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Formalising Formulas-as-Types-as-Objects

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Book cover Types for Proofs and Programs (TYPES 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1956))

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Abstract

We describe a formalisation of the Curry-Howard-Lawvere correspondence between the natural deduction system for minimal logic, the typed lambda calculus and Cartesian closed categories. We formalise the type of natural deduction proof trees as a family of sets Γ ⊢ A indexed by the current assumption list Γ and the conclusion A and organise numerous useful lemmas about proof trees categorically. We prove categorical properties about proof trees up to (syntactic) identity as well as up to ßη-convertibility. We prove that our notion of proof trees is equivalent in an appropriate sense to more traditional representations of lambda terms.

The formalisation is carried out in the proof assistant ALF for Martin- Löf type theory.

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Author information

Authors and Affiliations

  1. Department of Computing Science, Chalmers University of Technology and Göteborg University, 412 96, Göteborg

    Qiao Haiyan

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  1. Qiao Haiyan
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Editor information

Editors and Affiliations

  1. Department of Computing Science, Chalmers University of Technology and University of Göteborg, 41296, Göteborg, Sweden

    Thierry Coquand , Peter Dybjer , Bengt Nordström  & Jan Smith , ,  & 

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Haiyan, Q. (2000). Formalising Formulas-as-Types-as-Objects. In: Coquand, T., Dybjer, P., Nordström, B., Smith, J. (eds) Types for Proofs and Programs. TYPES 1999. Lecture Notes in Computer Science, vol 1956. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44557-9_11

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  • DOI: https://doi.org/10.1007/3-540-44557-9_11

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41517-6

  • Online ISBN: 978-3-540-44557-9

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