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

Thorsten Altenkirch Thorsten.Altenkirch at nottingham.ac.uk
Mon Sep 5 11:38:40 CEST 2022


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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