[TYPES] Equivalence invariance for MLTT (reference request)

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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