Break `abs` into two aspects

[Zemyla]

"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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