Abstract
We show that strictly positive inductive types, constructed from polynomial functors, constant exponentiation and arbitrarily nested inductive types exist in any Martin-Löf category (extensive locally cartesian closed category with W-types) by exploiting our work on container types. This generalises a result by Dybjer (1997) who showed that non-nested strictly positive inductive types can be represented using W-types. We also provide a detailed analysis of the categorical infrastructure needed to establish the result.
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References
Abbott, M.: Categories of Containers. PhD thesis, University of Leicester (2003)
Abbott, M., Altenkirch, T., Ghani, N.: Categories of containers. In: Gordon, A.D. (ed.) FOSSACS 2003. LNCS, vol. 2620, pp. 23–38. Springer, Heidelberg (2003)
Abbott, M., Altenkirch, T., Ghani, N.: Representing strictly positive types. Presented at APPSEM annual meeting, invited for submission to Theoretical Computer Science (2004)
Abel, A., Altenkirch, T.: Apredicative strong normalisation proof for a λ-calculus with interleaving inductive types. In: Coquand, T., Nordström, B., Dybjer, P., Smith, J. (eds.) TYPES 1999. LNCS, vol. 1956, pp. 21–40. Springer, Heidelberg (2000)
Dybjer, P.: Representing inductively defined sets by wellorderings in Martin-Löf’s type theory. Theoretical Computer Science 176, 329–335 (1997)
Dybjer, P., Setzer, A.: A finite axiomatization of inductive-recursive definitions. In: Typed Lambda Calculus and Applications, pp. 129–146 (1999)
Dybjer, P., Setzer, A.: Indexed induction-recursion. In: Kahle, R., Schroeder-Heister, P., Stärk, R.F. (eds.) PTCS 2001. LNCS, vol. 2183, p. 93. Springer, Heidelberg (2001)
Gambino, N., Hyland, M.: Wellfounded trees and dependent polynomial functors. In: Berardi, S., Coppo, M., Damiani, F. (eds.) TYPES 2003. LNCS, vol. 3085, pp. 210–225. Springer, Heidelberg (2004)
Hofmann, M.: On the interpretation of type theory in locally cartesian closed categories. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 427–441. Springer, Heidelberg (1995)
Jacobs, B.: Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics, vol. 141. Elsevier, Amsterdam (1999)
Johnstone, P.T.: Topos Theory. Academic Press, London (1977)
Martin-Löf, P.: Intuitionistic Type Theory. Bibliopolis, Napoli (1984)
Moerdijk, I., Palmgren, E.: Wellfounded trees in categories. Annals of Pure and Applied Logic 104, 189–218 (2000)
Nordström, B., Petersson, K., Smith, J.M.: Programming in Martin-Löf’s Type Theory. International Series of Monographs on Computer Science, vol. 7. Oxford University Press, Oxford (1990)
Streicher, T.: Semantics of Type Theory. Progress in Theoretical Computer Science. Birkhäuser, Basel (1991)
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Abbott, M., Altenkirch, T., Ghani, N. (2004). Representing Nested Inductive Types Using W-Types. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds) Automata, Languages and Programming. ICALP 2004. Lecture Notes in Computer Science, vol 3142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27836-8_8
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DOI: https://doi.org/10.1007/978-3-540-27836-8_8
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