Proofs and Reconstructions

  • Nik Sultana
  • Christoph Benzmüller
  • Lawrence C. Paulson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9322)


Implementing proof reconstruction is difficult because it involves symbolic manipulations of formal objects whose representation varies between different systems. It requires significant knowledge of the source and target systems. One cannot simply re-target to another logic. We present a modular proof reconstruction system with separate components, specifying their behaviour and describing how they interact. This system is demonstrated and evaluated through an implementation to reconstruct proofs generated by Leo-II and Satallax in Isabelle HOL, and is shown to work better than the current method of rediscovering proofs using a select set of provers.


Proof reconstruction Higher-order logic Abstract machines 


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  1. 1.
    Benzmüller, C.: Equality and Extensionality in Higher-Order Theorem Proving. PhD thesis, Naturwissenschaftlich-Technische Fakultät I, Saarland University (1999)Google Scholar
  2. 2.
    Benzmüller, C., Brown, C.E., Kohlhase, M.: Cut-Simulation and Impredicativity. Logical Methods in Computer Science 5(1:6), 1–21 (2009)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Benzmüller, C.E., Rabe, F., Sutcliffe, G.: THF0 – The core TPTP language for classical higher-order logic. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 491–506. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Benzmüller, C., Theiss, F., Paulson, L.C., Fietzke, A.: LEO-II – A cooperative automatic theorem prover for higher-order logic. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 162–170. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Blanchette, J.C.: Automatic Proofs and Refutations for Higher-Order Logic. PhD thesis, Institut für Informatik, Technische Universität München (2012)Google Scholar
  6. 6.
    Böhme, S., Weber, T.: Designing proof formats: A user’s perspective. In: Fontaine, P., Stump, A. (eds.) International Workshop on Proof Exchange for Theorem Proving, pp. 27–32 (2011)Google Scholar
  7. 7.
    Brown, C.E.: Satallax: An automatic higher-order prover. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 111–117. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Chihani, Z., Miller, D., Renaud, F.: Foundational proof certificates in first-order logic. In: Bonacina, M.P. (ed.) CADE 2013. LNCS, vol. 7898, pp. 162–177. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  9. 9.
    de Moura, L., Bjørner, N.S.: Z3: An efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    de Nivelle, H.: Extraction of proofs from clausal normal form transformation. In: Bradfield, J.C. (ed.) CSL 2002. LNCS, vol. 2471, pp. 584–598. Springer, Heidelberg (2002)Google Scholar
  11. 11.
    Dowek, G.: Skolemization in simple type theory: the logical and the theoretical points of view. In: Benzmüller, C., Brown, C.E., Siekmann, J., Statman, R. (eds.) Festschrift in Honour of Peter B. Andrews on his 70th Birthday. Studies in Logic and the Foundations of Mathematics. College Publications (2009)Google Scholar
  12. 12.
    Hurd, J.: First-order proof tactics in higher-order logic theorem provers. In: Archer, M., Di Vito, B., Muñoz, C. (eds.) Design and Application of Strategies/Tactics in Higher Order Logics, number CP-2003-212448 in NASA Technical Reports, pp. 56–68, September 2003Google Scholar
  13. 13.
    Keller, C.: A Matter of Trust: Skeptical Communication Between Coq and External Provers. PhD thesis, École Polytechnique, June 2013Google Scholar
  14. 14.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  15. 15.
    Paulson, L.C.: Isabelle. LNCS, vol. 828. Springer, Heidelberg (1994)zbMATHGoogle Scholar
  16. 16.
    Paulson, L.C., Blanchette, J.C.: Three years of experience with Sledgehammer, a practical link between automatic and interactive theorem provers. In: International Workshop on the Implementation of Logics. EasyChair (2010)Google Scholar
  17. 17.
    Schulz, S.: E – A Brainiac Theorem Prover. Journal of AI Communications 15(2/3), 111–126 (2002)zbMATHGoogle Scholar
  18. 18.
    Sultana, N., Blanchette, J.C., Paulson, L.C.: LEO-II and Satallax on the Sledgehammer test bench. Journal of Applied Logic (2012)Google Scholar
  19. 19.
    Sultana, N.: Higher-order proof translation. PhD thesis, Computer Laboratory, University of Cambridge, Available as Tech Report UCAM-CL-TR-867 (2015)Google Scholar
  20. 20.
    Sutcliffe, G.: The TPTP Problem Library and Associated Infrastructure: The FOF and CNF Parts, v3.5.0. Journal of Automated Reasoning 43(4), 337–362 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Weidenbach, C.: Combining superposition, sorts and splitting. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. 2, pp. 1965–2013. MIT Press (2001)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Nik Sultana
    • 1
  • Christoph Benzmüller
    • 2
  • Lawrence C. Paulson
    • 1
  1. 1.Computer LabCambridge UniversityCambridgeEngland
  2. 2.Department of Mathematics and Computer ScienceFreie Universität BerlinBerlinGermany

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