From: Andrew Polonsky <andrew....@gmail.com>
To: Jason Gross <jason...@gmail.com>
Cc: Homotopy Type Theory <HomotopyT...@googlegroups.com>
Subject: Re: [HoTT] Re: Meta-conjecture about MLTT
Date: Mon, 12 Sep 2016 11:55:43 +0200 [thread overview]
Message-ID: <CABcT7WCAoDsh_Jz_e-JRMEk4E0cLQ=kM2FgsLCPA5SdtURaLjA@mail.gmail.com> (raw)
In-Reply-To: <CAKObCaooaVyFoh-Xr5S8gMj--35FQNUGwQZ8t0CejRQhhRnNJA@mail.gmail.com>
> 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
next prev parent reply other threads:[~2016-09-12 9:55 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 [this message]
2016-09-12 10:07 ` Andrew Polonsky
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
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