Break `abs` into two aspects

[Andreas Abel]

 >         class Semiring a where
 >            zero  :: a
 >            plus  :: a -> a -> a
 >            one   :: a
 >            times :: a -> a -> a
 >            fromNatural :: Natural -> a

I think `fromNatural` should not be part of the `Semiring` class, but we 
could have an extension (NaturalSemiring) that adds this method.

In the Agda code base, we have, for lack of a standard, rolled our own 
semiring class,

   https://github.com/agda/agda/blob/master/src/full/Agda/Utils/SemiRing.hs

and we use it for several finite semirings, e.g.,

 
https://github.com/agda/agda/blob/64c0c2e813a84f91b3accd7c56efaa53712bc3f5/src/full/Agda/TypeChecking/Positivity/Occurrence.hs#L127-L155

Cheers,
Andreas

On 2020-02-03 22:34, Carter Schonwald wrote:
> Andrew: could you explain the algebra notation you were using for short 
> hand?  I think I followed, but for people the libraries list might be 
> their first exposure to advanced / graduate abstract algebra (which 
> winds up being simpler than most folks expect ;) )
> 
> On Fri, Jan 31, 2020 at 4:36 PM Carter Schonwald 
> <carter.schonwald at gmail.com <mailto:carter.schonwald at gmail.com>> wrote:
> 
>     that actually sounds pretty sane. I think!
> 
>     On Fri, Jan 31, 2020 at 3:38 PM Andrew Lelechenko
>     <andrew.lelechenko at gmail.com <mailto:andrew.lelechenko at gmail.com>>
>     wrote:
> 
>         On Tue, 28 Jan 2020, Dannyu NDos wrote:
> 
>          > Second, I suggest to move `abs` and `signum` from `Num` to
>         `Floating`
> 
>         I can fully relate your frustration with `abs` and `signum` (and
>         numeric type classes in Haskell altogether). But IMO breaking
>         both in `Num` and in `Floating` at once is not a promising way
>         to make things proper.
> 
>         I would rather follow the beaten track of Applicative Monad and
>         Semigroup Monoid proposals and - as a first step - introduce a
>         superclass (probably, borrowing the design from `semirings`
>         package):
> 
>         class Semiring a where
>            zero  :: a
>            plus  :: a -> a -> a
>            one   :: a
>            times :: a -> a -> a
>            fromNatural :: Natural -> a
>         class Semiring a => Num a where ...
> 
>         Tangible benefits in `base` include:
>         a) instance Semiring Bool,
>         b) a total instance Semiring Natural (in contrast to a partial
>         instance Num Natural),
>         c) instance Num a => Semiring (Complex a) (in contrast to
>         instance RealFloat a => Num (Complex a)),
>         d) newtypes Sum and Product would require only Semiring
>         constraint instead of Num.
> 
>         Best regards,
>         Andrew
> 
> 
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