Reason for equality definition in COQ and HOTT

Question

In HOTT and also in COQ one cannot prove UIP, i.e.
\Prod_{p:a=a} p = refl a

But one can prove:
\Prod_{p:a=a} (a,p) = (a, refl a)

Why is this defined as it is? Is it, because one wants to have a nice homotopy interpretation? Or is there some natural, deeper reason for this definition?

Answer 1

Today we know of a good reason for rejecting UIP: it is incompatible with the principle of univalence from homotopy type theory, which roughly says that isomorphic types can be identified. However, as far as I am aware, the reason that Coq's equality does not validate UIP is mostly a historical accident inherited from one of its ancestors: Martin-Löf's intensional type theory, which predates HoTT by many years.

The behavior of equality in ITT was originally motivated by the desire to keep type checking decidable. This is possible in ITT because it requires us to explicitly mark every rewriting step in a proof. (Formally, these rewriting steps correspond to the use of the equality eliminator eq_rect in Coq.) By contrast, Martin-Löf designed another system called extensional type theory where rewriting is implicit: whenever two terms a and b are equal, in the sense that we can prove that a = b, they can be used interchangeably. This relies on an equality reflection rule which says that propositionally equal elements are also definitionally equal. Unfortunately, there is a price to pay for this convenience: type checking becomes undecidable. Roughly speaking, the type-checking algorithm relies crucially on the explicit rewriting steps of ITT to guide its computation, whereas these hints are absent in ETT.

We can prove UIP easily in ETT because of the equality reflection rule; however, it was unknown for a long time whether UIP was provable in ITT. We had to wait until the 90's for the work of Hofmann and Streicher, which showed that UIP cannot be proved in ITT by constructing a model where UIP is not valid. (Check also these slides by Hofmann, which explain the issue from a historic perspective.)

Edit

This doesn' t mean that UIP is incompatible with decidable type checking: it was shown later that it can be derived in other decidable variants of Martin-Löf type theory (such as Agda), and it can be safely added as an axiom in a system like Coq.