List

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.

\(\mathcal{O}(n)\). 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]

>>> filter odd [1, 2, 3]
[1,3]

\(\mathcal{O}(\min(m,n))\). zip takes two lists and returns a list of corresponding pairs.

>>> zip [1, 2] ['a', 'b']
[(1, 'a'), (2, 'b')]

If one input list is shorter than the other, excess elements of the longer list are discarded, even if one of the lists is infinite:

>>> zip [1] ['a', 'b']
[(1, 'a')]
>>> zip [1, 2] ['a']
[(1, 'a')]
>>> zip [] [1..]
[]
>>> zip [1..] []
[]

zip is right-lazy:

>>> zip [] _|_
[]
>>> zip _|_ []
_|_

zip is capable of list fusion, but it is restricted to its first list argument and its resulting list.

\(\mathcal{O}(n)\). map f 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, ...]

>>> map (+1) [1, 2, 3]
[2,3,4]

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]

Sort a list by comparing the results of a key function applied to each element. sortOn f is equivalent to sortBy (comparing f), but has the performance advantage of only evaluating f once for each element in the input list. This is called the decorate-sort-undecorate paradigm, or Schwartzian transform.

Elements are arranged from lowest to highest, keeping duplicates in the order they appeared in the input.

>>> sortOn fst [(2, "world"), (4, "!"), (1, "Hello")]
[(1,"Hello"),(2,"world"),(4,"!")]

Since: base-4.8.0.0

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

>>> sortBy (\(a,_) (b,_) -> compare a b) [(2, "world"), (4, "!"), (1, "Hello")]
[(1,"Hello"),(2,"world"),(4,"!")]

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.

Elements are arranged from lowest to highest, keeping duplicates in the order they appeared in the input.

>>> sort [1,6,4,3,2,5]
[1,2,3,4,5,6]

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

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

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

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

\(\mathcal{O}(n)\). The tails function returns all final segments of the argument, longest first. For example,

>>> tails "abc"
["abc","bc","c",""]

Note that tails has the following strictness property: tails _|_ = _|_ : _|_

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

>>> inits "abc"
["","a","ab","abc"]

Note that inits has the following strictness property: inits (xs ++ _|_) = inits xs ++ _|_

In particular, inits _|_ = [] : _|_

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.

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

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

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

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

\(\mathcal{O}(n)\). 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.

>>> genericLength [1, 2, 3] :: Int
3
>>> genericLength [1, 2, 3] :: Float
3.0

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]]

If some of the rows are shorter than the following rows, their elements are skipped:

>>> transpose [[10,11],[20],[],[30,31,32]]
[[10,20,30],[11,31],[32]]

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 ", " ["Lorem", "ipsum", "dolor"]
"Lorem, ipsum, dolor"

\(\mathcal{O}(n)\). 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"

\(\mathcal{O}(\min(m,n))\). The isPrefixOf function takes two lists and returns True iff the first list is a prefix of the second.

>>> "Hello" `isPrefixOf` "Hello World!"
True

>>> "Hello" `isPrefixOf` "Wello Horld!"
False

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

>>> unzip3 []
([],[],[])
>>> unzip3 [(1, 'a', True), (2, 'b', False)]
([1,2],"ab",[True,False])

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

>>> unzip []
([],[])
>>> unzip [(1, 'a'), (2, 'b')]
([1,2],"ab")

\(\mathcal{O}(\min(m,n))\). zipWith generalises zip by zipping with the function given as the first argument, instead of a tupling function.

zipWith (,) xs ys == zip xs ys
zipWith f [x1,x2,x3..] [y1,y2,y3..] == [f x1 y1, f x2 y2, f x3 y3..]

For example, zipWith (+) is applied to two lists to produce the list of corresponding sums:

>>> zipWith (+) [1, 2, 3] [4, 5, 6]
[5,7,9]

zipWith is right-lazy:

>>> zipWith f [] _|_
[]

zipWith is capable of list fusion, but it is restricted to its first list argument and its resulting list.

zip3 takes three lists and returns a list of triples, analogous to zip. It is capable of list fusion, but it is restricted to its first list argument and its resulting list.

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

>>> reverse []
[]
>>> reverse [42]
[42]
>>> reverse [2,5,7]
[7,5,2]
>>> reverse [1..]
* Hangs forever *

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],[])

break p is equivalent to span (not . p).

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])

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

splitAt n 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) when n is not _|_ (splitAt _|_ xs = _|_). splitAt is an instance of the more general genericSplitAt, in which n may be of any integral type.

drop n 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 genericDrop, in which n may be of any integral type.

take n, 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 genericTake, in which n may be of any integral type.

dropWhile p xs returns the suffix remaining after takeWhile p 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]

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]
[]

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

>>> replicate 0 True
[]
>>> replicate (-1) True
[]
>>> replicate 4 True
[True,True,True,True]

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

>>> repeat 17
[17,17,17,17,17,17,17,17,17...

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

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

Note that iterate is lazy, potentially leading to thunk build-up if the consumer doesn't force each iterate. See iterate' for a strict variant of this function.

>>> iterate not True
[True,False,True,False...
>>> iterate (+3) 42
[42,45,48,51,54,57,60,63...

\(\mathcal{O}(n)\). scanr1 is a variant of scanr that has no starting value argument.

>>> scanr1 (+) [1..4]
[10,9,7,4]
>>> scanr1 (+) []
[]
>>> scanr1 (-) [1..4]
[-2,3,-1,4]
>>> scanr1 (&&) [True, False, True, True]
[False,False,True,True]
>>> scanr1 (||) [True, True, False, False]
[True,True,False,False]
>>> scanr1 (+) [1..]
* Hangs forever *

\(\mathcal{O}(n)\). scanr is the right-to-left dual of scanl. Note that the order of parameters on the accumulating function are reversed compared to scanl. Also note that

head (scanr f z xs) == foldr f z xs.

>>> scanr (+) 0 [1..4]
[10,9,7,4,0]
>>> scanr (+) 42 []
[42]
>>> scanr (-) 100 [1..4]
[98,-97,99,-96,100]
>>> scanr (\nextChar reversedString -> nextChar : reversedString) "foo" ['a', 'b', 'c', 'd']
["abcdfoo","bcdfoo","cdfoo","dfoo","foo"]
>>> scanr (+) 0 [1..]
* Hangs forever *

\(\mathcal{O}(n)\). A strict version of scanl.

\(\mathcal{O}(n)\). scanl1 is a variant of scanl that has no starting value argument:

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

>>> scanl1 (+) [1..4]
[1,3,6,10]
>>> scanl1 (+) []
[]
>>> scanl1 (-) [1..4]
[1,-1,-4,-8]
>>> scanl1 (&&) [True, False, True, True]
[True,False,False,False]
>>> scanl1 (||) [False, False, True, True]
[False,False,True,True]
>>> scanl1 (+) [1..]
* Hangs forever *

\(\mathcal{O}(n)\). 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

>>> scanl (+) 0 [1..4]
[0,1,3,6,10]
>>> scanl (+) 42 []
[42]
>>> scanl (-) 100 [1..4]
[100,99,97,94,90]
>>> scanl (\reversedString nextChar -> nextChar : reversedString) "foo" ['a', 'b', 'c', 'd']
["foo","afoo","bafoo","cbafoo","dcbafoo"]
>>> scanl (+) 0 [1..]
* Hangs forever *

\(\mathcal{O}(1)\). Decompose a list into its head and tail.

• If the list is empty, returns Nothing.
• If the list is non-empty, returns Just (x, xs), where x is the head of the list and xs its tail.

>>> uncons []
Nothing
>>> uncons [1]
Just (1,[])
>>> uncons [1, 2, 3]
Just (1,[2,3])

Since: base-4.8.0.0

Creates an infinite list from a finite list by appending the list to itself infinite times (i.e. by cycling the list). Unlike cycle from Data.List, this implementation doesn't throw error on empty lists, but returns an empty list instead.

>>> cycle []
[]
>>> take 10 $ cycle [1,2,3]
[1,2,3,1,2,3,1,2,3,1]

The sortWith function sorts a list of elements using the user supplied function to project something out of each element