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------------------------------------------------------------------------------ --- Library with some useful operations on lists. --- --- @author Michael Hanus, Bjoern Peemoeller --- @version November 2020 --- @category general ------------------------------------------------------------------------------ {-# OPTIONS_FRONTEND -Wno-incomplete-patterns #-} module Data.List ( elemIndex, elemIndices, find, findIndex, findIndices , nub, nubBy, delete, deleteBy, (\\), union, intersect , intersperse, intercalate, transpose, diagonal, permutations, partition , group, groupBy, splitOn, split, inits, tails, replace , isPrefixOf, isSuffixOf, isInfixOf , sort, sortBy, insertBy , unionBy, intersectBy , last, init , sum, product, maximum, minimum, maximumBy, minimumBy , scanl, scanl1, scanr, scanr1 , mapAccumL, mapAccumR , cycle, unfoldr ) where import Data.Maybe (listToMaybe) infix 5 \\ --- Returns the index `i` of the first occurrence of an element in a list --- as `(Just i)`, otherwise `Nothing` is returned. elemIndex :: Eq a => a -> [a] -> Maybe Int elemIndex x = findIndex (x ==) --- Returns the list of indices of occurrences of an element in a list. elemIndices :: Eq a => a -> [a] -> [Int] elemIndices x = findIndices (x ==) --- Returns the first element `e` of a list satisfying a predicate --- as `(Just e)`, --- otherwise `Nothing` is returned. find :: (a -> Bool) -> [a] -> Maybe a find p = listToMaybe . filter p --- Returns the index `i` of the first occurrences of a list element --- satisfying a predicate as `(Just i)`, otherwise `Nothing` is returned. findIndex :: (a -> Bool) -> [a] -> Maybe Int findIndex p = listToMaybe . findIndices p --- Returns the list of indices of list elements satisfying a predicate. findIndices :: (a -> Bool) -> [a] -> [Int] findIndices p xs = [ i | (x,i) <- zip xs [0..], p x ] --- Removes all duplicates in the argument list. nub :: Eq a => [a] -> [a] nub xs = nubBy (==) xs --- Removes all duplicates in the argument list according to an --- equivalence relation. nubBy :: (a -> a -> Bool) -> [a] -> [a] nubBy _ [] = [] nubBy eq (x:xs) = x : nubBy eq (filter (\y -> not (eq x y)) xs) --- Deletes the first occurrence of an element in a list. delete :: Eq a => a -> [a] -> [a] delete = deleteBy (==) --- Deletes the first occurrence of an element in a list --- according to an equivalence relation. deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a] deleteBy _ _ [] = [] deleteBy eq x (y:ys) = if eq x y then ys else y : deleteBy eq x ys --- Computes the difference of two lists. --- @param xs - a list --- @param ys - a list --- @return the list where the first occurrence of each element of --- `ys` has been removed from `xs` (\\) :: Eq a => [a] -> [a] -> [a] xs \\ ys = foldl (flip delete) xs ys --- Computes the union of two lists. union :: Eq a => [a] -> [a] -> [a] union [] ys = ys union (x:xs) ys = if x `elem` ys then union xs ys else x : union xs ys --- Computes the union of two lists according to the given equivalence relation unionBy :: (a -> a -> Bool) -> [a] -> [a] -> [a] unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs --- Computes the intersection of two lists. intersect :: Eq a => [a] -> [a] -> [a] intersect [] _ = [] intersect (x:xs) ys = if x `elem` ys then x : intersect xs ys else intersect xs ys --- Computes the intersection of two lists --- according to the given equivalence relation intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a] intersectBy _ [] _ = [] intersectBy _ (_:_) [] = [] intersectBy eq xs@(_:_) ys@(_:_) = [x | x <- xs, any (eq x) ys] --- Puts a separator element between all elements in a list. --- --- Example: `(intersperse 9 [1,2,3,4]) = [1,9,2,9,3,9,4]` intersperse :: a -> [a] -> [a] intersperse _ [] = [] intersperse _ [x] = [x] intersperse sep (x:xs@(_:_)) = x : sep : intersperse sep xs --- `intercalate xs xss` is equivalent to `(concat (intersperse xs xss))`. --- It inserts the list `xs` in between the lists in `xss` and --- concatenates the result. intercalate :: [a] -> [[a]] -> [a] intercalate xs xss = concat (intersperse xs xss) --- Transposes the rows and columns of the argument. --- --- Example: `(transpose [[1,2,3],[4,5,6]]) = [[1,4],[2,5],[3,6]]` transpose :: [[a]] -> [[a]] transpose [] = [] transpose ([] : xss) = transpose xss transpose ((x:xs) : xss) = (x : map head xss) : transpose (xs : map tail xss) --- Diagonalization of a list of lists. --- Fairly merges (possibly infinite) list of (possibly infinite) lists. --- --- @param xss - lists of lists --- @return fair enumeration of all elements of inner lists of given lists --- diagonal :: [[a]] -> [a] diagonal = concat . foldr diags [] where diags [] ys = ys diags (x:xs) ys = [x] : merge' xs ys merge' [] ys = ys merge' xs@(_:_) [] = map (:[]) xs merge' (x:xs) (y:ys) = (x:y) : merge' xs ys --- Returns the list of all permutations of the argument. permutations :: [a] -> [[a]] permutations xs0 = xs0 : perms xs0 [] where perms [] _ = [] perms (t:ts) is = foldr interleave (perms ts (t:is)) (permutations is) where interleave xs r = let (_, zs) = interleave' id xs r in zs interleave' _ [] r = (ts, r) interleave' f (y:ys) r = let (us, zs) = interleave' (f . (y:)) ys r in (y:us, f (t:y:us) : zs) --- Partitions a list into a pair of lists where the first list --- contains those elements that satisfy the predicate argument --- and the second list contains the remaining arguments. --- --- Example: `(partition (<4) [8,1,5,2,4,3]) = ([1,2,3],[8,5,4])` partition :: (a -> Bool) -> [a] -> ([a],[a]) partition p xs = foldr select ([],[]) xs where select x (ts,fs) = if p x then (x:ts,fs) else (ts,x:fs) --- Splits the list argument into a list of lists of equal adjacent --- elements. --- --- Example: `(group [1,2,2,3,3,3,4]) = [[1],[2,2],[3,3,3],[4]]` group :: Eq a => [a] -> [[a]] group = groupBy (==) --- Splits the list argument into a list of lists of related adjacent --- elements. --- @param eq - the relation to classify adjacent elements --- @param xs - the list of elements --- @return the list of lists of related adjacent elements groupBy :: (a -> a -> Bool) -> [a] -> [[a]] groupBy _ [] = [] groupBy eq (x:xs) = (x:ys) : groupBy eq zs where (ys,zs) = span (eq x) xs --- Breaks the second list argument into pieces separated by the first --- list argument, consuming the delimiter. An empty delimiter is --- invalid, and will cause an error to be raised. splitOn :: Eq a => [a] -> [a] -> [[a]] splitOn [] _ = error "splitOn called with an empty pattern" splitOn [x] xs = split (x ==) xs splitOn sep@(_:_:_) xs = go xs where go [] = [[]] go l@(y:ys) | sep `isPrefixOf` l = [] : go (drop len l) | otherwise = let (zs:zss) = go ys in (y:zs):zss len = length sep --- Splits a list into components delimited by separators, --- where the predicate returns True for a separator element. --- The resulting components do not contain the separators. --- Two adjacent separators result in an empty component in the output. --- --- split (=='a') "aabbaca" == ["","","bb","c",""] --- split (=='a') "" == [""] split :: (a -> Bool) -> [a] -> [[a]] split _ [] = [[]] split p (x:xs) | p x = [] : split p xs | otherwise = let (ys:yss) = split p xs in (x:ys):yss --- Returns all initial segments of a list, starting with the shortest. --- Example: `inits [1,2,3] == [[],[1],[1,2],[1,2,3]]` --- @param xs - the list of elements --- @return the list of initial segments of the argument list inits :: [a] -> [[a]] inits [] = [[]] inits (x:xs) = [] : map (x:) (inits xs) --- Returns all final segments of a list, starting with the longest. --- Example: `tails [1,2,3] == [[1,2,3],[2,3],[3],[]]` tails :: [a] -> [[a]] tails [] = [[]] tails xxs@(_:xs) = xxs : tails xs --- Replaces an element in a list. --- @param x - the new element --- @param p - the position of the new element (head = 0) --- @param ys - the old list --- @return the new list where the `p`. element is replaced by `x` replace :: a -> Int -> [a] -> [a] replace _ _ [] = [] replace x p (y:ys) | p==0 = x:ys | otherwise = y:(replace x (p-1) ys) --- Checks whether a list is a prefix of another. --- @param xs - a list --- @param ys - a list --- @return `True` if `xs` is a prefix of `ys` isPrefixOf :: Eq a => [a] -> [a] -> Bool isPrefixOf [] _ = True isPrefixOf (_:_) [] = False isPrefixOf (x:xs) (y:ys) = x==y && (isPrefixOf xs ys) --- Checks whether a list is a suffix of another. --- @param xs - a list --- @param ys - a list --- @return `True` if `xs` is a suffix of `ys` isSuffixOf :: Eq a => [a] -> [a] -> Bool isSuffixOf xs ys = isPrefixOf (reverse xs) (reverse ys) --- Checks whether a list is contained in another. --- @param xs - a list --- @param ys - a list --- @return True if xs is contained in ys isInfixOf :: Eq a => [a] -> [a] -> Bool isInfixOf xs ys = any (isPrefixOf xs) (tails ys) --- The default sorting operation, mergeSort, with standard ordering `<=`. sort :: Ord a => [a] -> [a] sort = sortBy (<=) --- Sorts a list w.r.t. an ordering relation by the insertion method. sortBy :: (a -> a -> Bool) -> [a] -> [a] sortBy = mergeSortBy --- Bottom-up mergesort with ordering as first parameter. mergeSortBy :: (a -> a -> Bool) -> [a] -> [a] mergeSortBy leq zs = mergeLists (genRuns zs) where -- generate runs of length 2: genRuns [] = [] genRuns [x] = [[x]] genRuns (x1:x2:xs) | leq x1 x2 = [x1,x2] : genRuns xs | otherwise = [x2,x1] : genRuns xs -- merge the runs: mergeLists [] = [] mergeLists [x] = x mergeLists (x1:x2:xs) = mergeLists (merge leq x1 x2 : mergePairs xs) mergePairs [] = [] mergePairs [x] = [x] mergePairs (x1:x2:xs) = merge leq x1 x2 : mergePairs xs --- Merges two lists with respect to an ordering predicate. merge :: (a -> a -> Bool) -> [a] -> [a] -> [a] merge _ [] ys = ys merge _ (x:xs) [] = x : xs merge leq (x:xs) (y:ys) | leq x y = x : merge leq xs (y:ys) | otherwise = y : merge leq (x:xs) ys --- Inserts an object into a list according to an ordering relation. --- @param le - an ordering relation (e.g., less-or-equal) --- @param x - an element --- @param xs - a list --- @return a list where the element has been inserted insertBy :: (a -> a -> Bool) -> a -> [a] -> [a] insertBy _ x [] = [x] insertBy le x (y:ys) = if le x y then x : y : ys else y : insertBy le x ys --- Returns the last element of a non-empty list. last :: [a] -> a last [x] = x last (_ : xs@(_:_)) = last xs --- Returns the input list with the last element removed. init :: [a] -> [a] init [_] = [] init (x:xs@(_:_)) = x : init xs --- Returns the sum of a list of integers. sum :: Num a => [a] -> a sum ns = foldl (+) 0 ns --- Returns the product of a list of integers. product :: Num a => [a] -> a product ns = foldl (*) 1 ns --- Returns the maximum of a non-empty list. maximum :: Ord a => [a] -> a maximum xs@(_:_) = foldl1 max xs --- Returns the maximum of a non-empty list --- according to the given comparison function maximumBy :: (a -> a -> Ordering) -> [a] -> a maximumBy cmp xs@(_:_) = foldl1 maxBy xs where maxBy x y = case cmp x y of GT -> x _ -> y --- Returns the minimum of a non-empty list. minimum :: Ord a => [a] -> a minimum xs@(_:_) = foldl1 min xs --- Returns the minimum of a non-empty list --- according to the given comparison function minimumBy :: (a -> a -> Ordering) -> [a] -> a minimumBy cmp xs@(_:_) = foldl1 minBy xs where minBy x y = case cmp x y of GT -> y _ -> x --- `scanl` is similar to `foldl`, but returns a list of successive --- reduced values from the left: --- scanl f z [x1, x2, ...] == [z, z `f` x1, (z `f` x1) `f` x2, ...] scanl :: (a -> b -> a) -> a -> [b] -> [a] scanl f q ls = q : (case ls of [] -> [] x:xs -> scanl f (f q x) xs) --- `scanl1` is a variant of `scanl` that has no starting value argument: --- scanl1 f [x1, x2, ...] == [x1, x1 `f` x2, ...] scanl1 :: (a -> a -> a) -> [a] -> [a] scanl1 _ [] = [] scanl1 f (x:xs) = scanl f x xs --- `scanr` is the right-to-left dual of `scanl`. scanr :: (a -> b -> b) -> b -> [a] -> [b] scanr _ q0 [] = [q0] scanr f q0 (x:xs) = f x q : qs where qs@(q:_) = scanr f q0 xs --- `scanr1` is a variant of `scanr` that has no starting value argument. scanr1 :: (a -> a -> a) -> [a] -> [a] scanr1 _ [] = [] scanr1 _ [x] = [x] scanr1 f (x:xs@(_:_)) = f x q : qs where qs@(q:_) = scanr1 f xs --- The `mapAccumL` function behaves like a combination of `map` and --- `foldl`; it applies a function to each element of a list, passing --- an accumulating parameter from left to right, and returning a final --- value of this accumulator together with the new list. mapAccumL :: (acc -> x -> (acc, y)) -> acc -> [x] -> (acc, [y]) mapAccumL _ s [] = (s, []) mapAccumL f s (x:xs) = (s'',y:ys) where (s', y ) = f s x (s'',ys) = mapAccumL f s' xs --- The `mapAccumR` function behaves like a combination of `map` and --- `foldr`; it applies a function to each element of a list, passing --- an accumulating parameter from right to left, and returning a final --- value of this accumulator together with the new list. mapAccumR :: (acc -> x -> (acc, y)) -> acc -> [x] -> (acc, [y]) mapAccumR _ s [] = (s, []) mapAccumR f s (x:xs) = (s'', y:ys) where (s'',y ) = f s' x (s', ys) = mapAccumR f s xs --- Builds an infinite list from a finite one. cycle :: [a] -> [a] cycle xs@(_:_) = ys where ys = xs ++ ys --- Builds a list from a seed value. unfoldr :: (b -> Maybe (a, b)) -> b -> [a] unfoldr f b = case f b of Just (a, new_b) -> a : unfoldr f new_b Nothing -> [] |