Remove Control.Parallel*, now in package parallel
[packages/random.git] / Data / Function.hs
1 -----------------------------------------------------------------------------
2 -- |
3 -- Module : Data.Function
4 -- Copyright : Nils Anders Danielsson 2006
5 -- License : BSD-style (see the LICENSE file in the distribution)
6 --
7 -- Maintainer : libraries@haskell.org
8 -- Stability : experimental
9 -- Portability : portable
10 --
11 -- Simple combinators working solely on and with functions.
12
13 module Data.Function
14 ( -- * "Prelude" re-exports
15 id, const, (.), flip, ($)
16 -- * Other combinators
17 , fix
18 , on
19 ) where
20
21 import Prelude
22
23 infixl 0 `on`
24
25 -- | @'fix' f@ is the least fixed point of the function @f@,
26 -- i.e. the least defined @x@ such that @f x = x@.
27 fix :: (a -> a) -> a
28 fix f = let x = f x in x
29
30 -- | @(*) \`on\` f = \\x y -> f x * f y@.
31 --
32 -- Typical usage: @'Data.List.sortBy' ('compare' \`on\` 'fst')@.
33 --
34 -- Algebraic properties:
35 --
36 -- * @(*) \`on\` 'id' = (*)@ (if @(*) ∉ {⊥, 'const' ⊥}@)
37 --
38 -- * @((*) \`on\` f) \`on\` g = (*) \`on\` (f . g)@
39 --
40 -- * @'flip' on f . 'flip' on g = 'flip' on (g . f)@
41
42 -- Proofs (so that I don't have to edit the test-suite):
43
44 -- (*) `on` id
45 -- =
46 -- \x y -> id x * id y
47 -- =
48 -- \x y -> x * y
49 -- = { If (*) /= _|_ or const _|_. }
50 -- (*)
51
52 -- (*) `on` f `on` g
53 -- =
54 -- ((*) `on` f) `on` g
55 -- =
56 -- \x y -> ((*) `on` f) (g x) (g y)
57 -- =
58 -- \x y -> (\x y -> f x * f y) (g x) (g y)
59 -- =
60 -- \x y -> f (g x) * f (g y)
61 -- =
62 -- \x y -> (f . g) x * (f . g) y
63 -- =
64 -- (*) `on` (f . g)
65 -- =
66 -- (*) `on` f . g
67
68 -- flip on f . flip on g
69 -- =
70 -- (\h (*) -> (*) `on` h) f . (\h (*) -> (*) `on` h) g
71 -- =
72 -- (\(*) -> (*) `on` f) . (\(*) -> (*) `on` g)
73 -- =
74 -- \(*) -> (*) `on` g `on` f
75 -- = { See above. }
76 -- \(*) -> (*) `on` g . f
77 -- =
78 -- (\h (*) -> (*) `on` h) (g . f)
79 -- =
80 -- flip on (g . f)
81
82 on :: (b -> b -> c) -> (a -> b) -> a -> a -> c
83 (*) `on` f = \x y -> f x * f y