skip to main content

The taming of the rew: a type theory with computational assumptions

Published:04 January 2021Publication History
Skip Abstract Section

Abstract

Dependently typed programming languages and proof assistants such as Agda and Coq rely on computation to automatically simplify expressions during type checking. To overcome the lack of certain programming primitives or logical principles in those systems, it is common to appeal to axioms to postulate their existence. However, one can only postulate the bare existence of an axiom, not its computational behaviour. Instead, users are forced to postulate equality proofs and appeal to them explicitly to simplify expressions, making axioms dramatically more complicated to work with than built-in primitives. On the other hand, the equality reflection rule from extensional type theory solves these problems by collapsing computation and equality, at the cost of having no practical type checking algorithm. This paper introduces Rewriting Type Theory (RTT), a type theory where it is possible to add computational assumptions in the form of rewrite rules. Rewrite rules go beyond the computational capabilities of intensional type theory, but in contrast to extensional type theory, they are applied automatically so type checking does not require input from the user. To ensure type soundness of RTT—as well as effective type checking—we provide a framework where confluence of user-defined rewrite rules can be checked modularly and automatically, and where adding new rewrite rules is guaranteed to preserve subject reduction. The properties of RTT have been formally verified using the MetaCoq framework and an implementation of rewrite rules is already available in the Agda proof assistant.

References

  1. Andreas Abel, Joakim Öhman, and Andrea Vezzosi. 2018. Decidability of Conversion for Type Theory in Type Theory. Proceedings of the ACM on Programming Languages 2, POPL, Article 23 ( Jan. 2018 ), 29 pages. https://doi.org/10.1145/ 3158111 Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Andreas Abel and Brigitte Pientka. 2011. Higher-Order Dynamic Pattern Unification for Dependent Types and Records. In Typed Lambda Calculi and Applications-10th International Conference, TLCA 2011, Novi Sad, Serbia, June 1-3, 2011. Proceedings (Lecture Notes in Computer Science), C.-H. Luke Ong (Ed.), Vol. 6690. Springer, 10-26. https://doi.org/10. 1007/978-3-642-21691-6_5 Google ScholarGoogle ScholarCross RefCross Ref
  3. Agda Development Team. 2020. Agda 2.6.1 documentation. http://agda.readthedocs. io/en/v2.6.1/Google ScholarGoogle Scholar
  4. Guillaume Allais, Conor McBride, and Pierre Boutillier. 2013. New Equations for Neutral Terms: A Sound and Complete Decision Procedure, Formalized. In Proceedings of the 2013 ACM SIGPLAN Workshop on Dependently-typed Programming (DTP '13). ACM, New York, NY, USA, 13-24. https://doi.org/10.1145/2502409.2502411 Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Claus Appel, Vincent van Oostrom, and Jakob Grue Simonsen. 2010. Higher-Order (Non-) Modularity. In Proceedings of the 21st International Conference on Rewriting Techniques and Applications, RTA 2010, July 11-13, 2010, Edinburgh, Scottland, UK (LIPIcs), Christopher Lynch (Ed.), Vol. 6. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 17-32. https://doi.org/10.4230/LIPIcs.RTA. 2010.17 Google ScholarGoogle ScholarCross RefCross Ref
  6. Ali Assaf and Guillaume Burel. 2015. Translating HOL to Dedukti. Electronic Proceedings in Theoretical Computer Science 186 ( Jul 2015 ), 74-88. https://doi.org/10.4204/EPTCS.186.8 Google ScholarGoogle ScholarCross RefCross Ref
  7. Franco Barbanera, Maribel Fernández, and Herman Geuvers. 1997. Modularity of Strong Normalization in the Algebraiclambda-Cube. Journal of Functional Programming 7, 6 ( 1997 ), 613-660.Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Henk Barendregt. 1984. The Lambda Calculus, Its Syntax and Semantics. Studies in Logic and the Foundations of Mathematics, Vol. 103. Elsevier. https://doi.org/10.1016/B978-0-444-87508-2. 50001-0 Google ScholarGoogle ScholarCross RefCross Ref
  9. Bruno Barras, Jean-Pierre Jouannaud, Pierre-Yves Strub, and Qian Wang. 2011. CoQMTU: A Higher-Order Type Theory with a Predicative Hierarchy of Universes Parametrized by a Decidable First-Order Theory. In Proceedings of the 26th Annual IEEE Symposium on Logic in Computer Science, LICS 2011, June 21-24, 2011, Toronto, Ontario, Canada. IEEE Computer Society, 143-151. https://doi.org/10.1109/LICS. 2011.37 Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Ulrich Berger, Ralph Matthes, and Anton Setzer. 2019. Martin Hofmann's Case for Non-Strictly Positive Data Types. In 24th International Conference on Types for Proofs and Programs (TYPES 2018 ), Peter Dybjer, José Espírito Santo, and Luís Pinto (Eds.). Leibniz International Proceedings in Informatics (LIPIcs), Vol. 130. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 1 : 1-1 : 22. https://doi.org/10.4230/LIPIcs.TYPES. 2018.1 Google ScholarGoogle ScholarCross RefCross Ref
  11. Frédéric Blanqui. 2004. A Type-Based Termination Criterion for Dependently-Typed Higher-Order Rewrite Systems. In Rewriting Techniques and Applications, 15th International Conference, RTA 2004, Aachen, Germany, June 3-5, 2004, Proceedings (Lecture Notes in Computer Science), Vincent van Oostrom (Ed.), Vol. 3091. Springer, 24-39. http://springerlink. metapress.com/openurl.asp?genre=article&issn= 0302-9743 &volume= 3091 &spage= 24Google ScholarGoogle Scholar
  12. Frédéric Blanqui. 2005. Definitions by rewriting in the Calculus of Constructions. Mathematical Structures in Computer Science 15, 1 ( 2005 ), 37-92. https://doi.org/10.1017/S0960129504004426 Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Frédéric Blanqui. 2018. Size-based termination of higher-order rewriting. Journal of Functional Programming 28 ( April 2018 ), e11. https://doi.org/10.1017/S0956796818000072 Google ScholarGoogle ScholarCross RefCross Ref
  14. Frédéric Blanqui. 2020. Type Safety of Rewrite Rules in Dependent Types. In 5th International Conference on Formal Structures for Computation and Deduction, FSCD 2020, June 29-July 6, 2020, Paris, France (Virtual Conference) (LIPIcs), Zena M. Ariola (Ed.), Vol. 167. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 13 : 1-13 : 14. https://doi.org/10.4230/LIPIcs.FSCD. 2020. 13 Google ScholarGoogle ScholarCross RefCross Ref
  15. Frédéric Blanqui, Jean-Pierre Jouannaud, and Albert Rubio. 2008. The Computability Path Ordering: The End of a Quest. In Computer Science Logic, 22nd International Workshop, CSL 2008, 17th Annual Conference of the EACSL, Bertinoro, Italy, September 16-19, 2008. Proceedings (Lecture Notes in Computer Science), Michael Kaminski and Simone Martini (Eds.), Vol. 5213. Springer, 1-14. https://doi.org/10.1007/978-3-540-87531-4_1 Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Mathieu Boespflug, Quentin Carbonneaux, and Olivier Hermant. 2012. The λΠ-calculus Modulo as a Universal Proof Language. In the Second International Workshop on Proof Exchange for Theorem Proving (PxTP 2012 ), Tjark Weber David Pichardie (Ed.), Vol. Vol. 878. Manchester, United Kingdom, pp. 28-43. https://hal-mines-paristech.archivesouvertes.fr/hal-00917845Google ScholarGoogle Scholar
  17. Jacek Chrzaszcz and Daria Walukiewicz-Chrzaszcz. 2007. Towards Rewriting in Coq. In Rewriting, Computation and Proof, Essays Dedicated to Jean-Pierre Jouannaud on the Occasion of His 60th Birthday (Lecture Notes in Computer Science), Hubert Comon-Lundh, Claude Kirchner, and Hélène Kirchner (Eds.), Vol. 4600. Springer, 113-131. https://doi.org/10.1007/978-3-540-73147-4_6 Google ScholarGoogle ScholarCross RefCross Ref
  18. Jesper Cockx. 2020. Type theory unchained: Extending Agda with user-defined rewrite rules (system description). ( 2020 ). To appear in the post-proceedings of the TYPES conference.Google ScholarGoogle Scholar
  19. The Coq Development Team. 2016. The Coq proof assistant reference manual. http://coq.inria. fr Version 8.6.Google ScholarGoogle Scholar
  20. Thierry Coquand and Christine Paulin. 1988. Inductively Defined Types. In Proceedings of the International Conference on Computer Logic (COLOG '88). Springer-Verlag, Berlin, Heidelberg, 50-66.Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Denis Cousineau and Gilles Dowek. 2007. Embedding Pure Type Systems in the Lambda-Pi-Calculus Modulo. In Typed Lambda Calculi and Applications, 8th International Conference, TLCA 2007, Paris, France, June 26-28, 2007, Proceedings (Lecture Notes in Computer Science), Simona Ronchi Della Rocca (Ed.), Vol. 4583. Springer, 102-117. https://doi.org/10. 1007/978-3-540-73228-0_9 Google ScholarGoogle ScholarCross RefCross Ref
  22. Bertram Felgenhauer and Vincent van Oostrom. 2013. Proof Orders for Decreasing Diagrams. In 24th International Conference on Rewriting Techniques and Applications, RTA 2013, June 24-26, 2013, Eindhoven, The Netherlands (LIPIcs), Femke van Raamsdonk (Ed.), Vol. 21. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 174-189. https://doi.org/10.4230/LIPIcs. RTA. 2013.174 Google ScholarGoogle ScholarCross RefCross Ref
  23. Gaspard Ferey and Jean-Pierre Jouannaud. 2019. Confluence in (Un)Typed Higher-Order Theories by means of Critical Pairs. (Dec. 2019 ). https://hal.archives-ouvertes.fr/hal-02096540 (under submission).Google ScholarGoogle Scholar
  24. Guillaume Genestier. 2019. SizeChangeTool: A Termination Checker for Rewriting Dependent Types. In HOR 2019-10th International Workshop on Higher-Order Rewriting (Joint Proceedings of HOR 2019 and IWC 2019 ), Mauricio Ayala-Rincón, Silvia Ghilezan, and Jakob Grue Simonsen (Eds.). Dortmund, Germany, 14-19. https://hal.archives-ouvertes. fr/hal02442465Google ScholarGoogle Scholar
  25. Gaëtan Gilbert, Jesper Cockx, Matthieu Sozeau, and Nicolas Tabareau. 2019. Definitional Proof-Irrelevance without K. Proceedings of the ACM on Programming Languages 3 ( Jan. 2019 ), 1-28. https://doi.org/10.1145/329031610.1145/3290316 Google ScholarGoogle ScholarCross RefCross Ref
  26. Martin Hofmann. 1993. Non Strictly Positive Datatypes in System F. Email on types mailing list. http://www.seas.upenn.edu/ sweirich/types/archive/1993/msg00027.html.Google ScholarGoogle Scholar
  27. Jean-Pierre Jouannaud and Jianqi Li. 2015. Termination of Dependently Typed Rewrite Rules. In 13th International Conference on Typed Lambda Calculi and Applications, TLCA 2015, July 1-3, 2015, Warsaw, Poland (LIPIcs), Thorsten Altenkirch (Ed.), Vol. 38. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 257-272. https://doi.org/10.4230/LIPIcs.TLCA. 2015.257 Google ScholarGoogle ScholarCross RefCross Ref
  28. Jean-Pierre Jouannaud and Mitsuhiro Okada. 1991. A Computation Model for Executable Higher-Order Algebraic Specification Languages. In Proceedings, Sixth Annual IEEE Symposium on Logic in Computer Science, 15-18 July, 1991, Amsterdam, The Netherlands. IEEE Computer Society, 350-361. https://doi.org/10.1109/LICS. 1991.151659 Google ScholarGoogle ScholarCross RefCross Ref
  29. Jean-Pierre Jouannaud and Albert Rubio. 1999. The Higher-Order Recursive Path Ordering. In Proceedings, 14th Annual IEEE Symposium on Logic in Computer Science, 2-5 July, 1999, Trento, Italy. IEEE Computer Society, 402-411. https: //doi.org/10.1109/LICS. 1999.782635 Google ScholarGoogle ScholarCross RefCross Ref
  30. Jean-Pierre Jouannaud and Pierre-Yves Strub. 2017. Coq without Type Casts: A Complete Proof of Coq Modulo Theory. In LPAR-21: 21st International Conference on Logic for Programming, Artificial Intelligence and Reasoning. EasyChair, Maun, Botswana, 474-489. https://hal.inria.fr/hal-01664457Google ScholarGoogle Scholar
  31. Chin Soon Lee, Neil D. Jones, and Amir M. Ben-Amram. 2001. The size-change principle for program termination. In Proceedings of the 28th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages. Association for Computing Machinery, 81-92. https://doi.org/10.1145/360204.360210 Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. Per Martin-Löf. 1975. An Intuitionistic Theory of Types: Predicative Part. In Logic Colloquium '73, H.E. Rose and J.C. Shepherdson (Eds.). Studies in Logic and the Foundations of Mathematics, Vol. 80. Elsevier, 73-118. https://doi.org/10. 1016/ S0049-237X( 08 ) 71945-1 Google ScholarGoogle ScholarCross RefCross Ref
  33. Christine Paulin-Mohring. 2015. Introduction to the Calculus of Inductive Constructions. In All about Proofs, Proofs for All, Bruno Woltzenlogel Paleo and David Delahaye (Eds.). Studies in Logic (Mathematical logic and foundations), Vol. 55. College Publications. https://hal.inria.fr/hal-01094195Google ScholarGoogle Scholar
  34. Pierre-Marie Pédrot and Nicolas Tabareau. 2018. Failure is Not an Option: An Exceptional Type Theory. In ESOP 2018-27th European Symposium on Programming (Lecture Notes in Computer Science), Vol. 10801. Springer, Thessaloniki, Greece, 245-271. https://doi.org/10.1007/978-3-319-89884-1_9 Google ScholarGoogle ScholarCross RefCross Ref
  35. Pierre-Marie Pédrot, Nicolas Tabareau, Hans Jacob Fehrmann, and Éric Tanter. 2019. A Reasonably Exceptional Type Theory. Proceedings of the ACM on Programming Languages 3, ICFP, Article 108 ( July 2019 ), 29 pages. https://doi.org/10.1145/ 3341712 Google ScholarGoogle ScholarDigital LibraryDigital Library
  36. Ronan Saillard. 2015. Typechecking in the lambda-Pi-Calculus Modulo: Theory and Practice. Ph.D. Dissertation. MINES ParisTech.Google ScholarGoogle Scholar
  37. Gert Smolka. 2015. Confluence and Normalization in Reduction Systems. ( 2015 ). https://www.ps.uni-saarland.de/courses/ sem-ws15/ars. pdf (lecture notes).Google ScholarGoogle Scholar
  38. Matthieu Sozeau, Abhishek Anand, Simon Boulier, Cyril Cohen, Yannick Forster, Fabian Kunze, Gregory Malecha, Nicolas Tabareau, and Théo Winterhalter. 2020. The MetaCoq Project. Journal of Automated Reasoning (Feb. 2020 ). https: //doi.org/10.1007/s10817-019-09540-0 Google ScholarGoogle ScholarCross RefCross Ref
  39. Matthieu Sozeau, Simon Boulier, Yannick Forster, Nicolas Tabareau, and Théo Winterhalter. 2019. Coq Coq Correct! Verification of Type Checking and Erasure for Coq, in Coq. Proceedings of the ACM on Programming Languages 4, POPL, Article 8 ( Dec. 2019 ), 28 pages. https://doi.org/10.1145/3371076 Google ScholarGoogle ScholarDigital LibraryDigital Library
  40. Mark-Oliver Stehr. 2005a. The Open Calculus of Constructions (Part I): An Equational Type Theory with Dependent Types for Programming, Specification, and Interactive Theorem Proving. Fundamenta Informaticae 68, 1-2 ( 2005 ), 131-174.Google ScholarGoogle Scholar
  41. Mark-Oliver Stehr. 2005b. The Open Calculus of Constructions (Part II): An Equational Type Theory with Dependent Types for Programming, Specification, and Interactive Theorem Proving. Fundamenta Informaticae 68, 3 ( 2005 ), 249-288.Google ScholarGoogle Scholar
  42. Pierre-Yves Strub. 2010. Coq Modulo Theory. In Computer Science Logic, 24th International Workshop, CSL 2010, 19th Annual Conference of the EACSL, Brno, Czech Republic, August 23-27, 2010. Proceedings (Lecture Notes in Computer Science), Anuj Dawar and Helmut Veith (Eds.), Vol. 6247. Springer, 529-543. https://doi.org/10.1007/978-3-642-15205-4_40 Google ScholarGoogle ScholarCross RefCross Ref
  43. M. Takahashi. 1995. Parallel Reductions in ń-Calculus. Information and Computation 118, 1 ( 1995 ), 120-127. https: //doi.org/10.1006/inco. 1995.1057 Google ScholarGoogle ScholarDigital LibraryDigital Library
  44. Val Tannen. 1988. Combining Algebra and Higher-Order Types. In Proceedings, Third Annual Symposium on Logic in Computer Science, 5-8 July 1988, Edinburgh, Scotland, UK. IEEE Computer Society, 82-90. https://doi.org/10.1109/LICS. 1988.5103 Google ScholarGoogle ScholarCross RefCross Ref
  45. Terese. 2003. Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, Vol. 55. Cambridge University Press.Google ScholarGoogle Scholar
  46. The Univalent Foundations Program. 2013. Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study.Google ScholarGoogle Scholar
  47. Vincent van Oostrom. 1994a. Confluence by Decreasing Diagrams. Theoretical Computer Science 126, 2 ( 1994 ), 259-280. https://doi.org/10.1016/ 0304-3975 ( 92 ) 00023-K Google ScholarGoogle ScholarDigital LibraryDigital Library
  48. Vincent van Oostrom. 1994b. Confluence for abstract and higher-order rewriting. Ph.D. Dissertation. Vrije Universiteit Amsterdam.Google ScholarGoogle Scholar
  49. Vincent van Oostrom. 1995. Development Closed Critical Pairs. In Higher-Order Algebra, Logic, and Term Rewriting, Second International Workshop, HOA '95, Paderborn, Germany, September 21-22, 1995, Selected Papers (Lecture Notes in Computer Science), Gilles Dowek, Jan Heering, Karl Meinke, and Bernhard Möller (Eds.), Vol. 1074. Springer, 185-200. https://doi.org/10.1007/3-540-61254-8_26 Google ScholarGoogle ScholarCross RefCross Ref
  50. Daria Walukiewicz-Chrzaszcz. 2003. Termination of rewriting in the Calculus of Constructions. Journal of Functional Programming 13, 2 ( 2003 ), 339-414. https://doi.org/10.1017/S0956796802004641 Google ScholarGoogle ScholarDigital LibraryDigital Library
  51. Daria Walukiewicz-Chrzaszcz and Jacek Chrzaszcz. 2006. Consistency and Completeness of Rewriting in the Calculus of Constructions. In Automated Reasoning, Third International Joint Conference, IJCAR 2006, Seattle, WA, USA, August 17-20, 2006, Proceedings (Lecture Notes in Computer Science), Ulrich Furbach and Natarajan Shankar (Eds.), Vol. 4130. Springer, 619-631. https://doi.org/10.1007/11814771_50 Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. The taming of the rew: a type theory with computational assumptions

    Recommendations

    Comments

    Login options

    Check if you have access through your login credentials or your institution to get full access on this article.

    Sign in

    Full Access

    PDF Format

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader
    About Cookies On This Site

    We use cookies to ensure that we give you the best experience on our website.

    Learn more

    Got it!