{-# LANGUAGE Trustworthy #-} {-# LANGUAGE NoImplicitPrelude #-} ----------------------------------------------------------------------------- -- | -- Module : Data.Function -- Copyright : Nils Anders Danielsson 2006 -- , Alexander Berntsen 2014 -- License : BSD-style (see the LICENSE file in the distribution) -- -- Maintainer : libraries@haskell.org -- Stability : experimental -- Portability : portable -- -- Simple combinators working solely on and with functions. -- ----------------------------------------------------------------------------- module Data.Function ( -- * "Prelude" re-exports id, const, (.), flip, (\$) -- * Other combinators , (&) , fix , on ) where import GHC.Base ( (\$), (.), id, const, flip ) infixl 0 `on` infixl 1 & -- | @'fix' f@ is the least fixed point of the function @f@, -- i.e. the least defined @x@ such that @f x = x@. -- -- For example, we can write the factorial function using direct recursion as -- -- >>> let fac n = if n <= 1 then 1 else n * fac (n-1) in fac 5 -- 120 -- -- This uses the fact that Haskellâ€™s @let@ introduces recursive bindings. We can -- rewrite this definition using 'fix', -- -- >>> fix (\rec n -> if n <= 1 then 1 else n * rec (n-1)) 5 -- 120 -- -- Instead of making a recursive call, we introduce a dummy parameter @rec@; -- when used within 'fix', this parameter then refers to 'fix'' argument, hence -- the recursion is reintroduced. fix :: (a -> a) -> a fix f = let x = f x in x -- | @((==) \`on\` f) x y = f x == f y@ -- -- Typical usage: @'Data.List.sortBy' ('compare' \`on\` 'fst')@. -- Algebraic properties: -- -- * @(*) \`on\` 'id' = (*)@ (if @(*) ∉ {⊥, 'const' ⊥}@) -- -- * @((*) \`on\` f) \`on\` g = (*) \`on\` (f . g)@ -- -- * @'flip' on f . 'flip' on g = 'flip' on (g . f)@ -- Proofs (so that I don't have to edit the test-suite): -- (*) `on` id -- = -- \x y -> id x * id y -- = -- \x y -> x * y -- = { If (*) /= _|_ or const _|_. } -- (*) -- (*) `on` f `on` g -- = -- ((*) `on` f) `on` g -- = -- \x y -> ((*) `on` f) (g x) (g y) -- = -- \x y -> (\x y -> f x * f y) (g x) (g y) -- = -- \x y -> f (g x) * f (g y) -- = -- \x y -> (f . g) x * (f . g) y -- = -- (*) `on` (f . g) -- = -- (*) `on` f . g -- flip on f . flip on g -- = -- (\h (*) -> (*) `on` h) f . (\h (*) -> (*) `on` h) g -- = -- (\(*) -> (*) `on` f) . (\(*) -> (*) `on` g) -- = -- \(*) -> (*) `on` g `on` f -- = { See above. } -- \(*) -> (*) `on` g . f -- = -- (\h (*) -> (*) `on` h) (g . f) -- = -- flip on (g . f) on :: (b -> b -> c) -> (a -> b) -> a -> a -> c (.*.) `on` f = \x y -> f x .*. f y -- | '&' is a reverse application operator. This provides notational -- convenience. Its precedence is one higher than that of the forward -- application operator '\$', which allows '&' to be nested in '\$'. -- -- >>> 5 & (+1) & show -- "6" -- -- @since 4.8.0.0 (&) :: a -> (a -> b) -> b x & f = f x -- \$setup -- >>> import Prelude