[Agda] Defining the Delay monad as a HIT in cubical agda

[Dan Doel]

It seems like what's going on here is kind of related to the somewhat
odd behavior of coinduction in Agda in general, where every set of
definitions involving coinduction is always νμ. This means that there
is a never _path_ that chains arbitrarily many `steps` together. But
what you really want for things to work out is for the paths to be well
founded, so that an `== now x` has to eventually reach the now.

Is this fixed by first defining the coinductive Delay, and then inductively
defining a HIT like:

    data Part (A : Set) : Set where
      include : Delay A -> Part A
      step : ∀ d -> include (later d) == include (force d)

since the steps no longer tie through a coinductive type?

On Tue, Apr 9, 2019 at 4:09 AM Andrea Vezzosi <sanzhiyan at gmail.com> wrote:

> On Mon, Apr 8, 2019 at 8:38 PM Nicolai Kraus <nicolai.kraus at gmail.com>
> wrote:
> >
> > Great!!  :)
> > That means we know what this type is: Delay(A) = 1.
> > It's not so clear (to me) though how we can justify these higher
> > coinductive types.
>
> It seems that this example should make sense as a cubical set in the
> same style as the construction of HITs in CTT ?
> Just defined as a mixed inductive-coinductive type rather than purely
> inductive.
>
> https://arxiv.org/abs/1802.01170
>
> >
> > On 08/04/19 19:34, Andrea Vezzosi wrote:
> > > Oh, yeah, I hadn't thought about it, but indeed there's nothing
> > > special about "now" here.
> > >
> > > mutual
> > >    never′≡ : ∀ (x : Delay′ A) → never′ ≡ x
> > >    never′≡ x i .force = never≡ (x .force) i
> > >
> > >    never≡ : ∀ (x : Delay A) → never ≡ x
> > >    never≡ x i = step (never′≡ (delay x) i) i
> > >
> > >
> > >
> > > On Mon, Apr 8, 2019 at 8:11 PM Nicolai Kraus <nicolai.kraus at gmail.com>
> wrote:
> > >> Wow, thanks, Andrea!
> > >> Can this proof be extended to show that Delay(A) is contractible?
> > >> -- Nicolai
> > >>
> > >> On 08/04/19 19:01, Andrea Vezzosi wrote:
> > >>> Here is a proof that "never" is equal to "now x", making use of path
> > >>> abstraction and copatterns.
> > >>>
> > >>> mutual
> > >>>     never′ : Delay′ A
> > >>>     never′ .force = never
> > >>>
> > >>>     never : Delay A
> > >>>     never = later never′
> > >>>
> > >>> mutual
> > >>>     never≡now′ : ∀ (x : A) → never′ ≡ delay (now x)
> > >>>     never≡now′ x i .force = never≡now x i
> > >>>
> > >>>     never≡now : ∀ (x : A) → never ≡ now x
> > >>>     never≡now x i = step (never≡now′ x i) i
> > >>>
> > >>> Of course one still wonders what Delay really is.
> > >>>
> > >>> This definition is accepted because the corecursion goes through the
> > >>> coinductive projection ".force" and both "step" and application
> to"i"
> > >>> are considered guardedness preserving.
> > >>>
> > >>> On Tue, Apr 2, 2019 at 11:11 PM Nicolai Kraus <
> nicolai.kraus at gmail.com> wrote:
> > >>>> On 02/04/19 21:23, Jesper Cockx wrote:
> > >>>>
> > >>>> Well, if `never` is equal to `now x`, then by transitivity `now x`
> is equal to `now y` for any `x` and `y`, which would mean I found a very
> complicated way to define the constant unit type :P
> > >>>>
> > >>>>
> > >>>> Right... the conjecture should be Delay(A) = Unit. I made a silly
> mistake before!
> > >>>> It's possible that the theory doesn't allow us to prove Delay(A) =
> 1, but I don't expect that we can show the negation of this.
> > >>>>
> > >>>> About terminology: Nisse informed me that `Delay` is used for the
> (non-truncated) coinductive type with two constructors `now` and `later`,
> while the properly truncated variant where `later^n x` = `now x` for any
> finite `n` is called the partiality monad.
> > >>>>
> > >>>>
> > >>>> This is also the terminology that I know. In addition, probably one
> would want to call something "partiality monad" only if it actually is a
> monad. The definition for this that I find most elegant is the one by Tarmo
> and Niccolò (iirc, this definition ends up being equivalent to our
> suggestion in the "Partiality revisited" paper).
> > >>>>
> > >>>> So my question is actually whether the partiality monad is
> definable as a higher coinductive type with two point constructors `now`
> and `later` plus some path constructor(s).
> > >>>>
> > >>>> The problem with defining such a higher coinductive type `D` is
> that all attempts at proving two of its elements are *not* equal seem to
> fail:
> > >>>>
> > >>>> - Pattern matching on an equality between two constructors with an
> absurd pattern () obviously doesn't work for higher inductive types.
> > >>>> - Defining a function `f : D -> Bool` or `D -> Set` which
> distinguishes the two elements doesn't work either because both `Bool` and
> `Set` are inductively defined, so `f` can only depend on a finite prefix of
> its input (i.e. f must be continuous).
> > >>>>
> > >>>>
> > >>>> `f : D -> Bool` shouldn't work even with a correct partiality
> monad, because it shouldn't be decidable whether an element is `never`. One
> could replace `Bool` by the Sierpinski space, which is by definition
> Partiality(1). (btw, `Set` is not inductively defined?)
> > >>>>
> > >>>> - Defining an indexed datatype `data P : D -> Set` that is empty at
> one index but not at another seems to work, but then we get into trouble
> when we actually want to prove that it is empty for that particular index
> (this is not really surprising because indexed datatypes can be explained
> with normal datatypes + the equality type, so this is essentially the same
> as the first option).
> > >>>>
> > >>>> This exhausts my bag of tricks when it comes to proving two
> constructor forms are not equal. This seems to be an essential problem that
> would pop up any time one tries to mix coinduction with higher
> constructors. It would be an interesting research topic to try and define a
> suitable notion of "higher coinductive type" which does not have this
> problem.
> > >>>>
> > >>>>
> > >>>> Right, but I think we current have no idea what cubical Agda's
> "higher coinductive types" are. It's interesting that Agda allows these,
> but they could as well be inconsistent. (That's why I asked about models
> before.)
> > >>>> -- Nicolai
> > >>>>
> > >>>>
> > >>>>
> > >>>> -- Jesper
> > >>>>
> > >>>> On Tue, Apr 2, 2019 at 10:06 PM Nicolai Kraus <
> nicolai.kraus at gmail.com> wrote:
> > >>>>> Interesting! So, in case Delay(Unit) does turn out to be
> contractible, we might also expect that Delay(A) = A. This doesn't seem
> intuitive to me, but it could still be true. Do you see a way to construct
> Delay(A) -> A? If there is such a function, it should be quite canonical,
> and maybe it's easier to write this function than to prove the
> contractibility. But if we can't do this, and we also can't distinguish
> 'now' and 'never', then I have no idea what Delay(A) actually is. Does any
> of the cubical models capture such constructions?
> > >>>>> (Maybe, at this point, we shouldn't call it "Delay" :)
> > >>>>> -- Nicolai
> > >>>>>
> > >>>>>
> > >>>>> On 02/04/19 15:08, Jesper Cockx wrote:
> > >>>>>
> > >>>>> Hi Nicolai,
> > >>>>>
> > >>>>> Yes, Christian and I suspected the same thing (that this
> definition of the delay monad is actually a unit type), but I haven't
> managed to prove that either because of some mysterious termination checker
> problem.
> > >>>>>
> > >>>>> I'm currently trying a different approach where I define the Delay
> type mutually with the ⇓ type so I can quotient by the relation "normalize
> to the same value in a finite number of steps". I'll let you know later if
> it works.
> > >>>>>
> > >>>>> -- Jesper
> > >>>>>
> > >>>>> On Tue, Apr 2, 2019 at 3:15 PM Nicolai Kraus <
> nicolai.kraus at gmail.com> wrote:
> > >>>>>> Hi Jesper,
> > >>>>>>
> > >>>>>> I find this construction very interesting since it's the first
> "cubical higher co-inductive type" that I've seen! Unfortunately, I don't
> know how these "CHCIT's" behave in Agda.
> > >>>>>>
> > >>>>>> If I had to guess, I would expect that you *cannot* distinguish
> now and never, which would mean that your construction doesn't give you
> what you wanted. The difference to the quotiented delay monad (or, for what
> it's worth, the QIT/QIIT partiality monad) is that your construction
> inserts the equations "coinductively" (usually, only the later-steps are
> coinductive).
> > >>>>>>
> > >>>>>> Thus, my guess is that Delay(Unit) could be contractible. One
> could try to prove that every element is equal to 'now'.
> > >>>>>>
> > >>>>>> Cheers,
> > >>>>>> Nicolai
> > >>>>>>
> > >>>>>>
> > >>>>>> On 29/03/19 18:40, Jesper Cockx wrote:
> > >>>>>>
> > >>>>>> Hi all,
> > >>>>>>
> > >>>>>> As an experiment with cubical agda, I was trying to define a
> quotiented version of the Delay monad as a higher inductive type. I'm using
> this definition:
> > >>>>>>
> > >>>>>> data Delay (A : Set ℓ) : Set ℓ
> > >>>>>>
> > >>>>>> record Delay′ (A : Set ℓ) : Set ℓ where
> > >>>>>>     coinductive
> > >>>>>>     field
> > >>>>>>       force : Delay A
> > >>>>>>
> > >>>>>> open Delay′ public
> > >>>>>>
> > >>>>>> data Delay A where
> > >>>>>>     now   : A → Delay A
> > >>>>>>     later : Delay′ A → Delay A
> > >>>>>>     step  : (x : Delay′ A) → later x ≡ x .force
> > >>>>>>
> > >>>>>> I managed to implement some basic functions on it but I got stuck
> on trying to prove the looping computation 'never' does not in fact
> evaluate to any value. My code is available here:
> https://github.com/jespercockx/cubical/commit/f1647a90c1b27aadd5da748f08e23630221cc3d9
> I looked at the problem together with Christian Sattler and we are not even
> sure it is actually provable. Does anyone have an idea how to proceed? Or
> has someone already experimented with coinductive types in cubical and
> encountered similar problems? (I looked at the paper "Partiality revisited"
> by Thorsten, Nisse and Nicolai but they use a very different definition of
> the partiality monad.)
> > >>>>>>
> > >>>>>> Cheers,
> > >>>>>> Jesper
> > >>>>>>
> > >>>>>>
> > >>>>>> _______________________________________________
> > >>>>>> Agda mailing list
> > >>>>>> Agda at lists.chalmers.se
> > >>>>>> https://lists.chalmers.se/mailman/listinfo/agda
> > >>>>>>
> > >>>>>>
> > >>>>>> _______________________________________________
> > >>>>>> Agda mailing list
> > >>>>>> Agda at lists.chalmers.se
> > >>>>>> https://lists.chalmers.se/mailman/listinfo/agda
> > >>>> _______________________________________________
> > >>>> Agda mailing list
> > >>>> Agda at lists.chalmers.se
> > >>>> https://lists.chalmers.se/mailman/listinfo/agda
> >
> _______________________________________________
> Agda mailing list
> Agda at lists.chalmers.se
> https://lists.chalmers.se/mailman/listinfo/agda
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.chalmers.se/pipermail/agda/attachments/20190410/520807b1/attachment.html>