[ add ] Relation.Binary._Reflects_⟶_ as a companion to _Preserves_⟶_

[jamesmckinna]

A preparatory PR ahead of tackling #1436 in earnest.
NB:

• illustrative refactoring of Function.Definitions.Injective
• surprisingly, attempting to do the same for Function.Definitions.Congruent in terms of Preserves results in many, many, unsolved metas... sigh. See [ refactor ] Function.Definitions and Function.Consequences? #2565 for details

[gallais]

What's the intuition for the Reflects name? Does it have any relationship to the ssreflect
style Reflects that we already have? http://agda.github.io/agda-stdlib/v2.1.1/Relation.Nullary.Reflects.html#1010

[jamesmckinna]

What's the intuition for the Reflects name? Does it have any relationship to the ssreflect style Reflects that we already have? http://agda.github.io/agda-stdlib/v2.1.1/Relation.Nullary.Reflects.html#1010

Nothing to do with Reflects as in ssreflect, nor Relation.Nullary.Reflects!

But absolutely conventional usage in first-order logic/relational algebra (eg. discussion in McKinna&Pollack 1999 on preserving and reflecting α-conversion in λ-calculus representations...): the aim is to [DRY] reconcile Function.Definitions.Injective and Algebra.Definitions.Cancellative as specialisations of the concept...

[MatthewDaggitt]

I'm kind of against this change personally. I read Injective f = ∀ {x y} → f x ≈₂ f y → x ≈₁ y and I understand exactly what it says.

I read Injective = _Reflects _≈₂_ ⟶ _≈₁_ and I have to do the following:

1. Expand the missing function to Injective f = f Reflects _≈₂_ ⟶ _≈₁_ (non-trivial for beginners)
2. Expand Reflects to (_≈₁_ on f) ⇒ _≈₂_ which is equally baffling.
3. Expand on ...
4. Expand _⇒_...
5. Only now do I get to the actual definition.

All of these definitions are in different modules and Agda currently provides no easy method of expanding them. In my mind the very small benefit of this change is outweighed by the quite large decrease in usability!

[jamesmckinna]

@MatthewDaggitt very sharp critique from the ux perspective (I think perhaps we should have ux as a new label category for tagging issues/PRs), for which all thanks.

As the preamble to this PR indicates, I wanted this concept/definition in stdlib so that (eventually) all the Cancellative properties could be rephrased in terms of it. Currently, under #1436 / #2573 we only treat the cancellative properties for equality... if you will, perhaps as another, more directly graspable concept, we should get rid of Cancellative and friends altogether in favour of saying that eg m +_ is Injective on Nat with _≡_ (that would be the other way to go).

It's perhaps the case that my ideological/reformatory zeal on [DRY] issues leads to (over-)generalisations like those considered here (or most recently, in #2581 ), and that reflects what is (probably!) the greater abstraction power of higher-order languages such as Agda's type theory relative to the abstraction power of users (esp. wrt definitional expansion across modules, as you indicate so forcefully).

Against your suggestion, my own preference would be to record as a comment, the 'conventional' usage as being a definitional expansion of the abstract one. Something like

  Injective : (A → B) → Set _
  Injective = _Reflects _≈₂_ ⟶ _≈₁_
-- i.e.  Injective f = ∀ {x y} → f x ≈₂ f y → x ≈₁ y

Of course, that would be a break from existing practice in the library, as well as opening the door to bit-rot if/when definitions change, but here I don't think we are at quite such a risk... precisely because the definition is so 'stable'?

[jamesmckinna]

All of that said, happy to revert this particular instance of the definition (it was introduced above as 'illustrative'), but then what about Cancellative in all its variety?

[MatthewDaggitt]

I think adding non-type checked comments is very susceptible to bit rot as you say. I would be more open to a private definition that performed an equality check.

However, I'm still not entirely sure what advantages the generalisation brings in this case. It's not like we have either a) a set of properties/theory about these kinds of definitions in general or b) lots of definitions in this form? Injective, Cancellative, anything else?

Ditto, I'm not sure that I would be keen on replacing the definitions of Cancellative for exactly the same reasons...

[jamesmckinna]

I guess I would still try (but ) to justify the addition

• on formal grounds, as the 'opposite' of Respects
• on abstraction grounds (I clearly prefer greater degrees of abstraction, yes ,for its own sake, so I'll try to keep that in mind in future)

but this particular stone seems to be getting harder to push uphill. I'll stop!