I'm not completely satisfied with Hugunin's technique, because it justifies only the "simple" elimination principle, rather than the general, "recursive-recursive" elimination principle implemented in Agda. As far as I can tell, realistic use cases for inductive-inductive families also need the recursive-recursive elimination principle, where the types of the maps out of later families depend on the maps out of earlier families. (I'm not sure how much of the other research on IIFs stops short of handling recursion-recursion, but I think it should be taken seriously.)

On Monday, May 20, 2019 at 7:26:12 PM UTC-4, Jon Sterling wrote:

Echoing Andras, I recall that a new encoding due to Jasper Hugunin enable us to interpret IITs without using UIP, and it is an open question to determine whether Jasper's ideas can be extended to QIITs. I hope they can, and someone should find out :)

Best,
Jon