containers: intersections for Set, along with Semigroup newtype

[Tom Ellis]

On Sun, Dec 20, 2020 at 11:05:39PM +0100, Ben Franksen wrote:
> Am 06.12.20 um 19:58 schrieb Sven Panne:
> > To me it's just the other way around: It violates aesthetics if it doesn't
> > follow the mathematical definition in all cases, which is why I don't like
> > NonEmpty here.
> 
> I think you've got that wrong.
> 
>   x `elem` intersections []
> = {definition}
>   forall xs in []. x `elem` xs
> = {vacuous forall}
>   true
> 
> Any proposition P(x) is true for all x in []. So the mathematical
> definition of intersections::[Set a]-> Set a would not be the empty set
> but the set of all x:a, which in general we have no way to construct.

Yes, and to bring this back to Sven's original claim

| Why NonEmpty? I would expect "intersections [] = Set.empty", because
| the result contains all the elements which are in all sets,
| i.e. none. That's at least my intuition, is there some law which
| this would violate?

the correct definition of "intersections []" should be "all elements
which are in all of no sets" i.e. _every_ value of the given type!

Tom