[Agda] Quotient types in Agda, if you Trust Me

[Thorsten Altenkirch]

Just a general remark: if we just want to have hsets we know how to
implement this so that canonicity holds based on our work on Observational
Type Theory. Conor and his team went some way to do when working on
Epigram 2, which alas never was finished.

At some point we discussed adding this to Agda at an AIM. One problem is
that you can¹t just allow pattern matching on inductive families which
implicitly use equality, i.e. which repeat variables (such as
propositional equality itself).

Maybe we should revisit this. However, it would be good to do this in a
way which is compatible with future extensions to capture HoTT properly.

Also we should¹t just have quotient types but 1st order HITs (I call the
quotient-inductive types or QITs). There are a number of applications
where QITs are essential and they generalise quotient types. E.g. This is
used to formalise the Reals in the book and Ambrus and I are working on
Type Theory in Type Theory based on this.

Thorsten





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