Hi,

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 on behalf of Thomas Streicher" <homotopyt...@googlegroups.com on behalf of stre...@mathematik.tu-darmstadt.de> 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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