLeaky :: (a -> a -> a) -> a -> [a] iterateLeaky op x = let merge (a:as) b = a : merge b as merge _ _ = error "iterateLeaky: an empty list cannot occur" sqrs = map (\y -> op y y) z z = x : merge sqrs (map (op x) sqrs) in z