notes/pl/hs/haskell-fun-prelude.txt

1. const

const     :: a -> b -> a
const x _ =  x


2. flip

flip       :: (a -> b -> c) -> b -> a -> c
flip f x y =  f y x


Parameters flipping

flip' :: (a -> b -> c) -> (b -> a -> c)
flip' f = g
  where g x y = f y x

or

flip' f y x = f x y




3. enumFromTo

class  Enum a   where
  enumFromTo     :: a -> a -> [a]
  enumFromTo x y =  map toEnum [fromEnum x .. fromEnum y]
  ...
  toEnum   :: Int -> a
  fromEnum :: a -> Int

-- Used in Haskell's translation of [n..m].


4. until

until                   :: (a -> Bool) -> (a -> a) -> a -> a
until p f x | p x       =  x
            | otherwise =  until p f (f x)

5. uncurry

uncurry     :: (a -> b -> c) -> ((a, b) -> c)
uncurry f p =  f (fst p) (snd p)


6. elem

elem          :: (Eq a) => a -> [a] -> Bool
elem _ []     = False
elem x (y:ys) = x==y || elem x ys

or

elem x        =  any (== x)


7. any, all

any          :: (a -> Bool) -> [a] -> Bool
any _ []     = False
any p (x:xs) = p x || any p xs

any p        = or . map p

all         :: (a -> Bool) -> [a] -> Bool
all _ []     =  True
all p (x:xs) =  p x && all p xs

all p        =  and . map p


8. and, or

and       :: [Bool] -> Bool
and []     =  True
and (x:xs) =  x && and xs

and        =  foldr (&&) True


or       :: [Bool] -> Bool
or []     =  False
or (x:xs) =  x || or xs

or        =  foldr (||) False


9. span, break

-- | '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)@

span                    :: (a -> Bool) -> [a] -> ([a],[a])
span _ xs@[]            =  (xs, xs)
span p xs@(x:xs')
         | p x          =  let (ys,zs) = span p xs' in (x:ys,zs)
         | otherwise    =  ([],xs)


10. group, groupBy

-- | 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.
group :: Eq a => [a] -> [[a]]
group  = groupBy (==)

-- | The 'groupBy' function is the non-overloaded version of 'group'.
groupBy          :: (a -> a -> Bool) -> [a] -> [[a]]
groupBy _  []     = []
groupBy eq (x:xs) = (x:ys) : groupBy eq zs
                    where (ys,zs) = span (eq x) xs


11. composition (.)

-- | Function composition.
-- Make sure it has TWO args only on the left, so that it inlines
-- when applied to two functions, even if there is no final argument
(.)    :: (b -> c) -> (a -> b) -> a -> c
(.) f g = \x -> f (g x) 


12. application ($)

-- | Application operator.  This operator is redundant, since ordinary
-- application @(f x)@ means the same as @(f '$' x)@. However, '$' has
-- low, right-associative binding precedence, so it sometimes allows
-- parentheses to be omitted; for example:
--
-- >     f $ g $ h x  =  f (g (h x))
--
-- It is also useful in higher-order situations, such as @'map' ('$' 0) xs@,
-- or @'Data.List.zipWith' ('$') fs xs@.
{-# INLINE ($) #-}
($)                     :: (a -> b) -> a -> b
f $ x 

Note: f(g(z x)) is equal to f $ g $ z x
      sum(filter (> 10)(map (*2) [2..10)) is equal to sum $ filter (> 10) $ map (*2) [2..10]
We can even map function application itself
> map ($ 3) [(4+), (10*), (^2), sqrt]
[7.0, 30.0, 9.0, 1.732050875688772]

> :i ($)
...
infixr 0 $


13. on

on :: (b -> b -> c) -> (a -> b) -> a -> a -> c
f 'on' g = \x y -> f (g x) (g y)


(.*.) `on` f = \x y -> f x .*. f y.

Typical usage: Data.List.sortBy (compare `on` fst). 


(==) 'on' (> 0)   --- (\x y -> (x > 0) == (y > 0))
compare 'on' length --- (\x y -> length x 'compare' length y)


14. repeat, iterate, replicate, cycle

-- | 'repeat' @x@ is an infinite list, with @x@ the value of every element.
repeat :: a -> [a]
{-# INLINE [0] repeat #-}
-- The pragma just gives the rules more chance to fire
repeat x = xs where xs = x : xs


-- | 'iterate' @f x@ returns an infinite list of repeated applications
-- of @f@ to @x@:
--
-- > iterate f x == [x, f x, f (f x), ...]

iterate :: (a -> a) -> a -> [a]
iterate f x =  x : iterate f (f x)


-- | 'replicate' @n 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.
{-# INLINE replicate #-}
replicate               :: Int -> a -> [a]
replicate n x           =  take n (repeat x)


-- | '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.

cycle                   :: [a] -> [a]
cycle []                = error "Prelude.cycle: empty list"
cycle xs                = xs' where xs' = xs ++ xs'


15. zipWith, zip, unzip

-- | '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.
zipWith :: (a->b->c) -> [a]->[b]->[c]
zipWith f (a:as) (b:bs) = f a b : zipWith f as bs
zipWith _ _      _      = []


-- | '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.
zip :: [a] -> [b] -> [(a,b)]
zip (a:as) (b:bs) = (a,b) : zip as bs
zip _      _      = []


-- | 'unzip' transforms a list of pairs into a list of first components
-- and a list of second components.
unzip    :: [(a,b)] -> ([a],[b])
{-# INLINE unzip #-}
unzip    =  foldr (\(a,b) ~(as,bs) -> (a:as,b:bs)) ([],[])