Notions of anonymous existence in Martin-Löf type theory

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Kraus, Nicolai, Escardo, Martin, Coquand, Thierry and Altenkirch, Thorsten (2016) Notions of anonymous existence in Martin-Löf type theory. Logical Methods in Computer Science . ISSN 1860-5974 (In Press)

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Abstract

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.

Item Type:

Article

RIS ID:

https://nottingham-repository.worktribe.com/output/823981

Keywords:

homotopy type theory, Hedberg’s theorem, anonymous existence, weakly constant functions, factorization, truncation, squash types, bracket types, coherence conditions

Schools/Departments:

University of Nottingham, UK > Faculty of Science > School of Computer Science

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