{-# LANGUAGE Safe #-} ----------------------------------------------------------------------------- -- | -- Module : Data.Function -- Copyright : Nils Anders Danielsson 2006 -- 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 Prelude infixl 0 `on` -- | @'fix' f@ is the least fixed point of the function @f@, -- i.e. the least defined @x@ such that @f x = x@. 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