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Inductively defined types in the Calculus of Constructions

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Mathematical Foundations of Programming Semantics (MFPS 1989)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 442))

Abstract

We define the notion of an inductively defined type in the Calculus of Constructions and show how inductively defined types can be represented by closed types. We show that all primitive recursive functionals over these inductively defined types are also representable. This generalizes work by Böhm & Berarducci on synthesis of functions on term algebras in the second-order polymorphic λ-calculus (F 2). We give several applications of this generalization, including a representation of F 2-programs in F 3, along with a definition of functions reify, reflect, and eval for F 2 in F 3. We also show how to define induction over inductively defined types and sketch some results that show that the extension of the Calculus of Construction by induction principles does not alter the set of functions in its computational fragment, F ω. This is because a proof by induction can be realized by primitive recursion, which is already definable in F ω.

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Author information

Authors and Affiliations

  1. School of Computer Science, Carnegie Mellon University, 5000 Forbes Avenue, 15213, Pittsburgh, Pennsylvania

    Frank Pfenning

  2. Ecole Normale Superiéure, INRIA and LIENS, URA CNRS 1327, 45 Rue d'Ulm, 75005, Paris, France

    Christine Paulin-Mohring

Authors
  1. Frank Pfenning
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  2. Christine Paulin-Mohring
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Editor information

M. Main A. Melton M. Mislove D. Schmidt

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© 1990 Springer-Verlag Berlin Heidelberg

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Pfenning, F., Paulin-Mohring, C. (1990). Inductively defined types in the Calculus of Constructions. In: Main, M., Melton, A., Mislove, M., Schmidt, D. (eds) Mathematical Foundations of Programming Semantics. MFPS 1989. Lecture Notes in Computer Science, vol 442. Springer, New York, NY. https://doi.org/10.1007/BFb0040259

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  • DOI: https://doi.org/10.1007/BFb0040259

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  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-97375-3

  • Online ISBN: 978-0-387-34808-7

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