Schematic Cut Elimination and the Ordered Pigeonhole Principle

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9706)


Schematic cut-elimination is a method of cut-elimination which can handle certain types of inductive proofs. In previous work, an attempt was made to apply the schematic CERES method to a formal proof with an arbitrary number of \(\varPi _{2}\) cuts (a recursive proof encapsulating the infinitary pigeonhole principle). However the derived schematic refutation for the characteristic clause set of the proof could not be expressed in the schematic resolution calculus developed so far. Without this formalization a Herbrand system cannot be algorithmically extracted. In this work, we provide a restriction of infinitary pigeonhole principle, the ECA-schema (Eventually Constant Assertion), or ordered infinitary pigeonhole principle, whose analysis can be completely carried out in the existing framework of schematic CERES. This is the first time the framework is used for proof analysis. From the refutation of the clause set and a substitution schema we construct a Herbrand system.


  1. 1.
    Baaz, M., Hetzl, S., Leitsch, A., Richter, C., Spohr, H.: Ceres: An analysis of Fürstenberg’s proof of the infinity of primes. Theoret. Comput. Sci. 403(2–3), 160–175 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Baaz, M., Leitsch, A.: On skolemization and proof complexity. Fundamenta Informaticae 20(4), 353–379 (1994)MathSciNetMATHGoogle Scholar
  3. 3.
    Baaz, M., Leitsch, A.: Cut-elimination and redundancy-elimination by resolution. J. Symbolic Comput. 29, 149–176 (2000)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Baaz, M., Leitsch, A.: Methods of Cut-Elimination. Springer Publishing Company, Incorporated (2013)Google Scholar
  5. 5.
    Brotherston, J.: Cyclic proofs for first-order logic with inductive definitions. In: Beckert, B. (ed.) TABLEAUX 2005. LNCS (LNAI), vol. 3702, pp. 78–92. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Brotherston, J., Simpson, A.: Sequent calculi for induction and infinite descent. J. Logic Comput. 22, 1177–1216 (2010)MathSciNetMATHGoogle Scholar
  7. 7.
    Bundy, A.: The automation of proof by mathematical induction. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 845–911. Elsevier, Amsterdam (2001)CrossRefGoogle Scholar
  8. 8.
    Cerna, D., Leitsch, A.: Analysis of clause set schema aided by automated theorem proving: Acase study (2015). arXiv:1503.08551v1 [cs.LO]
  9. 9.
    Cerna, D., Leitsch, A.: Schematic cut elimination and the ordered pigeonhole principle [extended version] (2016). arXiv:1601.06548 [math.LO]
  10. 10.
    Cerna, D.M.: Advances in schematic cut elimination. Ph.D. thesis, Technical University of Vienna (2015).
  11. 11.
    Comon, H.: Inductionless induction. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 913–962. Elsevier, Amsterdam (2001)CrossRefGoogle Scholar
  12. 12.
    Dunchev, C.: Automation of cut-elimination in proof schemata. Ph.D. thesis, Technical University of Vienna (2012)Google Scholar
  13. 13.
    Dunchev, C., Leitsch, A., Rukhaia, M., Weller, D.: Cut-elimination and proof schemata. In: Aher, M., Hole, D., Jeřábek, E., Kupke, C. (eds.) TbiLLC 2013. LNCS, vol. 8984, pp. 117–136. Springer, Heidelberg (2015)Google Scholar
  14. 14.
    Gentzen, G.: Untersuchungen über das logische Schließen I. Math. Z. 39(1), 176–210 (1935)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Girard, J.-Y.: Proof Theory and Logical Complexity. Studies in Proof Theory, vol. 1. Bibliopolis, Naples (1987)MATHGoogle Scholar
  16. 16.
    Mcdowell, R., Miller, D.: Cut-elimination for a logic with definitions and induction. Theoret. Comput. Sci. 232, 2000 (1997)MathSciNetGoogle Scholar
  17. 17.
    Takeuti, G.: Proof Theory. Studies in Logic and the Foundations of Mathematics, vol. 81. American Elsevier Pub., New York (1975)MATHGoogle Scholar
  18. 18.
    Weidenbach, C., Dimova, D., Fietzke, A., Kumar, R., Suda, M., Wischnewski, P.: SPASS version 3.5. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 140–145. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Research Institute for Symbolic Computation (RISC)Johannes Kepler UniversityLinzAustria
  2. 2.Logic and Theory GroupTechnical University of ViennaViennaAustria

Personalised recommendations