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 --]
next prev parent 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).