# List Comprehensions

# Basic List Comprehensions

Haskell has list comprehensions, which are a lot like set comprehensions in math and similar implementations in imperative languages such as Python and JavaScript. At their most basic, list comprehensions take the following form.

[ x | x <- someList ]

For example

[ x | x <- [1..4] ]    -- [1,2,3,4]

Functions can be directly applied to x as well:

[ f x | x <- someList ]

This is equivalent to:

map f someList

Example:

[ x+1 | x <- [1..4]]    -- [2,3,4,5]

# Patterns in Generator Expressions

However, x in the generator expression is not just variable, but can be any pattern. In cases of pattern mismatch the generated element is skipped over, and processing of the list continues with the next element, thus acting like a filter:

[x | Just x <- [Just 1, Nothing, Just 3]]     -- [1, 3]

A generator with a variable x in its pattern creates new scope containing all the expressions on its right, where x is defined to be the generated element.

This means that guards can be coded as

[ x | x <- [1..4], even x] ==
[ x | x <- [1..4], () <- [() | even x]] ==
[ x | x <- [1..4], () <- if even x then [()] else []]

# Guards

Another feature of list comprehensions is guards, which also act as filters. Guards are Boolean expressions and appear on the right side of the bar in a list comprehension.

Their most basic use is

[x    | p x]   ===   if p x then [x] else []

Any variable used in a guard must appear on its left in the comprehension, or otherwise be in scope. So,

[ f x | x <- list, pred1 x y, pred2 x]     -- `y` must be defined in outer scope

which is equivalent to

map f (filter pred2 (filter (\x -> pred1 x y) list))          -- or,

-- ($ list) (filter (`pred1` y) >>> filter pred2 >>> map f)     

-- list >>= (\x-> [x | pred1 x y]) >>= (\x-> [x | pred2 x]) >>= (\x -> [f x])

(the >>= operator is infixl 1, i.e. it associates (is parenthesized) to the left). Examples:

[ x       | x <- [1..4], even x]           -- [2,4]

[ x^2 + 1 | x <- [1..100], even x ]        -- map (\x -> x^2 + 1) (filter even [1..100])

# Parallel Comprehensions

With Parallel List Comprehensions language extension,

[(x,y) | x <- xs | y <- ys]

is equivalent to

zip xs ys

Example:

[(x,y) | x <- [1,2,3] | y <- [10,20]] 

-- [(1,10),(2,20)]

# Do Notation

Any list comprehension can be correspondingly coded with list monad's do notation.

[f x | x <- xs]                 f  <$> xs         do { x <- xs ; return (f x) }

[f x | f <- fs, x <- xs]        fs <*> xs         do { f <- fs ; x <- xs ; return (f x) }

[y   | x <- xs, y <- f x]       f  =<< xs         do { x <- xs ; y <- f x ; return y }

The guards can be handled using Control.Monad.guard:

[x   | x <- xs, even x]                           do { x <- xs ; guard (even x) ; return x }

# Nested Generators

List comprehensions can also draw elements from multiple lists, in which case the result will be the list of every possible combination of the two elements, as if the two lists were processed in the nested fashion. For example,

[ (a,b) | a <- [1,2,3], b <- ['a','b'] ]

-- [(1,'a'), (1,'b'), (2,'a'), (2,'b'), (3,'a'), (3,'b')]

# Local Bindings

List comprehensions can introduce local bindings for variables to hold some interim values:

[(x,y) | x <- [1..4], let y=x*x+1, even y]    -- [(1,2),(3,10)]

Same effect can be achieved with a trick,

[(x,y) | x <- [1..4], y <- [x*x+1], even y]   -- [(1,2),(3,10)]

The let in list comprehensions is recursive, as usual. But generator bindings are not, which enables shadowing:

[x | x <- [1..4], x <- [x*x+1], even x]       -- [2,10]