Re: [HoTT] Re: HITs in Agda

["Martín Hötzel Escardó" <escardo.martin@gmail.com>]

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Actually, I think it is not a priori clear how Agda's --without-K interacts 
with --cubical.

For one thing, the cubical identity type (derived from the cubical path 
type via Andrew Swan's technique) is not an Agda inductive family and it is 
not Agda's inductively defined identity type. And also, as far as I know, 
inductive families RE an open problem in cubical type theory / the cubical 
model(s).

Any development in Agda invoking --cubical that tries to be sound should, 
for the moment, refrain from using inductive families.

In fact, in would be good to discuss the precautions one should take when 
using --cubical in Agda so that one is guaranteed to be consistent, and 
better, be talking about something that is currently understood (such as 
entities in the cubical model((s)). It is not entirely clear to me which 
features of Agda we can use and which ones we should not use and which ones 
we could use if we knew more.

Martin

On Thursday, 10 January 2019 15:28:07 UTC, Nils Anders Danielsson wrote:
>
> On 10/01/2019 16.19, Ali Caglayan wrote: 
> > I was under the impression that this was in plain agda, that's why it 
> > was more suprising. I didn't realise it was about the cubical agda. 
>
> You get Cubical Agda by using the option --cubical (for instance in a 
> pragma: {-# OPTIONS --cubical #-}). The idea is that it should be sound 
> to import Agda code that uses --without-K from Cubical Agda. 
>
> -- 
> /NAD 
>

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