A Categorical Semantics for Inductive-Inductive Definitions

  • Thorsten Altenkirch
  • Peter Morris
  • Fredrik Nordvall Forsberg
  • Anton Setzer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6859)

Abstract

Induction-induction is a principle for defining data types in Martin-Löf Type Theory. An inductive-inductive definition consists of a set A, together with an A-indexed family B : A → Set, where both A and B are inductively defined in such a way that the constructors for A can refer to B and vice versa. In addition, the constructors for B can refer to the constructors for A. We extend the usual initial algebra semantics for ordinary inductive data types to the inductive-inductive setting by considering dialgebras instead of ordinary algebras. This gives a new and compact formalisation of inductive-inductive definitions, which we prove is equivalent to the usual formulation with elimination rules.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Thorsten Altenkirch
    • 1
  • Peter Morris
    • 1
  • Fredrik Nordvall Forsberg
    • 2
  • Anton Setzer
    • 2
  1. 1.School of Computer ScienceUniversity of NottinghamUK
  2. 2.Department of Computer ScienceSwansea UniversityUK

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