    -----------------------------------------------------------------------------
    -- |
    -- 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 @(*) &#x2209; {&#x22a5;, 'const' &#x22a5;}@)
    --
    -- * @((*) \`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