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The Interval Domain in Homotopy Type Theory

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Logics and Type Systems in Theory and Practice

Abstract

Even though the real numbers are the cornerstone of many fields in mathematics, it is challenging to formalize them in a constructive setting, and in particular, homotopy type theory. Several approaches have been established to define the real numbers, and the most prominent of them are based on Dedekind cuts and on Cauchy sequences. In this paper, we study a different approach towards defining the real numbers. Our approach is based on domain theory, and in particular, the interval domain, and we build forth on recent work on domain theory in univalent foundations. All the results in this paper have been formalized in Coq as part of the UniMath library.

This paper is dedicated to our supervisor and teacher, Herman Geuvers, on the occasion of his birthday. Herman, you teachings and research in domain theory, type theory, and mechanized proofs have greatly inspired us, and we hope that you enjoy this little expedition in the world of constructive real numbers, combining all the topics above.

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References

  1. Abramsky, S., Jung, A.: Domain theory (corrected and expanded version). In: Handbook of Logic in Computer Science, pp. 1–168. Oxford University Press (1994)

    Google Scholar 

  2. Ambridge, T.W.: Exact Real Search: Formalised Optimisation and Regression in Constructive Univalent Mathematics (2024). https://doi.org/10.48550/arXiv.2401.09270

  3. Audebaud, P., Paulin-Mohring, C.: Proofs of randomized algorithms in Coq. Sci. Comput. Program. 74(8), 568–589 (2009). https://doi.org/10.1016/j.scico.2007.09.002

    Article  MathSciNet  Google Scholar 

  4. Bauer, A., Kavkler, I.: A constructive theory of continuous domains suitable for implementation. Ann. Pure Appl. Logic 159(3), 251–267 (2009). https://doi.org/10.1016/j.apal.2008.09.025

    Article  MathSciNet  Google Scholar 

  5. Bauer, A., Taylor, P.: The Dedekind reals in abstract Stone duality. Math. Struct. Comput. Sci. 19(4), 757–838 (2009). https://doi.org/10.1017/S0960129509007695

    Article  MathSciNet  Google Scholar 

  6. Bidlingmaier, M.E., Faissole, F., Spitters, B.: Synthetic topology in homotopy type theory for probabilistic programming. Math. Struct. Comput. Sci. 31(10), 1301–1329 (2021). https://doi.org/10.1017/S0960129521000165

    Article  MathSciNet  Google Scholar 

  7. Bishop, E.: Foundations of Constructive Analysis. McGraw-Hill (1967)

    Google Scholar 

  8. Bishop, E., Bridges, D.: Constructive Analysis. Springer, Berlin (1985)

    Book  Google Scholar 

  9. Boldo, S., Lelay, C., Melquiond, G.: Formalization of real analysis: a survey of proof assistants and libraries. Math. Struct. Comput. Sci. 26(7), 1196–1233 (2016). https://doi.org/10.1017/S0960129514000437

    Article  MathSciNet  Google Scholar 

  10. Booij, A.B.: Extensional constructive real analysis via locators. Math. Struct. Comput. Sci. 31(1), 64–88 (2021). https://doi.org/10.1017/S0960129520000171

    Article  MathSciNet  Google Scholar 

  11. Booij, A.B.: The HoTT reals coincide with the Escardó-Simpson reals (2017). http://arxiv.org/abs/1706.05956

  12. Ciaffaglione, A., Di Gianantonio, P.: A co-inductive approach to real numbers. In: Coquand, T., Dybjer, P., Nordström, B., Smith, J. (eds.) TYPES 1999. LNCS, vol. 1956, pp. 114–130. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44557-9_7

    Chapter  Google Scholar 

  13. Cruz-Filipe, L.: A constructive formalization of the fundamental theorem of calculus. In: Geuvers, H., Wiedijk, F. (eds.) TYPES 2002. LNCS, vol. 2646, pp. 108–126. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-39185-1_7

    Chapter  Google Scholar 

  14. Cruz-Filipe, L., Geuvers, H., Wiedijk, F.: C-CoRN, the constructive Coq repository at Nijmegen. In: Asperti, A., Bancerek, G., Trybulec, A. (eds.) MKM 2004. LNCS, vol. 3119, pp. 88–103. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-27818-4_7

    Chapter  Google Scholar 

  15. de Jong, T.: The Scott model of PCF in univalent type theory. Math. Struct. Comput. Sci. 31(10), 1270–1300 (2021). https://doi.org/10.1017/S0960129521000153

    Article  MathSciNet  Google Scholar 

  16. de Jong, T.: Sharp elements and apartness in domains. Electron. Proc. Theor. Comput. Sci. 351, 134–151 (2021). https://doi.org/10.4204/EPTCS.351.9

    Article  MathSciNet  Google Scholar 

  17. de Jong, T.: Apartness, sharp elements, and the Scott topology of domains. Math. Struct. Comput. Sci. 1–32 (2023). https://doi.org/10.1017/S0960129523000282

  18. de Jong, T.: Domain theory in constructive and predicative univalent foundations. Ph.D. thesis, University of Birmingham (2023)

    Google Scholar 

  19. Escardó, M., Hofmann, M., Streicher, T.: On the non-sequential nature of the interval-domain model of real-number computation. Math. Struct. Comput. Sci. 14(6), 803–814 (2004). https://doi.org/10.1017/S0960129504004360

    Article  MathSciNet  Google Scholar 

  20. Escardó, M.H.: PCF extended with real numbers. Theor. Comput. Sci. 162(1), 79–115 (1996). https://doi.org/10.1016/0304-3975(95)00250-2

    Article  MathSciNet  Google Scholar 

  21. Escardó, M.H., Streicher, T.: Induction and recursion on the partial real line with applications to Real PCF. Theor. Comput. Sci. 210(1), 121–157 (1999). https://doi.org/10.1016/S0304-3975(98)00099-1

    Article  MathSciNet  Google Scholar 

  22. Geuvers, H., Niqui, M.: Constructive reals in Coq: axioms and categoricity. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R., Pollack, R. (eds.) TYPES 2000. LNCS, vol. 2277, pp. 79–95. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45842-5_6

    Chapter  Google Scholar 

  23. Geuvers, H., Niqui, M., Spitters, B., Wiedijk, F.: Constructive analysis, types and exact real numbers. Math. Struct. Comput. Sci. 17(1), 3–36 (2007). https://doi.org/10.1017/S0960129506005834

    Article  MathSciNet  Google Scholar 

  24. Ghica, D.R., Ambridge, T.W.: Global optimisation with constructive reals. In: 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2021, Rome, Italy, 29 June–2 July 2021, pp. 1–13. IEEE (2021). https://doi.org/10.1109/LICS52264.2021.9470549

  25. Gilbert, G.: Formalising real numbers in homotopy type theory. In: Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs, CPP 2017, pp. 112–124. Association for Computing Machinery, New York (2017). https://doi.org/10.1145/3018610.3018614

  26. Jones, C.: Completing the rationals and metric spaces in LEGO. Logical Environ. 297–316 (1993)

    Google Scholar 

  27. de Jong, T., Escardó, M.H.: Domain theory in constructive and predicative univalent foundations. In: 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), vol. 183, pp. 28:1–28:18. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2021). https://doi.org/10.4230/LIPIcs.CSL.2021.28

  28. Krebbers, R., Spitters, B.: Type classes for efficient exact real arithmetic in Coq. Logical Methods Comput. Sci. 9(1) (2013). https://doi.org/10.2168/LMCS-9(1:1)2013

  29. Leinster, T.: Higher Operads, Higher Categories (2003). https://doi.org/10.48550/arXiv.math/0305049

  30. Ljungstrom, A.: Symmetric Monoidal Smash Products in Homotopy Type Theory (2024). https://arxiv.org/abs/2402.03523

  31. Moore, R.E.: Interval Analysis, vol. 4. Prentice-Hall Englewood Cliffs (1966)

    Google Scholar 

  32. O’Connor, R.: A monadic, functional implementation of real numbers. Math. Struct. Comput. Sci. 17(1), 129–159 (2007). https://doi.org/10.1017/S0960129506005871

    Article  MathSciNet  Google Scholar 

  33. O’Connor, R.: Certified exact transcendental real number computation in Coq. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 246–261. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-71067-7_21

    Chapter  Google Scholar 

  34. Smyth, M.B.: The constructive maximal point space and partial metrizability. Ann. Pure Appl. Logic 137(1), 360–379 (2006). https://doi.org/10.1016/j.apal.2005.05.032

    Article  MathSciNet  Google Scholar 

  35. Univalent Foundations Program, T.: Homotopy Type Theory: Univalent Foundations of Mathematics (2013). https://homotopytypetheory.org/book, Institute for Advanced Study

  36. Voevodsky, V., Ahrens, B., Grayson, D., et al.: UniMath—a computer-checked library of univalent mathematics. http://unimath.org. https://doi.org/10.5281/zenodo.7848572. https://github.com/UniMath/UniMath

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Acknowledgements

We would like to thank the anonymous reviewers for their comments and suggestions. The authors also thank Andrej Bauer and Tom de Jong for useful pointers to the literature.

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Correspondence to Niels van der Weide .

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van der Weide, N., Frumin, D. (2024). The Interval Domain in Homotopy Type Theory. In: Capretta, V., Krebbers, R., Wiedijk, F. (eds) Logics and Type Systems in Theory and Practice. Lecture Notes in Computer Science, vol 14560. Springer, Cham. https://doi.org/10.1007/978-3-031-61716-4_16

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  • DOI: https://doi.org/10.1007/978-3-031-61716-4_16

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