Chapter 20
Data.List

20.1 Basic functions

(++) :: [a] -> [a] -> [a]

Append two lists, i.e.,

 [x1, ..., xm] ++ [y1, ..., yn] == [x1, ..., xm, y1, ..., yn]  
 [x1, ..., xm] ++ [y1, ...] == [x1, ..., xm, y1, ...]

If the first list is not finite, the result is the first list.

head :: [a] -> a

Extract the first element of a list, which must be non-empty.

last :: [a] -> a

Extract the last element of a list, which must be finite and non-empty.

tail :: [a] -> [a]

Extract the elements after the head of a list, which must be non-empty.

init :: [a] -> [a]

Return all the elements of a list except the last one. The list must be non-empty.

null :: [a] -> Bool

Test whether a list is empty.

length :: [a] -> Int

O(n). length returns the length of a finite list as an Int. It is an instance of the more general Data.List.genericLength, the result type of which may be any kind of number.

20.2 List transformations

map :: (a -> b) -> [a] -> [b]

mapf xs is the list obtained by applying f to each element of xs, i.e.,

 map f [x1, x2, ..., xn] == [f x1, f x2, ..., f xn]  
 map f [x1, x2, ...] == [f x1, f x2, ...]

reverse :: [a] -> [a]

reversexs returns the elements of xs in reverse order. xs must be finite.

intersperse :: a -> [a] -> [a]

The intersperse function takes an element and a list and ‘intersperses’ that element between the elements of the list. For example,

 intersperse ',' "abcde" == "a,b,c,d,e"

intercalate :: [a] -> [[a]] -> [a]

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.

transpose :: [[a]] -> [[a]]

The transpose function transposes the rows and columns of its argument. For example,

 transpose [[1,2,3],[4,5,6]] == [[1,4],[2,5],[3,6]]

subsequences :: [a] -> [[a]]

The subsequences function returns the list of all subsequences of the argument.

 subsequences "abc" == ["","a","b","ab","c","ac","bc","abc"]

permutations :: [a] -> [[a]]

The permutations function returns the list of all permutations of the argument.

 permutations "abc" == ["abc","bac","cba","bca","cab","acb"]

20.3 Reducing lists (folds)

foldl :: (a -> b -> a) -> a -> [b] -> a

foldl, applied to a binary operator, a starting value (typically the left-identity of the operator), and a list, reduces the list using the binary operator, from left to right:

 foldl f z [x1, x2, ..., xn] == (...((z ‘f‘ x1) ‘f‘ x2) ‘f‘...) ‘f‘ xn

The list must be finite.

foldl' :: (a -> b -> a) -> a -> [b] -> a

A strict version of foldl.

foldl1 :: (a -> a -> a) -> [a] -> a

foldl1 is a variant of foldl that has no starting value argument, and thus must be applied to non-empty lists.

foldl1' :: (a -> a -> a) -> [a] -> a

A strict version of foldl1

foldr :: (a -> b -> b) -> b -> [a] -> b

foldr, applied to a binary operator, a starting value (typically the right-identity of the operator), and a list, reduces the list using the binary operator, from right to left:

 foldr f z [x1, x2, ..., xn] == x1 ‘f‘ (x2 ‘f‘ ... (xn ‘f‘ z)...)

foldr1 :: (a -> a -> a) -> [a] -> a

foldr1 is a variant of foldr that has no starting value argument, and thus must be applied to non-empty lists.

20.3.1 Special folds

concat :: [[a]] -> [a]

Concatenate a list of lists.

concatMap :: (a -> [b]) -> [a] -> [b]

Map a function over a list and concatenate the results.

and :: [Bool] -> Bool

and returns the conjunction of a Boolean list. For the result to be True, the list must be finite; False, however, results from a False value at a finite index of a finite or infinite list.

or :: [Bool] -> Bool

or returns the disjunction of a Boolean list. For the result to be False, the list must be finite; True, however, results from a True value at a finite index of a finite or infinite list.

any :: (a -> Bool) -> [a] -> Bool

Applied to a predicate and a list, any determines if any element of the list satisfies the predicate. For the result to be False, the list must be finite; True, however, results from a True value for the predicate applied to an element at a finite index of a finite or infinite list.

all :: (a -> Bool) -> [a] -> Bool

Applied to a predicate and a list, all determines if all elements of the list satisfy the predicate. For the result to be True, the list must be finite; False, however, results from a False value for the predicate applied to an element at a finite index of a finite or infinite list.

sum :: Num a => [a] -> a

The sum function computes the sum of a finite list of numbers.

product :: Num a => [a] -> a

The product function computes the product of a finite list of numbers.

maximum :: Ord a => [a] -> a

maximum returns the maximum value from a list, which must be non-empty, finite, and of an ordered type. It is a special case of maximumBy, which allows the programmer to supply their own comparison function.

minimum :: Ord a => [a] -> a

minimum returns the minimum value from a list, which must be non-empty, finite, and of an ordered type. It is a special case of minimumBy, which allows the programmer to supply their own comparison function.

20.4 Building lists

20.4.1 Scans

scanl :: (a -> b -> a) -> a -> [b] -> [a]

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, ...]

Note that

 last (scanl f z xs) == foldl f z xs.

scanl1 :: (a -> a -> a) -> [a] -> [a]

scanl1 is a variant of scanl that has no starting value argument:

 scanl1 f [x1, x2, ...] == [x1, x1 ‘f‘ x2, ...]

scanr :: (a -> b -> b) -> b -> [a] -> [b]

scanr is the right-to-left dual of scanl. Note that

 head (scanr f z xs) == foldr f z xs.

scanr1 :: (a -> a -> a) -> [a] -> [a]

scanr1 is a variant of scanr that has no starting value argument.

20.4.2 Accumulating maps

mapAccumL :: (acc -> x -> (acc, y)) -> acc -> [x] -> (acc, [y])

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.

mapAccumR :: (acc -> x -> (acc, y)) -> acc -> [x] -> (acc, [y])

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.

20.4.3 Infinite lists

iterate :: (a -> a) -> a -> [a]

iteratef x returns an infinite list of repeated applications of f to x:

 iterate f x == [x, f x, f (f x), ...]

repeat :: a -> [a]

repeatx is an infinite list, with x the value of every element.

replicate :: Int -> a -> [a]

replicaten x is a list of length n with x the value of every element. It is an instance of the more general Data.List.genericReplicate, in which n may be of any integral type.

cycle :: [a] -> [a]

cycle ties a finite list into a circular one, or equivalently, the infinite repetition of the original list. It is the identity on infinite lists.

20.4.4 Unfolding

unfoldr :: (b -> Maybe (a, b)) -> b -> [a]

The unfoldr function is a ‘dual’ to foldr: while foldr reduces a list to a summary value, unfoldr builds a list from a seed value. The function takes the element and returns Nothing if it is done producing the list or returns Just(a,b), in which case, a is a prepended to the list and b is used as the next element in a recursive call. For example,

 iterate f == unfoldr (\x -> Just (x, f x))

In some cases, unfoldr can undo a foldr operation:

 unfoldr f' (foldr f z xs) == xs

if the following holds:

 f' (f x y) = Just (x,y)  
 f' z       = Nothing

A simple use of unfoldr:

 unfoldr (\b -> if b == 0 then Nothing else Just (b, b-1)) 10  
  [10,9,8,7,6,5,4,3,2,1]

20.5 Sublists

20.5.1 Extracting sublists

take :: Int -> [a] -> [a]

taken, applied to a list xs, returns the prefix of xs of length n, or xs itself if n > length xs:

 take 5 "Hello World!" == "Hello"  
 take 3 [1,2,3,4,5] == [1,2,3]  
 take 3 [1,2] == [1,2]  
 take 3 [] == []  
 take (-1) [1,2] == []  
 take 0 [1,2] == []

It is an instance of the more general Data.List.genericTake, in which n may be of any integral type.

drop :: Int -> [a] -> [a]

dropn xs returns the suffix of xs after the first n elements, or [] if n > length xs:

 drop 6 "Hello World!" == "World!"  
 drop 3 [1,2,3,4,5] == [4,5]  
 drop 3 [1,2] == []  
 drop 3 [] == []  
 drop (-1) [1,2] == [1,2]  
 drop 0 [1,2] == [1,2]

It is an instance of the more general Data.List.genericDrop, in which n may be of any integral type.

splitAt :: Int -> [a] -> ([a], [a])

splitAtn xs returns a tuple where first element is xs prefix of length n and second element is the remainder of the list:

 splitAt 6 "Hello World!" == ("Hello ","World!")  
 splitAt 3 [1,2,3,4,5] == ([1,2,3],[4,5])  
 splitAt 1 [1,2,3] == ([1],[2,3])  
 splitAt 3 [1,2,3] == ([1,2,3],[])  
 splitAt 4 [1,2,3] == ([1,2,3],[])  
 splitAt 0 [1,2,3] == ([],[1,2,3])  
 splitAt (-1) [1,2,3] == ([],[1,2,3])

It is equivalent to (take n xs, drop n xs). splitAt is an instance of the more general Data.List.genericSplitAt, in which n may be of any integral type.

takeWhile :: (a -> Bool) -> [a] -> [a]

takeWhile, applied to a predicate p and a list xs, returns the longest prefix (possibly empty) of xs of elements that satisfy p:

 takeWhile (< 3) [1,2,3,4,1,2,3,4] == [1,2]  
 takeWhile (< 9) [1,2,3] == [1,2,3]  
 takeWhile (< 0) [1,2,3] == []

dropWhile :: (a -> Bool) -> [a] -> [a]

dropWhilep xs returns the suffix remaining after takeWhilep xs:

 dropWhile (< 3) [1,2,3,4,5,1,2,3] == [3,4,5,1,2,3]  
 dropWhile (< 9) [1,2,3] == []  
 dropWhile (< 0) [1,2,3] == [1,2,3]

span :: (a -> Bool) -> [a] -> ([a], [a])

span, applied to a predicate p and a list xs, returns a tuple where first element is longest prefix (possibly empty) of xs of elements that satisfy p and second element is the remainder of the list:

 span (< 3) [1,2,3,4,1,2,3,4] == ([1,2],[3,4,1,2,3,4])  
 span (< 9) [1,2,3] == ([1,2,3],[])  
 span (< 0) [1,2,3] == ([],[1,2,3])

spanp xs is equivalent to (takeWhile p xs, dropWhile p xs)

break :: (a -> Bool) -> [a] -> ([a], [a])

break, applied to a predicate p and a list xs, returns a tuple where first element is longest prefix (possibly empty) of xs of elements that do not satisfy p and second element is the remainder of the list:

 break (> 3) [1,2,3,4,1,2,3,4] == ([1,2,3],[4,1,2,3,4])  
 break (< 9) [1,2,3] == ([],[1,2,3])  
 break (> 9) [1,2,3] == ([1,2,3],[])

breakp is equivalent to span (not . p).

stripPrefix :: Eq a => [a] -> [a] -> Maybe [a]

The stripPrefix function drops the given prefix from a list. It returns Nothing if the list did not start with the prefix given, or Just the list after the prefix, if it does.

 stripPrefix "foo" "foobar" == Just "bar"  
 stripPrefix "foo" "foo" == Just ""  
 stripPrefix "foo" "barfoo" == Nothing  
 stripPrefix "foo" "barfoobaz" == Nothing

group :: Eq a => [a] -> [[a]]

The group function takes a list and returns a list of lists such that the concatenation of the result is equal to the argument. Moreover, each sublist in the result contains only equal elements. For example,

 group "Mississippi" = ["M","i","ss","i","ss","i","pp","i"]

It is a special case of groupBy, which allows the programmer to supply their own equality test.

inits :: [a] -> [[a]]

The inits function returns all initial segments of the argument, shortest first. For example,

 inits "abc" == ["","a","ab","abc"]

tails :: [a] -> [[a]]

The tails function returns all final segments of the argument, longest first. For example,

 tails "abc" == ["abc", "bc", "c",""]

20.5.2 Predicates

isPrefixOf :: Eq a => [a] -> [a] -> Bool

The isPrefixOf function takes two lists and returns True iff the first list is a prefix of the second.

isSuffixOf :: Eq a => [a] -> [a] -> Bool

The isSuffixOf function takes two lists and returns True iff the first list is a suffix of the second. Both lists must be finite.

isInfixOf :: Eq a => [a] -> [a] -> Bool

The isInfixOf function takes two lists and returns True iff the first list is contained, wholly and intact, anywhere within the second.

Example:

isInfixOf "Haskell" "I really like Haskell." == True  
isInfixOf "Ial" "I really like Haskell." == False

20.6 Searching lists

20.6.1 Searching by equality

elem :: Eq a => a -> [a] -> Bool

elem is the list membership predicate, usually written in infix form, e.g., x ‘elem‘ xs. For the result to be False, the list must be finite; True, however, results from an element equal to x found at a finite index of a finite or infinite list.

notElem :: Eq a => a -> [a] -> Bool

notElem is the negation of elem.

lookup :: Eq a => a -> [(a, b)] -> Maybe b

lookupkey assocs looks up a key in an association list.

20.6.2 Searching with a predicate

find :: (a -> Bool) -> [a] -> Maybe a

The find function takes a predicate and a list and returns the first element in the list matching the predicate, or Nothing if there is no such element.

filter :: (a -> Bool) -> [a] -> [a]

filter, applied to a predicate and a list, returns the list of those elements that satisfy the predicate; i.e.,

 filter p xs = [ x | x <- xs, p x]

partition :: (a -> Bool) -> [a] -> ([a], [a])

The partition function takes a predicate a list and returns the pair of lists of elements which do and do not satisfy the predicate, respectively; i.e.,

 partition p xs == (filter p xs, filter (not . p) xs)

20.7 Indexing lists

These functions treat a list xs as a indexed collection, with indices ranging from 0 to length xs - 1.

(!!) :: [a] -> Int -> a

List index (subscript) operator, starting from 0. It is an instance of the more general Data.List.genericIndex, which takes an index of any integral type.

elemIndex :: Eq a => a -> [a] -> Maybe Int

The elemIndex function returns the index of the first element in the given list which is equal (by ==) to the query element, or Nothing if there is no such element.

elemIndices :: Eq a => a -> [a] -> [Int]

The elemIndices function extends elemIndex, by returning the indices of all elements equal to the query element, in ascending order.

findIndex :: (a -> Bool) -> [a] -> Maybe Int

The findIndex function takes a predicate and a list and returns the index of the first element in the list satisfying the predicate, or Nothing if there is no such element.

findIndices :: (a -> Bool) -> [a] -> [Int]

The findIndices function extends findIndex, by returning the indices of all elements satisfying the predicate, in ascending order.

20.8 Zipping and unzipping lists

zip :: [a] -> [b] -> [(a, b)]

zip takes two lists and returns a list of corresponding pairs. If one input list is short, excess elements of the longer list are discarded.

zip3 :: [a] -> [b] -> [c] -> [(a, b, c)]

zip3 takes three lists and returns a list of triples, analogous to zip.

zip4 :: [a] -> [b] -> [c] -> [d] -> [(a, b, c, d)]

The zip4 function takes four lists and returns a list of quadruples, analogous to zip.

zip5 :: [a] -> [b] -> [c] -> [d] -> [e] -> [(a, b, c, d, e)]

The zip5 function takes five lists and returns a list of five-tuples, analogous to zip.

zip6 :: [a]

-> [b] -> [c] -> [d] -> [e] -> [f] -> [(a, b, c, d, e, f)]

The zip6 function takes six lists and returns a list of six-tuples, analogous to zip.

zip7 :: [a]

-> [b]

-> [c] -> [d] -> [e] -> [f] -> [g] -> [(a, b, c, d, e, f, g)]

The zip7 function takes seven lists and returns a list of seven-tuples, analogous to zip.

zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]

zipWith generalises zip by zipping with the function given as the first argument, instead of a tupling function. For example, zipWith (+) is applied to two lists to produce the list of corresponding sums.

zipWith3 :: (a -> b -> c -> d) -> [a] -> [b] -> [c] -> [d]

The zipWith3 function takes a function which combines three elements, as well as three lists and returns a list of their point-wise combination, analogous to zipWith.

zipWith4 :: (a -> b -> c -> d -> e)

-> [a] -> [b] -> [c] -> [d] -> [e]

The zipWith4 function takes a function which combines four elements, as well as four lists and returns a list of their point-wise combination, analogous to zipWith.

zipWith5 :: (a -> b -> c -> d -> e -> f)

-> [a] -> [b] -> [c] -> [d] -> [e] -> [f]

The zipWith5 function takes a function which combines five elements, as well as five lists and returns a list of their point-wise combination, analogous to zipWith.

zipWith6 :: (a -> b -> c -> d -> e -> f -> g)

-> [a] -> [b] -> [c] -> [d] -> [e] -> [f] -> [g]

The zipWith6 function takes a function which combines six elements, as well as six lists and returns a list of their point-wise combination, analogous to zipWith.

zipWith7 :: (a -> b -> c -> d -> e -> f -> g -> h)

-> [a] -> [b] -> [c] -> [d] -> [e] -> [f] -> [g] -> [h]

The zipWith7 function takes a function which combines seven elements, as well as seven lists and returns a list of their point-wise combination, analogous to zipWith.

unzip :: [(a, b)] -> ([a], [b])

unzip transforms a list of pairs into a list of first components and a list of second components.

unzip3 :: [(a, b, c)] -> ([a], [b], [c])

The unzip3 function takes a list of triples and returns three lists, analogous to unzip.

unzip4 :: [(a, b, c, d)] -> ([a], [b], [c], [d])

The unzip4 function takes a list of quadruples and returns four lists, analogous to unzip.

unzip5 :: [(a, b, c, d, e)] -> ([a], [b], [c], [d], [e])

The unzip5 function takes a list of five-tuples and returns five lists, analogous to unzip.

unzip6 :: [(a, b, c, d, e, f)] -> ([a], [b], [c], [d], [e], [f])

The unzip6 function takes a list of six-tuples and returns six lists, analogous to unzip.

unzip7 :: [(a, b, c, d, e, f, g)]

-> ([a], [b], [c], [d], [e], [f], [g])

The unzip7 function takes a list of seven-tuples and returns seven lists, analogous to unzip.

20.9 Special lists

20.9.1 Functions on strings

lines :: String -> [String]

lines breaks a string up into a list of strings at newline characters. The resulting strings do not contain newlines.

words :: String -> [String]

words breaks a string up into a list of words, which were delimited by white space.

unlines :: [String] -> String

unlines is an inverse operation to lines. It joins lines, after appending a terminating newline to each.

unwords :: [String] -> String

unwords is an inverse operation to words. It joins words with separating spaces.

20.9.2 ”Set” operations

nub :: Eq a => [a] -> [a]

O(n^2). The nub function removes duplicate elements from a list. In particular, it keeps only the first occurrence of each element. (The name nub means ‘essence’.) It is a special case of nubBy, which allows the programmer to supply their own equality test.

delete :: Eq a => a -> [a] -> [a]

deletex removes the first occurrence of x from its list argument. For example,

 delete 'a' "banana" == "bnana"

It is a special case of deleteBy, which allows the programmer to supply their own equality test.

(\\) :: Eq a => [a] -> [a] -> [a]

The \\ function is list difference ((non-associative). In the result of xs\\ys, the first occurrence of each element of ys in turn (if any) has been removed from xs. Thus

 (xs ++ ys) \\ xs == ys.

It is a special case of deleteFirstsBy, which allows the programmer to supply their own equality test.

union :: Eq a => [a] -> [a] -> [a]

The union function returns the list union of the two lists. For example,

 "dog" ‘union‘ "cow" == "dogcw"

Duplicates, and elements of the first list, are removed from the the second list, but if the first list contains duplicates, so will the result. It is a special case of unionBy, which allows the programmer to supply their own equality test.

intersect :: Eq a => [a] -> [a] -> [a]

The intersect function takes the list intersection of two lists. For example,

 [1,2,3,4] ‘intersect‘ [2,4,6,8] == [2,4]

If the first list contains duplicates, so will the result.

 [1,2,2,3,4] ‘intersect‘ [6,4,4,2] == [2,2,4]

It is a special case of intersectBy, which allows the programmer to supply their own equality test.

20.9.3 Ordered lists

sort :: Ord a => [a] -> [a]

The sort function implements a stable sorting algorithm. It is a special case of sortBy, which allows the programmer to supply their own comparison function.

insert :: Ord a => a -> [a] -> [a]

The insert function takes an element and a list and inserts the element into the list at the last position where it is still less than or equal to the next element. In particular, if the list is sorted before the call, the result will also be sorted. It is a special case of insertBy, which allows the programmer to supply their own comparison function.

20.10 Generalized functions

20.10.1 The ”By” operations

By convention, overloaded functions have a non-overloaded counterpart whose name is suffixed with ‘By’.

20.10.1.1 User-supplied equality (replacing an Eq context)

The predicate is assumed to define an equivalence.

nubBy :: (a -> a -> Bool) -> [a] -> [a]

The nubBy function behaves just like nub, except it uses a user-supplied equality predicate instead of the overloaded == function.

deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]

The deleteBy function behaves like delete, but takes a user-supplied equality predicate.

deleteFirstsBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]

The deleteFirstsBy function takes a predicate and two lists and returns the first list with the first occurrence of each element of the second list removed.

unionBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]

The unionBy function is the non-overloaded version of union.

intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]

The intersectBy function is the non-overloaded version of intersect.

groupBy :: (a -> a -> Bool) -> [a] -> [[a]]

The groupBy function is the non-overloaded version of group.

20.10.1.2 User-supplied comparison (replacing an Ord context)

The function is assumed to define a total ordering.

sortBy :: (a -> a -> Ordering) -> [a] -> [a]

The sortBy function is the non-overloaded version of sort.

insertBy :: (a -> a -> Ordering) -> a -> [a] -> [a]

The non-overloaded version of insert.

maximumBy :: (a -> a -> Ordering) -> [a] -> a

The maximumBy function takes a comparison function and a list and returns the greatest element of the list by the comparison function. The list must be finite and non-empty.

minimumBy :: (a -> a -> Ordering) -> [a] -> a

The minimumBy function takes a comparison function and a list and returns the least element of the list by the comparison function. The list must be finite and non-empty.

20.10.2 The ”generic” operations

The prefix ‘generic’ indicates an overloaded function that is a generalized version of a Prelude function.

genericLength :: Num i => [b] -> i

The genericLength function is an overloaded version of length. In particular, instead of returning an Int, it returns any type which is an instance of Num. It is, however, less efficient than length.

genericTake :: Integral i => i -> [a] -> [a]

The genericTake function is an overloaded version of take, which accepts any Integral value as the number of elements to take.

genericDrop :: Integral i => i -> [a] -> [a]

The genericDrop function is an overloaded version of drop, which accepts any Integral value as the number of elements to drop.

genericSplitAt :: Integral i => i -> [b] -> ([b], [b])

The genericSplitAt function is an overloaded version of splitAt, which accepts any Integral value as the position at which to split.

genericIndex :: Integral a => [b] -> a -> b

The genericIndex function is an overloaded version of !!, which accepts any Integral value as the index.

genericReplicate :: Integral i => i -> a -> [a]

The genericReplicate function is an overloaded version of replicate, which accepts any Integral value as the number of repetitions to make.