Re: [HoTT] Re: Meta-conjecture about MLTT

[Nicolai Kraus <nicola...@gmail.com>]

On 12/09/16 10:55, Andrew Polonsky wrote:
> 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,

This is a consequence of the equality reflection rule, isn't it?
-- Nicolai

> 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
>