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For Finitary Induction-Induction, Induction Is Enough

Authors Ambrus Kaposi , András Kovács , Ambroise Lafont



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Author Details

Ambrus Kaposi
  • Eötvös Loránd University, Budapest, Hungary
András Kovács
  • Eötvös Loránd University, Budapest, Hungary
Ambroise Lafont
  • IMT Atlantique, Inria, LS2N CNRS, Nantes, France

Acknowledgements

The authors would like to thank Thorsten Altenkirch, Rafaël Bocquet, Simon Boulier, Fredrik Nordvall-Forsberg and Jakob von Raumer for discussions on the topics of this paper. We also thank the anonymous reviewers for their helpful comments and suggestions.

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Ambrus Kaposi, András Kovács, and Ambroise Lafont. For Finitary Induction-Induction, Induction Is Enough. In 25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 6:1-6:30, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.TYPES.2019.6

Abstract

Inductive-inductive types (IITs) are a generalisation of inductive types in type theory. They allow the mutual definition of types with multiple sorts where later sorts can be indexed by previous ones. An example is the Chapman-style syntax of type theory with conversion relations for each sort where e.g. the sort of types is indexed by contexts. In this paper we show that if a model of extensional type theory (ETT) supports indexed W-types, then it supports finitely branching IITs. We use a small internal type theory called the theory of signatures to specify IITs. We show that if a model of ETT supports the syntax for the theory of signatures, then it supports all IITs. We construct this syntax from indexed W-types using preterms and typing relations and prove its initiality following Streicher. The construction of the syntax and its initiality proof were formalised in Agda.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
Keywords
  • type theory
  • inductive types
  • inductive-inductive types

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References

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