Re: [HoTT] Re: Why do we need judgmental equality?

["Théo Winterhalter" <theo.winterhalter@gmail.com>]

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Hi,

To follow on the idea of 2-level type theory. It indeed seems enough to 
have a strict equality (meaning with UIP and Funext) to pull off an 
interpretation of weakTT.
In fact, I have updated an formalised the work of Nicolas Oury of 
translating Extensional Type Theory into Intensional Type Theory, and 
extended the proof so that it can be an instance of a translation from an 
extensional 2-level type theory into an intensional one. In this work, the 
treatment of conversion in the target seems to be orthogonal to the result 
itself. With Simon Boulier we plan to tweak the formalisation a bit so that 
it allows to target weakTT (and we already conducted informal reasoning to 
justify it). You can thus give an interpretation to a theory into its weak 
version, provided you have UIP, funext and enough equalities to interpret 
for instance β-reduction.

I hope this is clear.

Théo

Le samedi 9 février 2019 14:29:28 UTC+1, Thorsten Altenkirch a écrit :
>
> Even LF included beta eta hence has a non-trivial judgemental equality. 
> Not to mention the equations for substitution. 
>
> Thorsten 
>
> On 09/02/2019, 11:38, "homotopyt...@googlegroups.com <javascript:> on 
> behalf of Thomas Streicher" <homotopyt...@googlegroups.com <javascript:> 
> on behalf of stre...@mathematik.tu-darmstadt.de <javascript:>> wrote: 
>
>     Working without any judgemental equality was the aim of the original 
>     LF where elements of a type A correspond to formal derivations of A in 
>     abstract syntax. Also Isabelle works a bit like that. 
>     
>     So with a modicum of judgemental equality one uses dependent types for 
>     having a syntax for formal derivations. But, of course, this is 
>     absolutely useless when you want to execute your proofs! 
>     
>     Thomas 
>     
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