Re: [HoTT] Re: HITs in Agda

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

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On Thursday, 10 January 2019 21:05:09 UTC, E Cavallo wrote:
>
> Our recent paper explains how to handle inductive families in cartesian 
> cubical type theory: http://www.cs.cmu.edu/~ecavallo/works/popl19.pdf. 
>

Nice.
 

> We believe something similar will work for De Morgan cubical type theory, 
> and I think Andrea and Anders are planning to work that out and add it to 
> cubical Agda. 
>

This is good too.
 

> For the moment it looks like cubical Agda will let you declare inductive 
> families, but will not reduce the Kan operations in those types.
>

Right. For the moment, you can prove that Agda's identity type (defined as 
an inductive family) is equivalent to the cubical identity type (because 
both have refl and J). However, the computations get stuck.

In fact, I tried this in order to be able to use Agda's pattern macthing on 
refl, rather than J on the cubical identity type, by going back-and-forth, 
but, as you say, the computations get stuck. 

If you manage to get the computations to go through, as you discuss above, 
with the work of Andreas and Anders, then this means that we can start 
using pattern matching on refl in cubical Agda without having to change 
Agda in any way other than accounting for inductive families (by the back 
and forth trick).
That would be nice for me, because what is preventing me from migrating 
from Agda to cubical Agda in my univalent development is that fact that it 
is populated by definitions by pattern matching on refl (and everything 
else) everywhere (according to the Agda style). 

So I am looking forward to the outcome of the developments you are 
advertising.

Martin

 

> Evan
>
> 2019年1月10日(木) 15:54 Martín Hötzel Escardó <escardo...@gmail.com 
> <javascript:>>:
>
>> 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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