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

XX. fix

fix :: (a -> a) -> a
fix f = let x = f x
        in  x

fix f is the least fixed point of the function f, i.e. the least defined x such that f x = x.

fix f = f (fix f)

f (fix f) = fix f


fix g = h, where g h = h
fix g = g (fix g) = g (g (fix g)) = ... = g (g ... (fix g)...).

Because of Lazy evaluation the sequence is not infinite but has maximum the number of steps needed
for g to bring result for some arbitrary argument.




Least fixed-point combinator

let fac f x = if x==0 then 1 else x * f (x - 1)

let fix f = f (fix f)

>fix fac 5
120

fix :: (x -> x) -> x
fix f = f (fix f)


fact :: Int -> Int
ff :: (Int -> Int) -> (Int -> Int)


http://en.wikibooks.org/wiki/Haskell/Fix_and_recursion


Kholomiov - part 16

-- | '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

> repeat f = f : f : f : ...


fix :: (a -> a) -> a
fix = foldr ($) undefined . repeat

(replace (:) c-tor of repeat by application - ($)).