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A Dependent Type Theory with Names and Binding

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Computer Science Logic (CSL 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3210))

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Abstract

We consider the problem of providing formal support for working with abstract syntax involving variable binders. Gabbay and Pitts have shown in their work on Fraenkel-Mostowski (FM) set theory how to address this through first-class names: in this paper we present a dependent type theory for programming and reasoning with such names. Our development is based on a categorical axiomatisation of names, with freshness as its central notion. An associated adjunction captures constructions known from FM theory: the freshness quantifier И, name-binding, and unique choice of fresh names. The Schanuel topos — the category underlying FM set theory — is an instance of this axiomatisation. Working from the categorical structure, we define a dependent type theory which it models. This uses bunches to integrate the monoidal structure corresponding to freshness, from which we define novel multiplicative dependent products ∏ * and sums ∑ *, as well as a propositions-as-types generalisation H of the freshness quantifier.

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Author information

Authors and Affiliations

  1. LFCS, School of Informatics, University of Edinburgh JCMB, King’s Buildings, Edinburgh, EH9 3JZ, Germany

    Ulrich Schöpp & Ian Stark

Authors
  1. Ulrich Schöpp
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  2. Ian Stark
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Editor information

Editors and Affiliations

  1. Wroclaw University, Poland

    Jerzy Marcinkowski

  2. Polish Academy of Sciences, Institute of Computer Science, Poland

    Andrzej Tarlecki

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Schöpp, U., Stark, I. (2004). A Dependent Type Theory with Names and Binding. In: Marcinkowski, J., Tarlecki, A. (eds) Computer Science Logic. CSL 2004. Lecture Notes in Computer Science, vol 3210. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30124-0_20

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  • DOI: https://doi.org/10.1007/978-3-540-30124-0_20

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-23024-3

  • Online ISBN: 978-3-540-30124-0

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