Skip to main content
SpringerLink
Log in

FoCaLiZe and Dedukti to the Rescue for Proof Interoperability

  • Conference paper
Interactive Theorem Proving (ITP 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10499))

Included in the following conference series:

Abstract

Numerous contributions have been made for some years to allow users to exchange formal proofs between different provers. The main propositions consist in ad hoc pointwise translations, e.g. between HOL Light and Isabelle in the Flyspeck project or uses of more or less complete certificates. We propose in this paper a methodology to combine proofs coming from different theorem provers. This methodology relies on the Dedukti logical framework as a common formalism in which proofs can be translated and combined. To relate the independently developed mathematical libraries used in proof assistants, we rely on the structuring features offered by FoCaLiZe, in particular parameterized modules and inheritance to build a formal library of transfer theorems called MathTransfer. We finally illustrate this methodology on the Sieve of Eratosthenes, which we prove correct using HOL and Coq in combination.

This work has been supported in part by the VECOLIB project of the French national research organization ANR (grant ANR-14-CE28-0018).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The purpose is to illustrate the methodology previously presented. Of course, this example is simple enough to be completely realized within Coq or done by reusing e.g. the translation from Hol Light to Coq proposed by Keller and Werner  [1].

  2. 2.

    The Coq Init library is the part of Coq standard library defining logical connectives and basic datatypes such as natural numbers and lists.

References

  1. Armand, M., Faure, G., Grégoire, B., Keller, C., Théry, L., Werner, B.: A modular integration of SAT/SMT solvers to Coq through proof witnesses. In: Jouannaud, J.-P., Shao, Z. (eds.) CPP 2011. LNCS, vol. 7086, pp. 135–150. Springer, Heidelberg (2011). doi:10.1007/978-3-642-25379-9_12

    Chapter  Google Scholar 

  2. Assaf, A.: A framework for defining computational higher-order logics. Ph.D. thesis, École Polytechnique (2015)

    Google Scholar 

  3. Assaf, A., Burel, G.: Translating HOL to Dedukti. In: Kaliszyk, C., Paskevich, A. (eds.) Proceedings Fourth Workshop on Proof eXchange for Theorem Proving, EPTCS, Berlin, Germany, 2–3 August 2015, vol. 186, pp. 74–88 (2015)

    Google Scholar 

  4. Assaf, A., Burel, G., Cauderlier, R., Delahaye, D., Dowek, G., Dubois, C., Gilbert, F., Halmagrand, P., Hermant, O., Saillard, R.: Expressing theories in the \(\lambda \Pi \)-calculus modulo theory and in the Dedukti system (2016). http://www.lsv.ens-cachan.fr/dowek/Publi/expressing.pdf

  5. Assaf, A., Cauderlier, R.: Mixing HOL and Coq in Dedukti. In: Kaliszyk, C., Paskevich, A. (eds.) 4th Workshop on Proof eXchange for Theorem Proving, EPTCS, Berlin, Germany, 2–3 August 2015, vol. 186, pp. 89–96 (2015)

    Google Scholar 

  6. Blanchette, J.C., Bulwahn, L., Nipkow, T.: Automatic proof and disproof in Isabelle/HOL. In: Tinelli, C., Sofronie-Stokkermans, V. (eds.) FroCoS 2011. LNCS (LNAI), vol. 6989, pp. 12–27. Springer, Heidelberg (2011). doi:10.1007/978-3-642-24364-6_2

    Chapter  Google Scholar 

  7. Burel, G.: Experimenting with deduction modulo. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS (LNAI), vol. 6803, pp. 162–176. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22438-6_14

    Chapter  MATH  Google Scholar 

  8. Bury, G., Delahaye, D., Doligez, D., Halmagrand, P., Hermant, O.: Automated deduction in the B set theory using typed proof search and deduction modulo. In: LPAR 20 : 20th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, Suva, Fiji, November 2015

    Google Scholar 

  9. Cauderlier, R.: A rewrite system for proof constructivization. In: Proceedings of the 2016 International Workshop on Logical Frameworks and Meta-languages: Theory and Practice, pp. 2:1–2:7. ACM (2016)

    Google Scholar 

  10. Cauderlier, R., Dubois, C.: ML pattern-matching, recursion, and rewriting: from FoCaLiZe to Dedukti. In: Sampaio, A., Wang, F. (eds.) ICTAC 2016. LNCS, vol. 9965, pp. 459–468. Springer, Cham (2016). doi:10.1007/978-3-319-46750-4_26

    Chapter  Google Scholar 

  11. Cauderlier, R., Halmagrand, P.: Checking Zenon modulo proofs in Dedukti. In: Kaliszyk, C., Paskevich, A. (eds.) Proceedings 4th Workshop on Proof eXchange for Theorem Proving, EPTCS, Berlin, Germany, 2–3 August 2015, vol. 186, pp. 57–73 (2015)

    Google Scholar 

  12. Denney, E.: A prototype proof translator from HOL to Coq. In: Aagaard, M., Harrison, J. (eds.) TPHOLs 2000. LNCS, vol. 1869, pp. 108–125. Springer, Heidelberg (2000). doi:10.1007/3-540-44659-1_8

    Chapter  Google Scholar 

  13. Gauthier, T., Kaliszyk, C.: Matching concepts across HOL libraries. In: Watt, S.M., Davenport, J.H., Sexton, A.P., Sojka, P., Urban, J. (eds.) CICM 2014. LNCS (LNAI), vol. 8543, pp. 267–281. Springer, Cham (2014). doi:10.1007/978-3-319-08434-3_20

    Chapter  Google Scholar 

  14. Horozal, F., Rabe, F.: Representing model theory in a type-theoretical logical framework. Theor. Comput. Sci. 412, 4919–4945 (2011)

    Article  MathSciNet  Google Scholar 

  15. Howe, D.J.: Importing mathematics from HOL into Nuprl. In: Goos, G., Hartmanis, J., Leeuwen, J., Wright, J., Grundy, J., Harrison, J. (eds.) TPHOLs 1996. LNCS, vol. 1125, pp. 267–281. Springer, Heidelberg (1996). doi:10.1007/BFb0105410

    Chapter  Google Scholar 

  16. Huffman, B., Kunčar, O.: Lifting and transfer: a modular design for quotients in Isabelle/HOL. In: Gonthier, G., Norrish, M. (eds.) CPP 2013. LNCS, vol. 8307, pp. 131–146. Springer, Cham (2013). doi:10.1007/978-3-319-03545-1_9

    Chapter  MATH  Google Scholar 

  17. Hurd, J.: The opentheory standard theory library. In: Bobaru, M., Havelund, K., Holzmann, G.J., Joshi, R. (eds.) NFM 2011. LNCS, vol. 6617, pp. 177–191. Springer, Heidelberg (2011). doi:10.1007/978-3-642-20398-5_14

    Chapter  Google Scholar 

  18. Kaliszyk, C., Krauss, A.: Scalable LCF-style proof translation. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) Interactive Theorem Proving. number 7998 in LNCS, pp. 51–66. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  19. Keller, C., Werner, B.: Importing HOL light into Coq. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 307–322. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14052-5_22

    Chapter  Google Scholar 

  20. Miller, D., Certificates, F.P.: Making proof universal and permanent. In: Momigliano, A., Pientka, B., Pollack, R. (eds.) Proceedings of the Eighth ACM SIGPLAN International Workshop on Logical Frameworks & Meta-languages: Theory & Practice, LFMTP 2013, Boston, Massachusetts, USA, 23 September 2013, pp. 1–2. ACM (2013)

    Google Scholar 

  21. Pessaux, F.: FoCaLiZe: inside an F-IDE. In: Dubois, C., Giannakopoulou, D., Méry, D. (eds.) Proceedings 1st Workshop on Formal Integrated Development Environment, F-IDE 2014, EPTCS, Grenoble, France, 6 April 6 2014, vol. 149, pp. 64–78 (2014)

    Google Scholar 

  22. Prevosto, V., Jaume, M.: Making proofs in a hierarchy of mathematical structures. In: Proceedings of Calculemus, September 2003

    Google Scholar 

  23. Saillard, R.: Type checking in the Lambda-Pi-Calculus modulo: theory and practice. Ph.D. thesis, MINES Paritech (2015)

    Google Scholar 

  24. Schürmann, C., Stehr, M.-O.: An executable formalization of the HOL/Nuprl connection in the metalogical framework twelf. In: Hermann, M., Voronkov, A. (eds.) LPAR 2006. LNCS, vol. 4246, pp. 150–166. Springer, Heidelberg (2006). doi:10.1007/11916277_11

    Chapter  Google Scholar 

  25. Wiedijk, F.: Encoding the HOL light logic in Coq (2007, unpublished notes)

    Google Scholar 

  26. Zimmermann, T., Herbelin, H.: Automatic and transparent transfer of theorems along isomorphisms in the coq proof assistant. CoRR, abs/1505.05028 (2015)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University Paris Diderot, Irif, Paris, France

    Raphaël Cauderlier

  2. ENSIIE, Samovar, Évry, France

    Catherine Dubois

Authors
  1. Raphaël Cauderlier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Catherine Dubois
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raphaël Cauderlier .

Editor information

Editors and Affiliations

  1. University of Brasília, Brasília D.F., Brazil

    Mauricio Ayala-Rincón

  2. NASA, Hampton, VA, USA

    César A. Muñoz

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Cauderlier, R., Dubois, C. (2017). FoCaLiZe and Dedukti to the Rescue for Proof Interoperability. In: Ayala-Rincón, M., Muñoz, C.A. (eds) Interactive Theorem Proving. ITP 2017. Lecture Notes in Computer Science(), vol 10499. Springer, Cham. https://doi.org/10.1007/978-3-319-66107-0_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-66107-0_9

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66106-3

  • Online ISBN: 978-3-319-66107-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation