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

[Apostolis Xekoukoulotakis]

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
> >
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