```{- |
Copyright   :  (c) Henning Thielemann 2007-2010

Stability   :  stable

A type class for non-negative numbers.
Prominent instances are 'Numeric.NonNegative.Wrapper.T' and peano numbers.
This class cannot do any checks,
but it let you show to the user what arguments your function expects.
Thus you must define class instances with care.
In fact many standard functions ('take', '(!!)', ...)
should have this type class constraint.
-}
module Numeric.NonNegative.Class (
C(..),
splitDefault,

(-|),
zero,
sum,
maximum,

switchDifferenceNegative,
switchDifferenceOrdering,
) where

import Data.Monoid (Monoid, )
import qualified Data.Monoid as Monoid

import Prelude hiding (sum, maximum, )

{- |
Instances of this class must ensure non-negative values.
We cannot enforce this by types, but the type class constraint @NonNegative.C@
avoids accidental usage of types which allow for negative numbers.

The Monoid superclass contributes a zero and an addition.
-}
class (Ord a, Monoid a) => C a where
{- |
@split x y == (m,(b,d))@ means that
@b == (x<=y)@,
@m == min x y@,
@d == max x y - min x y@, that is @d == abs(x-y)@.

We have chosen this function as base function,
since it provides comparison and subtraction in one go,
which is important for replacing common structures like

> if x<=y
>   then f(x-y)
>   else g(y-x)

that lead to a memory leak for peano numbers.
We have choosen the simple check @x<=y@
since we want @Zero <= undefined@ for peano numbers.
Because of undefined values 'split' is in general
not commutative in the sense

> let (m0,(b0,d0)) = split x y
>     (m1,(b1,d1)) = split y x
> in  m0==m1 && d0==d1

The result values are in the order
in which they are generated for Peano numbers.
We have chosen the nested pair instead of a triple
in order to prevent a memory leak
that occurs if you only use @b@ and @d@ and ignore @m@.
This is demonstrated by test cases
Chunky.splitSpaceLeak3 and Chunky.splitSpaceLeak4.
-}
split :: a -> a -> (a, (Bool, a))

{- |
Default implementation for wrapped types of 'Ord' and 'Num' class.
-}
{-# INLINE splitDefault #-}
splitDefault ::
(Ord b, Num b) =>
(a -> b) -> (b -> a) -> a -> a -> (a, (Bool, a))
splitDefault unpack pack px py =
let x = unpack px
y = unpack py
in  if x<=y
then (pack x, (True,  pack (y-x)))
else (pack y, (False, pack (x-y)))

zero :: C a => a
zero = Monoid.mempty

-- like (+)

add :: C a => a -> a -> a

sum :: C a => [a] -> a
sum = Monoid.mconcat

{- |
Left biased maximum of a list of numbers that can also be empty.
It holds

> maximum [] == zero
-}
maximum :: C a => [a] -> a
maximum = foldl max zero

{- |
In @switchDifferenceNegative x y branchXminusY branchYminusX@
the function @branchXminusY@ is applied to @x-y@
if this difference is non-negative,
otherwise @branchYminusX@ is applied to @y-x@.
-}
switchDifferenceNegative ::
C a =>
a -> a -> (a -> b) -> (a -> b) -> b
switchDifferenceNegative x y branchXminusY branchYminusX =
let (b,d) = snd \$ split y x
in  if b
then branchXminusY d
else branchYminusX d

{- |
In @switchDifferenceOrdering x y branchZero branchXminusY branchYminusX@
-}
switchDifferenceOrdering ::
C a =>
a -> a -> b -> (a -> b) -> (a -> b) -> b
switchDifferenceOrdering x y branchZero branchXminusY branchYminusX =
let (b,d) = snd \$ split y x
in  if b
then
if d==zero
then branchZero
else branchXminusY d
else branchYminusX d

{- |
@x -| y == max 0 (x-y)@

The default implementation is not efficient,
because it compares the values and then subtracts, again, if safe.
@max 0 (x-y)@ is more elegant and efficient
but not possible in the general case,
since @x-y@ may already yield a negative number.
-}
(-|) :: C a => a -> a -> a
x -| y  =
let (b,d) = snd \$ split y x
in  if b then d else zero
```