Discussion of Homotopy Type Theory and Univalent Foundations
 help / color / mirror / Atom feed
From: Andrew Polonsky <andrew....@gmail.com>
To: Homotopy Type Theory <HomotopyT...@googlegroups.com>
Cc: jason...@gmail.com
Subject: Re: [HoTT] Re: Meta-conjecture about MLTT
Date: Mon, 12 Sep 2016 03:07:08 -0700 (PDT)	[thread overview]
Message-ID: <a616964a-d650-4b95-b395-aa8e8af0f1d8@googlegroups.com> (raw)
In-Reply-To: <CABcT7WCAoDsh_Jz_e-JRMEk4E0cLQ=kM2FgsLCPA5SdtURaLjA@mail.gmail.com>


[-- Attachment #1.1: Type: text/plain, Size: 1072 bytes --]

Hm, the two types are not provably isomorphic... Ok, this is a tricky 
question indeed!

On Monday, September 12, 2016 at 11:55:45 AM UTC+2, Andrew Polonsky wrote:
>
> > I suspect "one of them provably has but the other doesn't" should mean 
> "one 
> > of them provably has but the other provably does not have". 
>
> Type theory with reflection rule usually contains the rule equating 
> all proofs of equality: 
>
>        p : Id(A;x,y) 
> --------------------------- 
>   p == refl : Id(A;x,y) 
>
> If this is admitted, then the following modified example works: 
>
> A = Id(U;Bool,Bool) 
> B = Iso(Bool,Bool) 
> P(X) = Id(U;A,X) 
>
> Clearly, P(A).  Suppose P(B).  By equality reflection, A == B : U. 
> But then, for x : Iso(Bool,Bool), we have x : Id(U;Bool,Bool), whence x == 
> refl. 
> Since this holds for arbitrary x, we have x==y for arbitrary x,y : 
> Iso(Bool,Bool), a contradiction. 
>
> Generally, I think any proof that equality reflection is inconsistent 
> with univalence could be adapted to yield a counterexample as above. 
>
> Cheers, 
> Andrew 
>

[-- Attachment #1.2: Type: text/html, Size: 1319 bytes --]

  reply	other threads:[~2016-09-12 10:07 UTC|newest]

Thread overview: 17+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2016-09-11 20:47 Martin Escardo
2016-09-11 22:08 ` [HoTT] " Peter LeFanu Lumsdaine
2016-09-11 22:26   ` Martin Escardo
2016-09-12  9:11   ` Thomas Streicher
2016-09-12  5:20 ` Andrew Polonsky
2016-09-12  5:59   ` [HoTT] " Jason Gross
2016-09-12  9:55     ` Andrew Polonsky
2016-09-12 10:07       ` Andrew Polonsky [this message]
2016-09-12 10:35       ` Nicolai Kraus
2016-09-12 10:16   ` Peter LeFanu Lumsdaine
2016-09-12 10:44 ` [HoTT] " Nicolai Kraus
2016-09-12 11:02 ` Andrej Bauer
2016-09-12 11:14   ` Thomas Streicher
2016-09-12 11:23     ` Andrew Polonsky
2016-09-12 11:41       ` Thomas Streicher
2016-09-12 12:47 ` Thomas Streicher
2016-09-12 13:01   ` Martin Escardo

Reply instructions:

You may reply publicly to this message via plain-text email
using any one of the following methods:

* Save the following mbox file, import it into your mail client,
  and reply-to-all from there: mbox

  Avoid top-posting and favor interleaved quoting:
  https://en.wikipedia.org/wiki/Posting_style#Interleaved_style

* Reply using the --to, --cc, and --in-reply-to
  switches of git-send-email(1):

  git send-email \
    --in-reply-to=a616964a-d650-4b95-b395-aa8e8af0f1d8@googlegroups.com \
    --to="andrew...."@gmail.com \
    --cc="HomotopyT..."@googlegroups.com \
    --cc="jason..."@gmail.com \
    /path/to/YOUR_REPLY

  https://kernel.org/pub/software/scm/git/docs/git-send-email.html

* If your mail client supports setting the In-Reply-To header
  via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).