Generalizations of Hedberg's theorem

Abstract

As the groupoid interpretation by Hofmann and Streicher shows, uniqueness of identity proofs (UIP) is not provable. Generalizing a theorem by Hedberg, we give new characterizations of types that satisfy UIP. It turns out to be natural in this context to consider constant endofunctions. For such a function, we can look at the type of its fixed points. We show that this type has at most one element, which is a nontrivial lemma in the absence of UIP. As an application, a new notion of anonymous existence can be defined. One further main result is that, if every type has a constant endofunction, then all equalities are decidable. All the proofs have been formalized in Agda. © 2013 Springer-Verlag.