Data.List.Infinite

Modern lightweight library for infinite lists with fusion:

• API similar to Data.List.
• No non-boot dependencies.
• Top performance, driven by fusion.
• Avoid dangerous instances like Foldable.
• Use NonEmpty where applicable.
• Use Word for indices.
• Be lazy, but not too lazy.

{-# LANGUAGE PostfixOperators #-}
import Data.List.Infinite (Infinite(..), (...), (....))
import qualified Data.List.Infinite as Inf

Construction

data Infinite a

Type of infinite lists.

In terms of recursion schemes, Infinite a is a fix point of the base functor (a,), foldr is a catamorphism and unfoldr is an anamorphism.

Constructors

a :< (Infinite a) infixr 5

Instances

Applicative Infinite

This instance operates pointwise, similar to ZipList.

Defined in Data.List.Infinite

Methods

pure :: a -> Infinite a

(<*>) :: Infinite (a -> b) -> Infinite a -> Infinite b

liftA2 :: (a -> b -> c) -> Infinite a -> Infinite b -> Infinite c

(*>) :: Infinite a -> Infinite b -> Infinite b

(<*) :: Infinite a -> Infinite b -> Infinite a

Functor Infinite

Just a pointwise map.

Defined in Data.List.Infinite

Methods

fmap :: (a -> b) -> Infinite a -> Infinite b

(<$) :: a -> Infinite b -> Infinite a

Monad Infinite

ZipList cannot be made a lawful Monad, but Infinite, being a Representable, can. Namely, join picks up a diagonal of an infinite matrix of Infinite (Infinite a). Bear in mind that this instance gets slow very soon because of linear indexing, so it is not recommended to be used in practice.

Defined in Data.List.Infinite

Methods

(>>=) :: Infinite a -> (a -> Infinite b) -> Infinite b

(>>) :: Infinite a -> Infinite b -> Infinite b

return :: a -> Infinite a

Elimination

head :: Infinite a -> a

Get the first elements of an infinite list.

tail :: Infinite a -> Infinite a

Get the elements of an infinite list after the first one.

uncons :: Infinite a -> (a, Infinite a)

Split an infinite list into its head and tail.

toList :: Infinite a -> [a]

Convert to a list. Use cycle to go in the opposite direction.

foldr :: (a -> b -> b) -> Infinite a -> b

Right-associative fold of an infinite list, necessarily lazy in the accumulator. Any unconditional attempt to force the accumulator even to the weak head normal form (WHNF) will hang the computation. E. g., the following definition isn't productive:

import Data.List.NonEmpty (NonEmpty(..))
toNonEmpty = foldr (\a (x :| xs) -> a :| x : xs) :: Infinite a -> NonEmpty a

One should use lazy patterns, e. g.,

toNonEmpty = foldr (\a ~(x :| xs) -> a :| x : xs)

This is a catamorphism on infinite lists.

Traversals

map :: (a -> b) -> Infinite a -> Infinite b

Apply a function to every element of an infinite list.

scanl :: (b -> a -> b) -> b -> Infinite a -> Infinite b

Fold an infinite list from the left and return a list of successive reductions, starting from the initial accumulator:

scanl f acc (x1 :< x2 :< ...) = acc :< f acc x1 :< f (f acc x1) x2 :< ...

scanl' :: (b -> a -> b) -> b -> Infinite a -> Infinite b

Same as scanl, but strict in accumulator.

scanl1 :: (a -> a -> a) -> Infinite a -> Infinite a

Fold an infinite list from the left and return a list of successive reductions, starting from the first element:

scanl1 f (x0 :< x1 :< x2 :< ...) = x0 :< f x0 x1 :< f (f x0 x1) x2 :< ...

mapAccumL :: (acc -> x -> (acc, y)) -> acc -> Infinite x -> Infinite y

Fold an infinite list from the left and return a list of successive reductions, keeping accumulator in a state:

mapAccumL f acc0 (x1 :< x2 :< ...) =
  let (acc1, y1) = f acc0 x1 in
    let (acc2, y2) = f acc1 x2 in
      ...
        y1 :< y2 :< ...

If you are looking how to traverse with a state, look no further.

Transformations

concat :: Infinite (NonEmpty a) -> Infinite a

Flatten out an infinite list of non-empty lists.

The peculiar type with NonEmpty is to guarantee that concat is productive and results in an infinite list. Otherwise the concatenation of infinitely many [a] could still be a finite list.

concatMap :: (a -> NonEmpty b) -> Infinite a -> Infinite b

First map every element, then concat.

The peculiar type with NonEmpty is to guarantee that concatMap is productive and results in an infinite list. Otherwise the concatenation of infinitely many [b] could still be a finite list.

intersperse :: a -> Infinite a -> Infinite a

Insert an element between adjacent elements of an infinite list.

intercalate :: NonEmpty a -> Infinite [a] -> Infinite a

Insert a non-empty list between adjacent elements of an infinite list, and subsequently flatten it out.

The peculiar type with NonEmpty is to guarantee that intercalate is productive and results in an infinite list. If separator is an empty list, concatenation of infinitely many [a] could still be a finite list.

interleave :: Infinite a -> Infinite a -> Infinite a

Interleave two infinite lists.

transpose :: Functor f => f (Infinite a) -> Infinite (f a)

Transpose rows and columns of an argument.

This is actually distribute from Distributive type class in disguise.

subsequences :: Infinite a -> Infinite [a]

Generate an infinite list of all subsequences of the argument.

subsequences1 :: Infinite a -> Infinite (NonEmpty a)

Generate an infinite list of all non-empty subsequences of the argument.

permutations :: Infinite a -> Infinite (Infinite a)

Generate an infinite list of all permutations of the argument.

Building

(...) :: Enum a => a -> Infinite a infix 0

Generate an infinite progression, starting from a given element, similar to [x..]. For better user experience consider enabling {-# LANGUAGE PostfixOperators #-}:

>>> :set -XPostfixOperators
>>> Data.List.Infinite.take 10 (0...)
[0,1,2,3,4,5,6,7,8,9]

Beware that for finite types (...) applies cycle atop of [x..]:

>>> :set -XPostfixOperators
>>> Data.List.Infinite.take 10 (EQ...)
[EQ,GT,EQ,GT,EQ,GT,EQ,GT,EQ,GT]

Remember that Int is a finite type as well. One is unlikely to hit this on a 64-bit architecture, but on a 32-bit machine it's fairly possible to traverse ((0 :: Int) ...) far enough to encounter 0 again.

(....) :: Enum a => (a, a) -> Infinite a infix 0

Generate an infinite arithmetic progression, starting from given elements, similar to [x,y..]. For better user experience consider enabling {-# LANGUAGE PostfixOperators #-}:

>>> :set -XPostfixOperators
>>> Data.List.Infinite.take 10 ((1,3)....)
[1,3,5,7,9,11,13,15,17,19]

Beware that for finite types (....) applies cycle atop of [x,y..]:

>>> :set -XPostfixOperators
>>> Data.List.Infinite.take 10 ((EQ,GT)....)
[EQ,GT,EQ,GT,EQ,GT,EQ,GT,EQ,GT]

Remember that Int is a finite type as well: for a sufficiently large step of progression y - x one may observe ((x :: Int, y)....) cycling back to emit x fairly soon.

iterate :: (a -> a) -> a -> Infinite a

Generate an infinite list of repeated applications.

iterate' :: (a -> a) -> a -> Infinite a

Same as iterate, but strict in accumulator.

unfoldr :: (b -> (a, b)) -> b -> Infinite a

Build an infinite list from a seed value.

This is an anamorphism on infinite lists.

tabulate :: (Word -> a) -> Infinite a

Generate an infinite list of f 0, f 1, f 2...

tabulate and (!!) witness that Infinite is Representable.

repeat :: a -> Infinite a

Repeat the same element ad infinitum.

cycle :: NonEmpty a -> Infinite a

Repeat a non-empty list ad infinitum. If you were looking for something like fromList :: [a] -> Infinite a, look no further.

It would be less annoying to take [a] instead of NonEmpty a, but we strive to avoid partial functions.

Sublists

prependList :: [a] -> Infinite a -> Infinite a

Prepend a list to an infinite list.

take :: Int -> Infinite a -> [a]

Take a prefix of given length.

drop :: Int -> Infinite a -> Infinite a

Drop a prefix of given length.

splitAt :: Int -> Infinite a -> ([a], Infinite a)

Split an infinite list into a prefix of given length and the rest.

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

Take the longest prefix satisfying a predicate.

dropWhile :: (a -> Bool) -> Infinite a -> Infinite a

Drop the longest prefix satisfying a predicate.

This function isn't productive (e. g., head . dropWhile f won't terminate), if all elements of the input list satisfy the predicate.

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

Split an infinite list into the longest prefix satisfying a predicate and the rest.

This function isn't productive in the second component of the tuple (e. g., head . snd . span f won't terminate), if all elements of the input list satisfy the predicate.

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

Split an infinite list into the longest prefix not satisfying a predicate and the rest.

This function isn't productive in the second component of the tuple (e. g., head . snd . break f won't terminate), if no elements of the input list satisfy the predicate.

group :: Eq a => Infinite a -> Infinite (NonEmpty a)

Group consecutive equal elements.

inits :: Infinite a -> Infinite [a]

Generate all prefixes of an infinite list.

inits1 :: Infinite a -> Infinite (NonEmpty a)

Generate all non-empty prefixes of an infinite list.

tails :: Infinite a -> Infinite (Infinite a)

Generate all suffixes of an infinite list.

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

Check whether a list is a prefix of an infinite list.

stripPrefix :: Eq a => [a] -> Infinite a -> Maybe (Infinite a)

If a list is a prefix of an infinite list, strip it and return the rest. Otherwise return Nothing.

Searching

filter :: (a -> Bool) -> Infinite a -> Infinite a

Filter an infinite list, removing elements which does not satisfy a predicate.

This function isn't productive (e. g., head . filter f won't terminate), if no elements of the input list satisfy the predicate.

A common objection is that since it could happen that no elements of the input satisfy the predicate, the return type should be [a] instead of Infinite a. This would not however make filter any more productive. Note that such hypothetical filter could not ever generate [] constructor, only (:), so we would just have a more lax type gaining nothing instead. Same reasoning applies to other filtering / partitioning / searching functions.

lookup :: Eq a => a -> Infinite (a, b) -> b

Find the first pair, whose first component is equal to the first argument, and return the second component. If there is nothing to be found, this function will hang indefinitely.

find :: (a -> Bool) -> Infinite a -> a

Find the first element, satisfying a predicate. If there is nothing to be found, this function will hang indefinitely.

mapMaybe :: (a -> Maybe b) -> Infinite a -> Infinite b

Apply a function to every element of an infinite list and collect Just results.

This function isn't productive (e. g., head . mapMaybe f won't terminate), if no elements of the input list result in Just.

Since: 0.1.1

catMaybes :: Infinite (Maybe a) -> Infinite a

Keep only Just elements.

This function isn't productive (e. g., head . catMaybes won't terminate), if no elements of the input list are Just.

Since: 0.1.1

partition :: (a -> Bool) -> Infinite a -> (Infinite a, Infinite a)

Split an infinite list into two infinite lists: the first one contains elements, satisfying a predicate, and the second one the rest.

This function isn't productive in the first component of the tuple (e. g., head . fst . partition f won't terminate), if no elements of the input list satisfy the predicate. Same for the second component, if all elements of the input list satisfy the predicate.

mapEither :: (a -> Either b c) -> Infinite a -> (Infinite b, Infinite c)

Apply a function to every element of an infinite list and separate Left and Right results.

This function isn't productive (e. g., head . fst . mapEither f won't terminate), if no elements of the input list result in Left or Right.

Since: 0.1.1

partitionEithers :: Infinite (Either a b) -> (Infinite a, Infinite b)

Separate Left and Right elements.

This function isn't productive (e. g., head . fst . partitionEithers won't terminate), if no elements of the input list are Left or Right.

Since: 0.1.1

Indexing

(!!) :: Infinite a -> Word -> a infixl 9

Return n-th element of an infinite list. On contrary to Data.List.!!, this function takes Word instead of Int to avoid error on negative arguments.

This is actually index from Representable type class in disguise.

elemIndex :: Eq a => a -> Infinite a -> Word

Return an index of the first element, equal to a given. If there is nothing to be found, this function will hang indefinitely.

elemIndices :: Eq a => a -> Infinite a -> Infinite Word

Return indices of all elements, equal to a given.

This function isn't productive (e. g., head . elemIndices f won't terminate), if no elements of the input list are equal the given one.

findIndex :: (a -> Bool) -> Infinite a -> Word

Return an index of the first element, satisfying a predicate. If there is nothing to be found, this function will hang indefinitely.

findIndices :: (a -> Bool) -> Infinite a -> Infinite Word

Return indices of all elements, satisfying a predicate.

This function isn't productive (e. g., head . elemIndices f won't terminate), if no elements of the input list satisfy the predicate.

Zipping

zip :: Infinite a -> Infinite b -> Infinite (a, b)

Zip two infinite lists.

zipWith :: (a -> b -> c) -> Infinite a -> Infinite b -> Infinite c

Zip two infinite lists with a given function.

zip3 :: Infinite a -> Infinite b -> Infinite c -> Infinite (a, b, c)

Zip three infinite lists.

zipWith3 :: (a -> b -> c -> d) -> Infinite a -> Infinite b -> Infinite c -> Infinite d

Zip three infinite lists with a given function.

zip4 :: Infinite a -> Infinite b -> Infinite c -> Infinite d -> Infinite (a, b, c, d)

Zip four infinite lists.

zipWith4 :: (a -> b -> c -> d -> e) -> Infinite a -> Infinite b -> Infinite c -> Infinite d -> Infinite e

Zip four infinite lists with a given function.

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

Zip five infinite lists.

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

Zip five infinite lists with a given function.

zip6 :: Infinite a -> Infinite b -> Infinite c -> Infinite d -> Infinite e -> Infinite f -> Infinite (a, b, c, d, e, f)

Zip six infinite lists.

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

Zip six infinite lists with a given function.

zip7 :: Infinite a -> Infinite b -> Infinite c -> Infinite d -> Infinite e -> Infinite f -> Infinite g -> Infinite (a, b, c, d, e, f, g)

Zip seven infinite lists.

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

Zip seven infinite lists with a given function.

unzip :: Infinite (a, b) -> (Infinite a, Infinite b)

Unzip an infinite list of tuples.

unzip3 :: Infinite (a, b, c) -> (Infinite a, Infinite b, Infinite c)

Unzip an infinite list of triples.

unzip4 :: Infinite (a, b, c, d) -> (Infinite a, Infinite b, Infinite c, Infinite d)

Unzip an infinite list of quadruples.

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

Unzip an infinite list of quintuples.

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

Unzip an infinite list of sextuples.

unzip7 :: Infinite (a, b, c, d, e, f, g) -> (Infinite a, Infinite b, Infinite c, Infinite d, Infinite e, Infinite f, Infinite g)

Unzip an infinite list of septuples.

Functions on strings

lines :: Infinite Char -> Infinite [Char]

Split an infinite string into lines, by \n. Empty lines are preserved.

In contrast to their counterparts from Data.List, it holds that unlines . lines = id.

words :: Infinite Char -> Infinite (NonEmpty Char)

Split an infinite string into words, by any isSpace symbol. Leading spaces are removed and, as underlined by the return type, repeated spaces are treated as a single delimiter.

unlines :: Infinite [Char] -> Infinite Char

Concatenate lines together with \n.

In contrast to their counterparts from Data.List, it holds that unlines . lines = id.

unwords :: Infinite (NonEmpty Char) -> Infinite Char

Concatenate words together with a space.

The function is meant to be a counterpart of with words. If you need to concatenate together Infinite [Char], use intercalate (pure ' ').

Set operations

nub :: Eq a => Infinite a -> Infinite a

Remove duplicate from a list, keeping only the first occurrence of each element.

delete :: Eq a => a -> Infinite a -> Infinite a

Remove all occurrences of an element from an infinite list.

(\\) :: Eq a => Infinite a -> [a] -> Infinite a

Take an infinite list and remove the first occurrence of every element of a finite list.

union :: Eq a => [a] -> Infinite a -> Infinite a

Union of a finite and an infinite list. It contains the finite list as a prefix and afterwards all non-duplicate elements of the infinite list, which are not members of the finite list.

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

Return all elements of an infinite list, which are simultaneously members of a finite list.

Ordered lists

insert :: Ord a => a -> Infinite a -> Infinite a

Insert an element at the first position where it is less than or equal to the next one. If the input was sorted, the output remains sorted as well.

Generalized functions

nubBy :: (a -> a -> Bool) -> Infinite a -> Infinite a

Overloaded version of nub.

deleteBy :: (a -> b -> Bool) -> a -> Infinite b -> Infinite b

Overloaded version of delete.

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

Overloaded version of (\\).

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

Overloaded version of union.

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

Overloaded version of intersect.

groupBy :: (a -> a -> Bool) -> Infinite a -> Infinite (NonEmpty a)

Overloaded version of group.

insertBy :: (a -> a -> Ordering) -> a -> Infinite a -> Infinite a

Overloaded version of insert.

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

Take a prefix of given length.

genericDrop :: Integral i => i -> Infinite a -> Infinite a

Drop a prefix of given length.

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

Split an infinite list into a prefix of given length and the rest.

Orphan instances

Applicative Infinite

This instance operates pointwise, similar to ZipList.

Methods

pure :: a -> Infinite a

(<*>) :: Infinite (a -> b) -> Infinite a -> Infinite b

liftA2 :: (a -> b -> c) -> Infinite a -> Infinite b -> Infinite c

(*>) :: Infinite a -> Infinite b -> Infinite b

(<*) :: Infinite a -> Infinite b -> Infinite a

Functor Infinite

Just a pointwise map.

Methods

fmap :: (a -> b) -> Infinite a -> Infinite b

(<$) :: a -> Infinite b -> Infinite a

Monad Infinite

ZipList cannot be made a lawful Monad, but Infinite, being a Representable, can. Namely, join picks up a diagonal of an infinite matrix of Infinite (Infinite a). Bear in mind that this instance gets slow very soon because of linear indexing, so it is not recommended to be used in practice.

Methods

(>>=) :: Infinite a -> (a -> Infinite b) -> Infinite b

(>>) :: Infinite a -> Infinite b -> Infinite b

return :: a -> Infinite a