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 on > behalf of Thomas Streicher" > 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 > > -- > You received this message because you are subscribed to the Google > Groups "Homotopy Type Theory" group. > To unsubscribe from this group and stop receiving emails from it, send > an email to HomotopyTypeTheory+unsubscribe@googlegroups.com . > > For more options, visit https://groups.google.com/d/optout. > > > This message and any attachment are intended solely for the addressee > and may contain confidential information. If you have received this > message in error, please contact the sender and delete the email and > attachment. > > Any views or opinions expressed by the author of this email do not > necessarily reflect the views of the University of Nottingham. Email > communications with the University of Nottingham may be monitored > where permitted by law. > > > -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout.