Notions of Anonymous Existence in MartinLöf Type Theory
Abstract
As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional MartinLöf type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several nontrivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda.
 Publication:

arXiv eprints
 Pub Date:
 October 2016
 arXiv:
 arXiv:1610.03346
 Bibcode:
 2016arXiv161003346K
 Keywords:

 Computer Science  Logic in Computer Science;
 03B15;
 F.4.1
 EPrint:
 36 pages, to appear in the special issue of TLCA'13 (LMCS)