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Impredicative Encodings of Inductive-Inductive Data in Cedille

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Trends in Functional Programming (TFP 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13868))

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Abstract

Cedille is a dependently typed programming language known for expressive and efficient impredicative encodings. In this work, we show that encodings of induction-induction are also possible by employing a standard technique from other encodings in Cedille, where a type representing the shape of data is intersected with a predicate that further constrains. Thus, just as with indexed inductive data, Cedille can encode a notion that is often axiomatically postulated or directly implemented in other dependent type theories without sacrificing efficiency.

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References

  1. Allen, S.F., Bickford, M., Constable, R.L., Eaton, R., Kreitz, C., Lorigo, L., Moran, E.: Innovations in computational type theory using nuprl. J. Appl. Logic 4(4), 428–469 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Altenkirch, T., Capriotti, P., Dijkstra, G., Kraus, N., Nordvall Forsberg, F.: Quotient inductive-inductive types. In: Baier, C., Dal Lago, U. (eds.) FoSSaCS 2018. LNCS, vol. 10803, pp. 293–310. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-89366-2_16

    Chapter  Google Scholar 

  3. Altenkirch, T., Morris, P., Nordvall Forsberg, F., Setzer, A.: A categorical semantics for inductive-inductive definitions. In: Corradini, A., Klin, B., Cîrstea, C. (eds.) CALCO 2011. LNCS, vol. 6859, pp. 70–84. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22944-2_6

    Chapter  Google Scholar 

  4. Atkey, R.: Syntax and semantics of quantitative type theory. In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, pp. 56–65 (2018)

    Google Scholar 

  5. Dijkstra, G.: Quotient inductive-inductive definitions. Ph.D. thesis, University of Nottingham (2017)

    Google Scholar 

  6. Dybjer, P., Setzer, A.: A finite axiomatization of inductive-recursive definitions. In: Girard, J.-Y. (ed.) TLCA 1999. LNCS, vol. 1581, pp. 129–146. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48959-2_11

    Chapter  Google Scholar 

  7. Dybjer, P., Setzer, A.: Induction-recursion and initial algebras. Ann. Pure Appl. Logic 124(1–3), 1–47 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dybjer, P., Setzer, A.: Indexed induction-recursion. J. Logic Algebr. Program. 66(1), 1–49 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Firsov, D., Blair, R., Stump, A.: Efficient Mendler-style lambda-encodings in cedille. In: Avigad, J., Mahboubi, A. (eds.) ITP 2018. LNCS, vol. 10895, pp. 235–252. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94821-8_14

    Chapter  Google Scholar 

  10. Firsov, D., Blair, R., Stump, A.: Efficient mendler-style lambda-encodings in cedille. In: Interactive Theorem Proving: 9th International Conference, ITP 2018, Held as Part of the Federated Logic Conference, FloC 2018, Oxford, UK, 9–12 July 2018, Proceedings 9, pp. 235–252. Springer (2018)

    Google Scholar 

  11. Forsberg, F.N.: Inductive-inductive definitions. Ph.D. thesis, Swansea University (2013). http://login.proxy.lib.uiowa.edu/login?url=https://www.proquest.com/dissertations-theses/inductive-definitions/docview/2041902169/se-2 copyright - Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works. Accessed 21 Oct 2022

  12. Forsberg, F.N., Setzer, A.: A finite axiomatisation of inductive-inductive definitions. Logic Constr. Comput. 3, 259–287 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Geuvers, H.: Induction is not derivable in second order dependent type theory. In: Abramsky, S. (ed.) TLCA 2001. LNCS, vol. 2044, pp. 166–181. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45413-6_16

    Chapter  Google Scholar 

  14. Hancock, P., McBride, C., Ghani, N., Malatesta, L., Altenkirch, T.: Small induction recursion. In: Hasegawa, M. (ed.) TLCA 2013. LNCS, vol. 7941, pp. 156–172. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38946-7_13

    Chapter  Google Scholar 

  15. Jenkins, C., Marmaduke, A., Stump, A.: Simulating large eliminations in cedille. In: Basold, H., Cockx, J., Ghilezan, S. (eds.) 27th International Conference on Types for Proofs and Programs (TYPES 2021). Leibniz International Proceedings in Informatics (LIPIcs), vol. 239, pp. 9:1–9:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2022). https://doi.org/10.4230/LIPIcs.TYPES.2021.9,https://drops.dagstuhl.de/opus/volltexte/2022/16778

  16. Kaposi, A., Kovács, A., Altenkirch, T.: Constructing quotient inductive-inductive types. Proc. ACM Program. Lang. 3(POPL), 1–24 (2019)

    Google Scholar 

  17. Kaposi, A., Kovács, A., Lafont, A.: For finitary induction-induction, induction is enough. In: TYPES 2019: 25th International Conference on Types for Proofs and Programs, vol. 175, pp. 6–1. Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2019)

    Google Scholar 

  18. Kaposi, A., von Raumer, J.: A syntax for mutual inductive families (2020)

    Google Scholar 

  19. Kopylov, A.: Dependent intersection: a new way of defining records in type theory. In: Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science, LICS 2003, pp. 86–95.IEEE Computer Society, Washington, DC (2003)

    Google Scholar 

  20. Marmaduke, A., Jenkins, C., Stump, A.: Quotients by idempotent functions in cedille. In: Bowman, W.J., Garcia, R. (eds.) TFP 2019. LNCS, vol. 12053, pp. 1–20. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-47147-7_1

    Chapter  MATH  Google Scholar 

  21. Miquel, A.: The implicit calculus of constructions extending pure type systems with an intersection type binder and subtyping. In: Abramsky, S. (ed.) TLCA 2001. LNCS, vol. 2044, pp. 344–359. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45413-6_27

    Chapter  MATH  Google Scholar 

  22. Nordvall Forsberg, F., Setzer, A.: Inductive-inductive definitions. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 454–468. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15205-4_35

    Chapter  Google Scholar 

  23. Parigot, M.: Programming with proofs: a second order type theory. In: Ganzinger, H. (ed.) ESOP 1988. LNCS, vol. 300, pp. 145–159. Springer, Heidelberg (1988). https://doi.org/10.1007/3-540-19027-9_10

    Chapter  Google Scholar 

  24. Parigot, M.: On the representation of data in lambda-calculus. In: Börger, E., Büning, H.K., Richter, M.M. (eds.) CSL 1989. LNCS, vol. 440, pp. 309–321. Springer, Heidelberg (1990). https://doi.org/10.1007/3-540-52753-2_47

    Chapter  Google Scholar 

  25. Stump, A.: The calculus of dependent lambda eliminations. J. Funct. Program. 27, e14 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Stump, A.: From realizability to induction via dependent intersection. Ann. Pure Appl. Logic 169(7), 637–655 (2018). https://doi.org/10.1016/j.apal.2018.03.002

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. The University of Iowa, Iowa City, IA, USA

    Andrew Marmaduke, Larry Diehl & Aaron Stump

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  1. Andrew Marmaduke
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  2. Larry Diehl
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  3. Aaron Stump
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Correspondence to Andrew Marmaduke .

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  1. University of Massachusetts Boston, Boston, MA, USA

    Stephen Chang

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Marmaduke, A., Diehl, L., Stump, A. (2023). Impredicative Encodings of Inductive-Inductive Data in Cedille. In: Chang, S. (eds) Trends in Functional Programming. TFP 2023. Lecture Notes in Computer Science, vol 13868. Springer, Cham. https://doi.org/10.1007/978-3-031-38938-2_1

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  • DOI: https://doi.org/10.1007/978-3-031-38938-2_1

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