Discussion of Homotopy Type Theory and Univalent Foundations
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From: Michael Shulman <shulman@sandiego.edu>
To: Valery Isaev <valery.isaev@gmail.com>
Cc: Jon Sterling <jon@jonmsterling.com>,
	 "HomotopyTypeTheory@googlegroups.com"
	<homotopytypetheory@googlegroups.com>
Subject: Re: [HoTT] New theorem prover Arend is released
Date: Sat, 10 Aug 2019 16:37:07 -0700
Message-ID: <CAOvivQxKMPvpm9Vwkt4ARR7R5qLmzJMUkFCrf=rFUL5sd4nXLQ@mail.gmail.com> (raw)
In-Reply-To: <CAA520ftTacC4iegu2UM887nbyJWQTByMKhrcsftKPXCadH06kQ@mail.gmail.com>

On Sat, Aug 10, 2019 at 5:25 AM Valery Isaev <valery.isaev@gmail.com> wrote:
> The document is slightly outdated. We do not have the rule iso A B (λx ⇒ x) (λx ⇒ x) idp idp i ⇒β A in the actual implementation since univalence is true even without it. This rule has another problem. It seems that the theory as presented in the document introduces a quasi-equivalence between A = B and Equiv A B, which means that there are some true statements which are not provable in it.

I don't understand.  By "quasi-equivalence" do you mean an incoherent
equivalence (what the book calls a map with a quasi-inverse)?  If so,
then every quasi-equivalence can of course be promoted to a strong
equivalence.

However, as I said, I'm more worried about the fourth rule coe_{λ k ⇒
iso A B f g p q k} a right ⇒β f a.  That's the one that I have trouble
seeing how to interpret in a model category.  Can you say anything
about that?

> If you can prove that some \data or \record satisfies isSet (or, more generally, that it is an n-type), then you can put this proof in \use \level function corresponding to this definition and it will be put in the corresponding universe.

What does it mean for it to be "put in" the corresponding universe?
The documentation for \use \level makes it sound as though the
definition *itself*, rather than something equivalent to it, ends up
in the corresponding universe.  How is the equivalence between A and
F(A,p) accessed inside the proof assistant?

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  reply index

Thread overview: 20+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-08-06 22:16 Валерий Исаев
2019-08-07 15:01 ` Andrej Bauer
2019-08-07 22:13 ` Nicolai Kraus
2019-08-08  9:55   ` Valery Isaev
2019-08-10  9:47     ` Michael Shulman
2019-08-10 12:30       ` Valery Isaev
2019-08-10 12:37       ` Valery Isaev
2019-08-08 12:20 ` Jon Sterling
2019-08-08 12:29   ` Bas Spitters
2019-08-08 14:44     ` Valery Isaev
2019-08-08 15:11       ` Jon Sterling
2019-08-08 15:22         ` Valery Isaev
2019-08-10  9:42           ` Michael Shulman
2019-08-10 12:24             ` Valery Isaev
2019-08-10 23:37               ` Michael Shulman [this message]
2019-08-11 10:46                 ` Valery Isaev
2019-08-11 12:39                   ` Michael Shulman
2019-08-11 16:55                     ` Michael Shulman
2019-08-12 14:44                       ` Daniel R. Grayson
2019-08-12 17:32                         ` Michael Shulman

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