From: Robin Adams <robin....@gmail.com>
To: Homotopy Type Theory <HomotopyT...@googlegroups.com>
Subject: Re: A puzzle about "univalent equality"
Date: Mon, 19 Sep 2016 05:40:18 -0700 (PDT) [thread overview]
Message-ID: <cea5c0b8-bfc9-4967-8b76-8e6fb5088b8b@googlegroups.com> (raw)
In-Reply-To: <876efd6d-5c0b-488d-a72a-0a2d14ecb0ec@googlegroups.com>
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Very interesting thread. I'd like to add an observation: in cubical type
theory, the terms X and Y are definitionally. equal.
Proof: Let us use s == t to denote that s and t are the same expression,
and s = t for definitional equality.
Let e0 be the term such that x : Bool, y : Bool, i : I |- e0 : Path Bool (f
x y) (g x y), so that e is the result of applying functional extensionality
to e0, that is
e == <i> \ x y . e0
Note that
e0[x := tt, y := tt] = Bool
e0[x := tt, y := ff] = Unit
e0[x := ff, y := tt] = Unit
e0[x := ff, y := ff] = Unit
Then we have
Y == comp^i (psi (e i)) [] X
= comp^i (phi (e i tt tt) x phi(e i tt ff) x phi(e i ff tt) x phi(e i ff
ff)) [] X
= comp^i (Bool x Unit x Unit x Unit) [] X
= < comp^i Bool [] tt , comp^i Unit [] *, comp^i Unit [] *, comp^i Unit
[] * > (computation rule for comp with Sigma)
= < tt, *, *, * > (computation rule for comp with Bool and Unit)
== X
Note that this only works because X is a canonical object; it doesn't hold
for an arbitrary term of type Psi f. (Because we don't, in general, have
that comp maps 1_A to 1_A up to definitional equality; that is, we don't
have comp^i A [] t = t where i does not occur free in A.)
--
Robin
On Monday, 5 September 2016 18:54:14 UTC+2, Andrew Polonsky wrote:
>
> Hi,
>
> There is a common understanding that the "right" concept of equality in
> Martin-Lof type theory is not the intensional identity type, but is a
> different notion of equality, which is extensional. The adjunction of the
> univalence axiom to standard MLTT makes the identity type behave like this
> "intended" equality --- but it breaks the computational edifice of type
> theory.
>
> More precisely, this "Hott book" approach only nails down the concept of
> equality with respect to its *logical properties* --- the things you can
> prove about it. But not its actual computational behavior.
>
> Since computational behavior can often be "seen" on the logical level, I
> am trying to get a better understanding of the sense in which this "Hott
> book" equality type is really (in)complete. How would a "truly
> computational" equality type be different from it? (Other than satisfying
> canonicity, etc.)
>
> One precise example is that, for the "right" notion of equality, the
> equivalences characterizing path types of standard type constructors proved
> in Chapter 2 of the book could perhaps hold definitionally. (The theorems
> proved there are thus "seeing" these equalities on the logical level.)
>
> I tried to look for a more interesting example, and came up with the
> following puzzle.
>
> f, g : Bool -> Bool -> Bool
> f x y = if x then y else ff
> g x y = if y then x else ff
>
> e : f = g
> e = FE(...) [using UA to get Function Extensionality]
>
> Phi : Bool -> Type
> Phi tt = Bool
> Phi ff = Unit
>
> Psi : (Bool->Bool->Bool)->Type
> Psi = \u. (u tt tt) x (u tt ff) x (u ff tt) x (u ff ff)
>
> X : Psi f
> X = (tt,*,*,*)
>
> Y : Psi g
> Y = subst Psi e X
>
> QUESTION.
> Can we prove, in "book Hott", that "proj1 Y = tt" is inhabited?
>
> Cheers!
> Andrew
>
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prev parent reply other threads:[~2016-09-19 12:40 UTC|newest]
Thread overview: 18+ messages / expand[flat|nested] mbox.gz Atom feed top
2016-09-05 16:54 Andrew Polonsky
2016-09-05 21:40 ` [HoTT] " Michael Shulman
2016-09-05 21:51 ` Dan Licata
2016-09-06 7:30 ` Andrew Polonsky
2016-09-06 12:32 ` Michael Shulman
2016-09-06 12:56 ` Dan Licata
2016-09-06 12:57 ` Peter LeFanu Lumsdaine
2016-09-06 13:44 ` Andrew Polonsky
2016-09-06 22:14 ` Martin Escardo
2016-09-07 23:18 ` Matt Oliveri
2016-09-08 4:14 ` Michael Shulman
2016-09-08 6:06 ` Jason Gross
2016-09-08 9:11 ` Martin Escardo
2016-09-08 6:34 ` Matt Oliveri
2016-09-08 6:45 ` Michael Shulman
2016-09-08 9:07 ` Martin Escardo
2016-09-08 9:51 ` Thomas Streicher
2016-09-19 12:40 ` Robin Adams [this message]
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