Path induction implied

Question

This is a follow-up question to Getting path induction to work in Agda

I wonder when that construct may be more expressive. It seems to me we can always express the

Answer 1

To provide a short answer: you're right, Agda's pattern matching implies the existence of a path-induction primitive. In fact, it has been shown that in a type theory with universes, dependent pattern matching is equivalent to the existence of induction primitives for inductive types and the so-called K axiom:

http://link.springer.com/chapter/10.1007/11780274_27

More recently, it has been shown that (the latest implementation of) Agda's --without-K option restricts pattern matching such that it is only equivalent with the existence of induction primitives for inductive types:

http://dl.acm.org/citation.cfm?id=2628136.2628139

Full disclosure: I'm a co-author of the latter work.