blob: 54eabbb6e71e8e91078b97a50fad7509c0e1b055 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
|
{-# 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
|
