[Agda] «Extensionally but not definitionally equal» — can I say that?

[Nicolai Kraus]

Apostolis, what you wrote is right, but here's how I suggest looking at
this: There are two levels, the object theory and the meta-theory. In our
case, the object theory is Agda. Everything you implement lives on that
level. The meta-theory is the level "above" (say, the level at which we
live), which allows you to reason about how Agda behaves. "Definitionally
equal" and "type-checks" are concepts of the meta-level. The object theory
doesn't know about them and can't talk about them. Thus, the object theory
can't say whether two implementations are definitionally distinct.
Thorsten's solution essentially shifts everything by one level. The
function codes become the new "object level", and Agda can now serve as
meta-theory to that new level. That's why Agda can suddenly talk about
intensional features of the addition function codes.
Nicolai



On Tue, Sep 6, 2022 at 11:45 AM Apostolis Xekoukoulotakis <
apostolis.xekoukoulotakis at gmail.com> wrote:

> I was thinking that something that can not be shown definitionally equal
> by the type checker does not tell us anything about whether it is
> propositionally equal or not. ( I was assuming that this would be the way
> to encode "definitional" equality in Agda)
>
> But now I understand your point. Any data structure that would indicate
> that two functions are intensionally distinct would be impossible since
> extensionally equal functions are equal in cubical theory.
>
> Thus we would be able to show that a function f is intensionally distinct
> with itself.
>
>
> On Tue, Sep 6, 2022 at 12:36 AM Thorsten Altenkirch <
> Thorsten.Altenkirch at nottingham.ac.uk> wrote:
>
>> How does this matter?
>>
>> Get Outlook for iOS <https://aka.ms/o0ukef>
>> ------------------------------
>> *From:* Apostolis Xekoukoulotakis <apostolis.xekoukoulotakis at gmail.com>
>> *Sent:* Monday, September 5, 2022 8:35:28 PM
>> *To:* Thorsten Altenkirch (staff) <psztxa at exmail.nottingham.ac.uk>
>> *Cc:* Ignat Insarov <kindaro at gmail.com>; agda at lists.chalmers.se <
>> agda at lists.chalmers.se>
>> *Subject:* Re: [Agda] «Extensionally but not definitionally equal» — can
>> I say that?
>>
>> *I think that one missing piece of information here is that the fact that
>> you cant prove that two things are equal doesnt not mean that they are not
>> equal.*
>>
>> On Mon, Sep 5, 2022 at 12:38 PM Thorsten Altenkirch <
>> Thorsten.Altenkirch at nottingham.ac.uk> wrote:
>>
>> No you cannot distinguish extensionally equal object in type theory.
>> Otherwise extensionality as provided by cubical agda would be inconsistent.
>> This is a feature, not a bug.
>>
>>
>>
>> If you want to talk about intensional aspects of functions you need to
>> talk about function codes not functions. That is you need to implement a
>> representation of functions that reveals the intensional aspects you want
>> to talk about. In your case you may want to use a monad (I think it is
>> called the writer monad) which counts the number of steps and then work in
>> this monad.
>>
>>
>>
>> Cheers,
>>
>> Thorsten
>>
>>
>>
>> *From: *Agda <agda-bounces at lists.chalmers.se> on behalf of Ignat Insarov
>> <kindaro at gmail.com>
>> *Date: *Saturday, 3 September 2022 at 10:45
>> *To: *agda at lists.chalmers.se <agda at lists.chalmers.se>
>> *Subject: *[Agda] «Extensionally but not definitionally equal» — can I
>> say that?
>>
>> Hello!
>>
>> Suppose I have two definitions of addition — one works on Peano
>> numbers and the other works in binary representation. Can I express in
>> Agda that these two definitions are extensionally equal but
>> definitionally distinct?
>>
>> Ideally in the future I want to proceed to reasoning about their
>> asymptotic performance (linear versus logarithmic). So, I want to have
>> several notions of equality, finer than the commonly postulated
>> functional extensionality.
>>
>> The way I imagine this could go is by reifying the definition of said
>> functions as a syntactic tree or another appropriate encoding of the
>> way Agda sees them. Then I should say «these two functions are
>> extensionally equal × their representation as syntactic trees is
>> distinct». Is this realistic? Are there other approaches?
>>
>> See also on Zulip:
>> <
>> https://agda.zulipchat.com/#narrow/stream/238741-general/topic/Extensionally.20but.20not.20definitionally.20equal.20.E2.80.94.20can.20I.20say.20that.3F/near/296889550
>> >
>> _______________________________________________
>> Agda mailing list
>> Agda at lists.chalmers.se
>> https://lists.chalmers.se/mailman/listinfo/agda
>>
>> This message and any attachment are intended solely for the addressee
>> and may contain confidential information. If you have received this
>> message in error, please contact the sender and delete the email and
>> attachment.
>>
>> Any views or opinions expressed by the author of this email do not
>> necessarily reflect the views of the University of Nottingham. Email
>> communications with the University of Nottingham may be monitored
>> where permitted by law.
>>
>>
>>
>>
>> _______________________________________________
>> Agda mailing list
>> Agda at lists.chalmers.se
>> https://lists.chalmers.se/mailman/listinfo/agda
>>
>> This message and any attachment are intended solely for the addressee
>> and may contain confidential information. If you have received this
>> message in error, please contact the sender and delete the email and
>> attachment.
>>
>> Any views or opinions expressed by the author of this email do not
>> necessarily reflect the views of the University of Nottingham. Email
>> communications with the University of Nottingham may be monitored
>> where permitted by law.
>>
>>
>>
>>
>> _______________________________________________
> 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/20220906/a679f7d6/attachment.html>