Inductive-Inductive Definitions
Abstract
We present a principle for introducing new types in type theory which generalises strictly positive indexed inductive data types. In this new principle a set A is defined inductively simultaneously with an A-indexed set B, which is also defined inductively. Compared to indexed inductive definitions, the novelty is that the index set A is generated inductively simultaneously with B. In other words, we mutually define two inductive sets, of which one depends on the other.
Instances of this principle have previously been used in order to formalise type theory inside type theory. However the consistency of the framework used (the theorem prover Agda) is not so clear, as it allows the definition of a universe containing a code for itself. We give an axiomatisation of the new principle in such a way that the resulting type theory is consistent, which we prove by constructing a set-theoretic model.
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References
- 1.Aczel, P.: On relating type theories and set theories. In: Altenkirch, T., Naraschewski, W., Reus, B. (eds.) TYPES 1998. LNCS, vol. 1657, pp. 1–18. Springer, Heidelberg (1999)CrossRefGoogle Scholar
- 2.Backhouse, R., Chisholm, P., Malcolm, G., Saaman, E.: Do-it-yourself type theory. Formal Aspects of Computing 1(1), 19–84 (1989)CrossRefGoogle Scholar
- 3.Benke, M., Dybjer, P., Jansson, P.: Universes for generic programs and proofs in dependent type theory. Nordic Journal of Computing 10, 265–269 (2003)zbMATHMathSciNetGoogle Scholar
- 4.Chapman, J.: Type theory should eat itself. Electronic Notes in Theoretical Computer Science 228, 21–36 (2009)CrossRefGoogle Scholar
- 5.Danielsson, N.: A formalisation of a dependently typed language as an inductive-recursive family. In: Altenkirch, T., McBride, C. (eds.) TYPES 2006. LNCS, vol. 4502, pp. 93–109. Springer, Heidelberg (2007)CrossRefGoogle Scholar
- 6.Dybjer, P.: Inductive families. Formal aspects of computing 6(4), 440–465 (1994)zbMATHCrossRefGoogle Scholar
- 7.Dybjer, P.: Internal type theory. In: Berardi, S., Coppo, M. (eds.) TYPES 1995. LNCS, vol. 1158, pp. 120–134. Springer, Heidelberg (1996)Google Scholar
- 8.Dybjer, P.: A general formulation of simultaneous inductive-recursive definitions in type theory. Journal of Symbolic Logic 65(2), 525–549 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
- 9.Dybjer, P., Setzer, A.: A finite axiomatization of inductive-recursive definitions. In: Girard, J. (ed.) TLCA 1999. LNCS, vol. 1581, pp. 129–146. Springer, Heidelberg (1999)CrossRefGoogle Scholar
- 10.Dybjer, P., Setzer, A.: Induction–recursion and initial algebras. Annals of Pure and Applied Logic 124(1-3), 1–47 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
- 11.Dybjer, P., Setzer, A.: Indexed induction–recursion. Journal of logic and algebraic programming 66(1), 1–49 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
- 12.Martin-Löf, P.: Intuitionistic type theory. Bibliopolis Naples (1984)Google Scholar
- 13.Morris, P.: Constructing Universes for Generic Programming. Ph.D. thesis, University of Nottingham (2007)Google Scholar
- 14.Nordvall Forsberg, F., Setzer, A.: Induction-induction: Agda development and extended version (2010), http://cs.swan.ac.uk/~csfnf/induction-induction/
- 15.Palmgren, E.: On universes in type theory. In: Sambin, G., Smith, J. (eds.) Twenty five years of constructive type theory, pp. 191–204. Oxford University Press, Oxford (1998)Google Scholar
- 16.Streicher, T.: Investigations into intensional type theory. Habilitiation Thesis (1993)Google Scholar
- 17.The Agda Team: The Agda wiki (2010), http://wiki.portal.chalmers.se/agda/