Break `abs` into two aspects

[Carter Schonwald]

Derp. :)

On Wed, Feb 5, 2020 at 9:08 PM Zemyla <zemyla at gmail.com> wrote:

> "i" is an Integer there, which has an Ord instance.
>
> On Wed, Feb 5, 2020, 19:20 Carter Schonwald <carter.schonwald at gmail.com>
> wrote:
>
>> Num doesn't have Ord as a parent constraint any more ... Though i suppose
>> that works as a default signature instance?
>>
>> On Wed, Feb 5, 2020 at 3:19 PM Mario Blažević <mblazevic at stilo.com>
>> wrote:
>>
>>> On 2020-02-04 11:32 a.m., Zemyla wrote:
>>> > It really doesn't matter if it's not "interesting" or not surjective
>>> for
>>> > some Semirings. It should be included, because:
>>>
>>>         I fully agree, and I'll add another reason you left out. The
>>> presence
>>> of fromNatural would allow defaulting of Num's fromInteger as
>>>
>>>  > fromInteger i
>>>  >     | i >= 0 = fromNatural (fromInteger i)
>>>  >     | otherwise = negate . fromInteger . negate $ i
>>>
>>>
>>> > (a) Even for semirings where it is "interesting", it's not surjective
>>> > (for instance, Rational or Double)
>>> > (b) It's a method with a default definition, so you don't have to
>>> expend
>>> > any mental effort on it
>>> > (c) A lot of instances have uninteresting methods: for instance, (*>)
>>> > and (<*) for Applicative ((->) e) are const id and const respectively.
>>> > Haskell adds methods to classes when they're always possible and
>>> > sometimes useful/interesting/faster, rather than when they're always
>>> > interesting.
>>> > (d) It's useful for Semiring-generic methods and instances.
>>> > (e) It can achieve an asymptotic speedup on some instances. Like, if
>>> you
>>> > have Semiring a => Semiring (f a) for some type f, then you can have
>>> > fromNatural n = pure (fromNatural n) instead of doing the whole O(log
>>> n)
>>> > song and dance with the default definition. Also, your example admits
>>> a
>>> > simple definition:
>>> >    fromNatural n = if n == 0 then S.empty else S.singleton True
>>> > (f) "zero" and "one" can be defined in terms of fromNatural, for
>>> > programmers who only need to define that:
>>> >    zero = fromNatural 0
>>> >    one = fromNatural 1
>>> > This leads to the MINIMAL pragma on Semiring being {-# MINIMAL plus,
>>> > times, (zero, one | fromNatural) #-}
>>> > (g) If it's not included in the class, but in some subclass
>>> > (NaturalSemiring, you proposed), but it's possible from the class,
>>> then
>>> > people will just define and use the O(log n) version instead of
>>> > requiring the subclass, leading to wasted effort and duplicated code.
>>>
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