A Focused Sequent Calculus for Higher-Order Logic

  • Fredrik Lindblad
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8562)


We present a focused intuitionistic sequent calculus for higher-order logic. It has primitive support for equality and mixes λ-term conversion with equality reasoning. Classical reasoning is enabled by extending the system with rules for reductio ad absurdum and the axiom of choice. The resulting system is proved sound with respect to Church’s simple type theory. The soundness proof has been formalized in Agda. A theorem prover based on bottom-up search in the calculus has been implemented. It has been tested on the TPTP higher-order problem set with good results. The problems for which the theorem prover performs best require higher-order unification more frequently than the average higher-order TPTP problem. Being strong at higher-order unification, the system may serve as a complement to other theorem provers in the field.


Inference Rule Theorem Prover Natural Deduction Sequent Calculus Derivation Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Sutcliffe, G.: The TPTP problem library and associated infrastructure. J. Autom. Reason. 43(4), 337–362 (2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    Sutcliffe, G., Benzmüller, C.: Automated reasoning in higher-order logic using the TPTP THF infrastructure. Journal of Formalized Reasoning 3(1), 1–27 (2010)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bove, A., Dybjer, P., Norell, U.: A brief overview of Agda — a functional language with dependent types. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 73–78. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Lindblad, F.: Higher-order proof construction based on first-order narrowing. Electron. Notes Theor. Comput. Sci. 196, 69–84 (2008)CrossRefGoogle Scholar
  5. 5.
    Middeldorp, A., Okui, S., Ida, T.: Lazy narrowing: Strong completeness and eager variable elimination. Theoretical Computer Science 167, 95–130 (1995)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Herbelin, H.: A lambda-calculus structure isomorphic to Gentzen-style sequent calculus structure. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 61–75. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  7. 7.
    Lengrand, S., Dyckhoff, R., McKinna, J.: A focused sequent calculus framework for proof search in pure type systems. Logical Methods in Computer Science 7(1) (2011)Google Scholar
  8. 8.
    Andreoli, J.: Logic programming with focusing proofs in linear logic. Journal of Logic and Computation 2, 297–347 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Miller, D., Nadathur, G., Pfenning, F., Scedrov, A.: Uniform proofs as a foundation for logic programming. Annals of Pure and Applied Logic 51(12), 125–157 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Girard, J.Y.: A new constructive logic: Classical logic. Mathematical Structures in Computer Science 1(3), 255–296 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Henkin, L.: Completeness in the theory of types. J. Symb. Log. 15(2), 81–91 (1950)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Lindblad, F.: AgsyHOL source code and Agda formalization (2012),
  13. 13.
    Lengrand, S.: Normalisation & Equivalence in Proof Theory & Type Theory. PhD thesis, Université Paris 7 & University of St Andrews (2006)Google Scholar
  14. 14.
    Lindblad, F.: Property directed generation of first-order test data. In: Trends in Functional Programming. Intellect, vol. 8, pp. 105–123 (2008)Google Scholar
  15. 15.
    Hanus, M.: Curry: An integrated functional logic language. Language report (March 2006),

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Fredrik Lindblad
    • 1
  1. 1.Chalmers University of TechnologyUniversity of GothenburgGothenburgSweden

Personalised recommendations