First-Order Logic with Dependent Types

  • Florian Rabe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4130)


We present DFOL, an extension of classical first-order logic with dependent types, i.e., as in Martin-Löf type theory, signatures may contain type-valued function symbols. A model theory for the logic is given that stays close to the established first-order model theory. The logic is presented as an institution, and the logical framework LF is used to define signatures, terms and formulas. We show that free models over Horn theories exist, which facilitates its use as an algebraic specification language, and show that the classical first-order axiomatization is complete for DFOL, too, which implies that existing first-order theorem provers can be extended. In particular, the axiomatization can be encoded in LF.


Free Variable Theorem Prove Type Theory Function Symbol Atomic Formula 
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.


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  1. [BC04]
    Bertot, Y., Castéran, P.: Coq’Art: The Calculus of Inductive Constructions. Springer, Heidelberg (2004)MATHGoogle Scholar
  2. [BM03]
    Bruni, R., Meseguer, J.: Generalized rewrite theories. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. [BM04]
    Bidoit, M., Mosses, P.D. (eds.): CASL User Manual. LNCS, vol. 2900. Springer, Heidelberg (2004)MATHGoogle Scholar
  4. [CAB+86]
    Constable, R., Allen, S., Bromley, H., Cleaveland, W., Cremer, J., Harper, R., Howe, D., Knoblock, T., Mendler, N., Panangaden, P., Sasaki, J., Smith, S.: Implementing Mathematics with the Nuprl Development System. Prentice-Hall, Englewood Cliffs (1986)Google Scholar
  5. [Car86]
    Cartmell, J.: Generalized algebraic theories and contextual category. Annals of Pure and Applied Logic 32 (1986)Google Scholar
  6. [CELM96]
    Clavel, M., Eker, S., Lincoln, P., Meseguer, J.: Principles of Maude. In: Meseguer, J. (ed.) Proceedings of the First International Workshop on Rewriting Logic, vol. 4, pp. 65–89 (1996)Google Scholar
  7. [Dia05]
    Diaconescu, R.: Institution-independent Model Theory (2005)Google Scholar
  8. [Dyb95]
    Dybjer, P.: Internal type theory. In: TYPES, pp. 120–134 (1995)Google Scholar
  9. [Gal86]
    Gallier, J.: Foundations of Automatic Theorem Proving. Wiley, Chichester (1986)MATHGoogle Scholar
  10. [GB92]
    Goguen, J.A., Burstall, R.M.: Institutions: Abstract model theory for specification and programming. Journal of the Association for Computing Machinery 39(1), 95–146 (1992)MATHMathSciNetGoogle Scholar
  11. [GWM+93]
    Goguen, J., Winkler, T., Meseguer, J., Futatsugi, K., Jouannaud, J.: Introducing OBJ. In: Goguen, J. (ed.) Applications of Algebraic Specification using OBJ, Cambridge (1993)Google Scholar
  12. [Hen49]
    Henkin, L.: The completeness of the first-order functional calculus. Journal of Symbolic Logic 14, 159–166 (1949)MATHCrossRefMathSciNetGoogle Scholar
  13. [Hof94]
    Hofmann, M.: On the interpretation of type theory in locally cartesian closed categories. In: CSL, pp. 427–441 (1994)Google Scholar
  14. [HS98]
    Huet, G., Saïbi, A.: Constructive category theory. In: Plotkin, G., Stirling, C., Tofte, M. (eds.) Proof, Language and Interaction: Essays in Honour of Robin Milner., MIT Press, Cambridge (1998)Google Scholar
  15. [HST94]
    Harper, R., Sannella, D., Tarlecki, A.: Structured presentations and logic representations. Annals of Pure and Applied Logic 67, 113–160 (1994)MATHCrossRefMathSciNetGoogle Scholar
  16. [Mak]
    Makkai, M.: First order logic with dependent sorts (FOLDS) (Unpublished)Google Scholar
  17. [ML74]
    Martin-Löf, P.: An intuitionistic theory of types: Predicative part. In: Proceedings of the 1973 Logic Colloquium, North-Holland, Amsterdam (1974)Google Scholar
  18. [NR01]
    Nieuwenhuis, R., Rubio, A.: Paramodulation-Based theorem proving. In: Handbook of Automated Reasoning, pp. 371–443. Elsevier Science Publishers, Amsterdam (2001)CrossRefGoogle Scholar
  19. [ORS92]
    Owre, S., Rushby, J.M., Shankar, N.: PVS: A prototype verification system. In: Kapur, D. (ed.) CADE 1992. LNCS, vol. 607, pp. 748–752. Springer, Heidelberg (1992)Google Scholar
  20. [OS97]
    Owre, S., Shankar, N.: The formal semantics of PVS. Technical Report SRI-CSL-97-2, SRI International (1997)Google Scholar
  21. [Pfe01]
    Pfenning, F.: Logical frameworks. In: Handbook of automated reasoning, pp. 1063–1147. Elsevier, Amsterdam (2001)CrossRefGoogle Scholar
  22. [PP03]
    Pientka, B., Pfenning, F.: Optimizing higher-order pattern unification. In: 19th International Conference on Automated Deduction, pp. 473–487. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  23. [PS99]
    Pfenning, F., Schürmann, C.: System description: Twelf - a meta-logical framework for deductive systems. In: Ganzinger, H. (ed.) CADE 1999. LNCS (LNAI), vol. 1632, pp. 202–206. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  24. [RV02]
    Riazanov, A., Voronkov, A.: The design and implementation of Vampire. AI Communications 15, 91–110 (2002)MATHGoogle Scholar
  25. [See84]
    Seely, R.: Locally cartesian closed categories and type theory. Math. Proc. Cambridge Philos. Soc. 95, 33–48 (1984)MATHCrossRefMathSciNetGoogle Scholar
  26. [SP96]
    Schürmann, C., Pfenning, F.: Automated theorem proving in a simple meta-logic for LF. In: Kirchner, C., Kirchner, H. (eds.) Proceedings of the 15th International Conference on Automated Deduction, pp. 286–300. Springer, Heidelberg (1996)Google Scholar
  27. [Tar85]
    Tarlecki, A.: On the existence of free models in abstract algebraic institutions. Theoretical Computer Science 37, 269–301 (1985)MATHCrossRefMathSciNetGoogle Scholar
  28. [Vir96]
    Virga, R.: Higher-order superposition for dependent types. In: Ganzinger, H. (ed.) Proceedings of the 7th International Conference on Rewriting Techniques and Applications, pp. 123–137. Springer, Heidelberg (1996)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Florian Rabe
    • 1
  1. 1.Carnegie Mellon University and International University Bremen 

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