[Agda] Why is propositional equality "Set a" instead of just Set?
Dan Licata
drl at cs.cmu.edu
Sun Aug 18 22:49:34 CEST 2013
The impredicativity also shows up if you have function extensionality or
univalence. E.g.
Set0 -> Set0 : Set1
so if equality is always in Set0, then
_==_{Set0 -> Set0} A B : Set0
But by function extensionality, A == B is equivalent to
(X : Set0) -> A X =={Set0} B X
So now there is a type in Set0 equivalent to a Pi whose domain is Set0,
which seems fishy on predicative grounds.
With univalence, you could
smuggle any bijection at any size into Set0.
But then equality is no longer a "proposition", so it's a different
setting than the original question asked about.
-Dan
On Aug18, Andreas Abel wrote:
> Putting propositional equality in a lower universe than the type it speaks
> about is at least problematic when you want to construct a PER model of
> type theory predicatively.
>
> Usually you have a rule like
>
> a = b : A
> -----------------------
> refl : _==_ {A = A} a b
>
> thus, you need to know the semantics of A before you can define the
> semantics of _==_ {A = A}, i.e., before you can define the proofs of
> equality of elements of type A.
>
> Now if you allow equality to be in a lower universe than the type it refers
> to, say
>
> A : Set 1
> -----------------------
> _==_ {A = A} : Set 0
>
> then I do not see a direct way to define the universes Set n by induction
> on n (since you need to know Set 1 before you can define Set 0, and vice
> versa).
>
> Frankly, I have not encountered a model of type theory that would allow
> generalized _==_; if you find one, let me know, I'd like to study it.
>
> Best,
> Andreas
>
> On 18.08.13 4:49 PM, Jonathan Leivent wrote:
>> On 08/17/2013 11:56 PM, Wolfram Kahl wrote:
>>> We define a fully |Level|-polymorphic variant of propositional equality
>>> in order to be able to define fully |Level|-polymorphic identity
>>> relations.
>>> (Thorsten Altenkirch%
>>> \footnote{at AIM XIII, April 2011}
>>> and others
>>> have expressed reservations whether such a definition
>>> at levels |k| lower than |a| should be legal.)
>>>
>>> \begin{code}
>>> data _≡≡_ {k a : Level} {A : Set a} (x : A) : A → Set k where
>>> ≡≡-refl : x ≡≡ x
>>> \end{code}
>>> ----------------------------------------------------------------------------
>
>>
>> I didn't expect a controversial aspect of type theory to turn up in such
>> a ubiquitous case. I was just basing my level assignments on the
>> constraints that the type checker finds.
>>
>> Is there some expectation as to what legal behavior Agda would
>> eventually adopt? Would every definition have to have a level at least
>> as high as all of its parameters/indexes?
>
>
>
> Theoretical Computer Science, University of Munich
> Oettingenstr. 67, D-80538 Munich, GERMANY
>
> andreas.abel at ifi.lmu.de
> http://www2.tcs.ifi.lmu.de/~abel/
> _______________________________________________
> Agda mailing list
> Agda at lists.chalmers.se
> https://lists.chalmers.se/mailman/listinfo/agda
More information about the Agda
mailing list