[Agda] What style of proof irrelevance do you prefer: annotated or inferred?

[James Chapman]

Dear Jesper,

DON'T STOP THE PROP!

Personally, I like having Prop as I think it is theoretically and
practically interesting in its own right.

I share Wolfgang's sentiments to an extent but restrict them to the
'.dot' notation (sorry Andreas). In the past I used it primarily out of
desperation when formalizing category theoretic stuff. I dotted the
fields that contained equality proofs in the hope of reducing memory
consumption below the amount of memory my computer had. In this case I
wasn't interested in any changes to the theory and it introduced
limitations such as not being able to substitute (transport) via an
irrelevant equation in a type. Later due to improvements in Agda,
I was happy to remove the dots and return to thinking that I am working in
a more standard type theory without '.dot'.

With Prop I am more interested in things like using the opportunity to
squash (truncate) types down to one element and I would be interested
in being able to turn on impredicative quantification for Prop. Agda
is supposed to be based on UTT after all :)

Having a different universe where different rules apply seems nicer to
me than having a different function space. There is also a nice
separation, if you don't like it don't use it.

On a general note I think one can draw a parallel between being a
programmer and a proof engineer. As a programmer I would like to have
a good optimizing compiler but I still expect that different
implementations of a function can have different run time performance
and I might need to hand optimize (and potentially obfuscate) my
program. I think it is the same with proofs, different approaches
might have different type checking performance and I might need to
optimize/consider the trade offs (inc. readability or writability(is
that a word?)).
If my proof is also a program, I care again about run time performance
as well as type checking performance but both are important.

Regards,

James
On Mon, 29 Oct 2018 at 11:34, Andreas Nuyts
<andreas.nuyts at cs.kuleuven.be> wrote:
>
> Dear everyone,
>
> Another dimension (or is it again the compile/run-time dimension?) is that of type-level vs. value-level annotations.
> We can consider `IrrFun = .(x : A) -> B x` a subtype of `RelFun = (x : A) -> B x`. In this view, RelFun may be inhabited by lambdas that are in fact known to be irrelevant and whose argument can therefore be erased, both at compile and at run-time.
>
> My (draft of a) suggestion is the following: when the Agda user types a lambda-expression `λ (x : A) -> b[x]`, it is inferred from `b[x]` whether this is an irrelevant lambda. Of course, if the type given by the user is `IrrFun`, then this only type-checks if the lambda is inferred to be irrelevant. However, if the type given by the user is `RelFun` but the lambda is irrelevant, then this fact is remembered in the form of an internal annotation on the lambda, as it allows erasure of the argument both at compile and at runtime. Without the annotation on the lambda, erasure could only happen after beta-reducing the lambda, which is probably less efficient.
> This behaviour would be in accordance to what happens if you explicitly coerce a function `f : IrrFun` to `RelFun` by lambda-expanding. It then becomes `λ (x : A) -> f _`.
>
> When it comes to inferring type-level annotations: this is a problem quite similar to inferring universe levels. When a user does not specify a universe level, do you want to infer the smallest one, or do you want to complain? (I think Agda currently complains about an unresolved meta, although C-c C-= will fill out the smallest option?) Similarly, when a user does not specify a modality, do you want to infer the most informative one (irrelevance, if possible), or do you want to complain? (I don't have any answers here, just pointing out the similarity of the problems.)
>
> Best regards,
>
> Andreas Nuyts
> (not the Andreas mentioned in Jesper's emails)
>
> On 29/10/2018 09:28, Jesper Cockx wrote:
> Dear Jon,
>
> Thanks for your answer. I see these two not so much as two different kinds of proof irrelevance, but as two separate dimensions: compile-time vs run-rime irrelevance and annotated vs inferred irrelevance. For example, Agda has annotated compile-time irrelevance (irrelevant functions and Prop), inferred run-time irrelevance (type erasure, forcing analysis, detection of unused arguments), Andreas recently added annotated run-time irrelevance (the @erased annotation), but it has very little inferred compile-time irrelevance (the only thing that comes to mind is the unit type, which is automatically proof-irrelevant because of eta).
>
> It seems to me you like a combination of annotated compile-time irrelevance and inferred run-time irrelevance, is that correct?
>
> What Wolfram seemed to suggest however is to infer more *compile-time* irrelevance. For example, we could infer that the empty type is proof-irrelevant and automatically discard any equation at the empty type during conversion (i.e. have eta for the empty type). Or we could detect that the identity type has a single constructor and (assuming --with-K) replace all proofs of identity by primTrustMe during elaboration.
>
> The thing is, such automatically inferred irrelevance wouldn't work very well in some cases, and it would certainly not help for the fancy applications of irrelevance you describe (enforcing parametricity or coherence). However it *would* give some benefits of irrelevance to people who use Agda but don't want to add extra irrelevance annotations, or indeed even to people who don't know about irrelevance at all. So it has a much greater potential impact than the annotated style of irrelevance.
>
> Of course, much depends on how often we could actually detect irrelevance automatically, which in turn would depend greatly on the concrete codebase. But perhaps doing the experiment could give us a better idea.
>
> -- Jesper
>
> On Mon, Oct 29, 2018 at 1:33 AM Jon Sterling <jon at jonmsterling.com<mailto:jon at jonmsterling.com>> wrote:
> Dear Jesper,
>
> I feel that these are two separate issues getting conflated -- on the one hand, a compiler may infer that something is irrelevant, and then erase it in order to achieve a more efficient execution. On the other hand, there is proof irrelevance which is used for *semantic* reasons: irrelevance is part of the specification of some function, and whether or not it executes in an irrelevant or erased way is really beside the point (of course, in such a case, the compiler *should* erase it because it is low hanging fruit).
>
> The latter (semantic) kind of proof irrelevance could be used, for instance, in order to achieve the following things:
>
> - to specify that some function is parametric in its indices: for instance, operations on vectors
>
> - relatedly, to force *coherence*: for instance, when defining some interpretation of some language, you can use proof irrelevence + some inversion lemmas to formalize the old pattern (from Streicher) of defining your function by recursion on the labels of the derivation rather than on the derivation itself. In old school math, this required defining a partial function and then establishing that it terminates, but in very fancy type theory we can use proof irrelevance to do this simultaneously and get a total function all at once.
>
> - and of course, to obtain more definitional equivalences where possible
>
> Let me unleash a kind of important point: as I mentioned, in all these cases, erasure should obviously be done, but there are cases where a good compiler would perform erasure even when it is not possible to type the term using the irrelevant types. Something might be erasable for global reasons. Anyway, it's not so important to me that Agda catch all of these cases, because I'm not using Agda for executing code (but others might be).
>
> What I'm saying is that inferring when something is computationally irrelevant must be treated as orthogonal to whether something can be typed with a proof irrelevance modality. Computational irrelevance is a property of code in the compilation target language (which could be anything), not a property of code in the Agda language.
>
> We should take compilation seriously, and not conflate it with typechecking and elaboration. Dependently typed languages have *two* computational semantics: the one which generates definitional equivalence of terms, and the one which executes.
>
> Best,
> Jon
>
>
>
>
> On Sun, Oct 28, 2018, at 2:01 PM, Jesper Cockx wrote:
> > Hi Agda people,
> >
> > In the comments on issue #3334
> > <https://github.com/agda/agda/issues/3334#issuecomment-433730080>, Wolfram
> > wrote:
> >
> > I consider it counterproductive to be able to declare irreleveance, and
> > > even more counterproductive to have to do it for efficiency reasons,
> > > because that encourages premature optimisations. The compiler (including
> > > the type checker) should detect it whenever possible, and optimise it away
> > > as soon as possible.
> >
> >
> > To which I replied:
> >
> > Often the fact that a certain term is irrelevant is obvious to *you* as the
> > > Agda programmer but there's no way for Agda to figure this out from the
> > > code. If it's possible that you will use the term at any point in the
> > > future, Agda cannot (and should not) erase it. So any automatic detection
> > > of irrelevance would probably be very disappointing. (The alternative where
> > > Agda would eagerly erase things without asking and then complaining later
> > > when you try to use them is arguably worse). Another advantage of
> > > annotating irrelevant things, is that Agda can warn you when you use it
> > > accidentally in a non-erased position, instead of removing the irrelevance
> > > silently and causing hard-to-explain performance regressions.
> > >
> >
> > But I realized this is based mostly on speculation and not hard facts.
> > Historically, Agda has focused on the annotated style of proof-irrelevance
> > (Andreas' irrelevant functions and irrelevant fields, and more recently my
> > implementation of Prop). Maybe inferred proof irrelevance would work better
> > than expected?
> >
> > So now my question to *you*, the Agda community is this: do you prefer
> > Agda's current style of annotated proof-irrelevance, or would you rather
> > have Agda infer things for you (and perhaps fail to do so in some cases)?
> >
> > I won't promise I will remove Prop and focus all my attention on inferring
> > irrelevance even if everyone votes for the latter option, but it would be
> > interesting to know going forward what you think.
> >
> > -- Jesper
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