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Erschienen in:
2013 | OriginalPaper | Buchkapitel
Generalizations of Hedberg’s Theorem
verfasst von : Nicolai Kraus, Martín Escardó, Thierry Coquand, Thorsten Altenkirch
Erschienen in: Typed Lambda Calculi and Applications
Verlag: Springer Berlin Heidelberg
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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.