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Dear Andrej, I am not completely sure because it is a while since I had watched this, but I think this might be related to the topic of Per Martin-Löf's 2013 Earnest Nagel Lecture “Invariance Under Isomorphism and Definability”, which you can find here: https://urldefense.com/v3/__https://www.cmu.edu/dietrich/philosophy/events/nagel-lectures/past-lectures.html__;!!IBzWLUs!WbQ8Sxq9uOhc-H40-I8IgA8jFEsX9FK7kI0GWPnCsjWqEW3A9Thx7H15rSmUBZyHOvIZExnwb8TJrd9CE7zzeP2HWx8$ . But I don't recall if Per proves the exact theorem you want; it might be a little different, judging from the abstract. Best, Jon On 12 Jun 2023, at 20:56, andrej.ba...@andrej.com wrote: > [ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list ] > > Dear all, > > in preparation for my TYPES 2023 talk I realized I don’t actually know of > anyone having proved the following about MLTT (Σ + Π + Id + Nat). > > EQUIVALENCE INVARIANCE: Let P be a well-formed type expression with a type > meta-variable X. If A and B are closed type expressions and e : A ≃ B an > equivalence between them, then the type of equivalences P[A/X] ≃ P[B/X] is > inhabited. > > There are many possible variants, of course, and I’d be interested in > learning about any results in this direction, especially ones that don’t > throw in any axioms. > > I am vaguely remebering that it has been done for Church’s simple type > theory, which actually sounds, well, simple. Does anyone know a reference? > > I think there might have been some work by Bob Harper & Dan Licata > (https://urldefense.com/v3/__https://www.cs.cmu.edu/*drl/pubs/lh112tt/lh122tt-final.pdf__;fg!!IBzWLUs!Ww4XrFxjdSUVFGjTgD3MToleD85TRwLRGFIy_xjkQqvSw3nuqa9fWxMNxAWYPT5FyS0Rr9hCcWv8p05bA6Xc3ZdFibwiK0ulYns$ > ), and another by Nicolas Tabareau & Matthieu Sozeau > (https://urldefense.com/v3/__https://doi.org/10.1145/3236787__;!!IBzWLUs!Ww4XrFxjdSUVFGjTgD3MToleD85TRwLRGFIy_xjkQqvSw3nuqa9fWxMNxAWYPT5FyS0Rr9hCcWv8p05bA6Xc3ZdFibwin1ZSLzI$ > ), which cuts thing off at the groupoid level. I am not even sure if they > really prove an analogue of the principle stated above. > > But how about pure MLTT, has anyone done it? > > With kind regards, > > Andrej
