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Notions of anonymous existence in Martin-Löf type theory

Kraus, Nicolai; Escardo, Martin; Coquand, Thierry; Altenkirch, Thorsten

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Martin Escardo

Thierry Coquand


As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of 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 non-trivial 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.


Kraus, N., Escardo, M., Coquand, T., & Altenkirch, T. (in press). Notions of anonymous existence in Martin-Löf type theory. Logical Methods in Computer Science, 13(1),

Journal Article Type Article
Acceptance Date Mar 24, 2017
Online Publication Date Mar 24, 2017
Deposit Date Nov 1, 2016
Publicly Available Date Mar 24, 2017
Journal Logical Methods in Computer Science
Print ISSN 1860-5974
Electronic ISSN 1860-5974
Publisher Logical Methods in Computer Science
Peer Reviewed Peer Reviewed
Volume 13
Issue 1
Keywords homotopy type theory, Hedberg’s theorem, anonymous existence, weakly constant functions, factorization, truncation, squash types, bracket types, coherence conditions
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Volume: Volume 13, Issue 1
Published on: March 24, 2017
Submitted on: March 24, 2017
Keywords: Computer Science - Logic in Computer Science,03B15,F.4.1

2020-09-21 NW


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