[ refactor ] (Re)define (Is)TightApartness and (Is)HeytingCommutativeRing/(Is)HeytingField

[jamesmckinna]

Fixes #2587

Adds:

• Algebra.Apartness.Properties.HeytingField, superseding Algebra.Apartness.Properties.HeytingCommutativeRing

NB:

• latter module should now be deprecated (?) in favour of moving its one remaining property to Algebra.Properties.Ring (or even Algebra.Properties.AbelianGroup...) DONE properties moved; module NOT deprecated
• Data.Rational.Unnormalised.Properties should be refactored to make use of Relation.Binary.Properties.DecSetoid for its (Is)*ApartnessRelation definitions DONE

Issue:

• is this v2.3 (as a bug fix of the various APIs involved?) or v3.0 (as a large breaking change?)
• re: @MatthewDaggitt 's comment below about the mathematics, perhaps if someone could check what is said in eg. Troelstra and van Dalen, that might shape how this goes forward, but my own view is: authors seem to disagree about Apartness vs. TightApartness, etc. but that the fundamental distinction should be between *CommutativeRing and *Field (and yes: perhaps HeytingLocalRing would have been better, but see also Define LocalRing #2219 ) in that the latter should have inverses, while the former need not. The existing APIs make the distinction turn on whether the apartness is tight or not, which seems 'wrong'... even to the point of being a bug?

RFC from @cspollard and @bsaul ...

[MatthewDaggitt]

Otherwise, code-wise it looks good. Can't comment on the mathematics 😄

[MatthewDaggitt]

I know this is a draft, but in the final version it would be good to explain broadly what the refactorings actually are here?

[jamesmckinna]

See latest commits, hopefully enough, but not too much, detail now!

[jamesmckinna]

I think this is now ready for review! Many thanks to @MatthewDaggitt for the pre-review, hopefully the latest round of (many!) commits have put things on a more sound footing. As in updated preamble above:

• is this v2.3 or v3.0? (maybe the latter is safer? but the former is justifiable if we think the old APIs were a bug or not?)
• I've left Algebra.Apartness.Properties.HeytingCommutativeRing as an empty stub, with two deprecations, rather than deprecate the module altogether, against possible future additions to this? But in any case, as a consequence it's no longer re-exported from the new Algebra.Apartness.Properties.HeytingField module.

[JacquesCarette]

Spent a lot of time reviewing this... and no comment to speak of. I did spot several proofs that I'd want to change, but I should add some reasoning combinators first that would show how these embody repeated patterns.

[jamesmckinna]

Otherwise, code-wise it looks good. Can't comment on the mathematics 😄

Hopefully the discussion on #2587 clarifies this?

[jamesmckinna]

I'll return to the merge conflicts after v2.3, given that this is a breaking change.