Journal of Automated Reasoning

, Volume 10, Issue 1, pp 79–93 | Cite as

What holds in a context?

  • F. Corella


In order to establish that ℒ[A] = ℒ[B] follows from a set of assumptions Γ often one provesA =B and then invokes the principle of substitution of equals for equals. It has been observed that in the ancillary proof ofA =B one is allowed to use, in addition to those assumptions of Γ which are free for ℒ, certain (open) sentencesP which may not be part of Γ and may not follow from Γ, but are related to the context ℒ. We show that in an appropriate formal system there is a closed form solution to the problem of determining precisely what sentencesP can be used. We say that those sentenceshold in the context ℒ under the set of assumptions Γ. We suggest how the solution could be exploited in an interactive theorem prover.

Key words

Context sensitive substitution interactive theorem proving rewriting typed lambda calculus higher order logic 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andrews, Peter B.,An Introduction to Mathematical Logic and Type Theory, Academic Press (1986).Google Scholar
  2. 2.
    Andrews, P. B., Miller, D. A., Cohen, E. L., and Pfenning, F., ‘Automating higher-order logic’,Contemporary Mathematics 29, 169–192 (1984).Google Scholar
  3. 3.
    Andrews, P. B., Issar, S., Nesmith, D., and Pfenning, F., ‘The TPS theorem proving system’, in:Proceedings of the Ninth International Conference on Automated Deduction, Argonne, Illinois, U.S.A., Springer-Verlag (1988).Google Scholar
  4. 4.
    Bourbaki, Nicolas,Eléments de Mathématique, I: Théorie des Ensembles. Diffusion C.C.L.S., Paris (1970). English translation published by Addison-Wesley, 2nd edn. (1974).Google Scholar
  5. 5.
    Church, Alonzo, ‘A formulation of the simple theory of types’,J. Symbolic Logic 5(1), 56–68 (1940).Google Scholar
  6. 6.
    Corella, Francisco, ‘Context sensitive substitution’, September 1986 (revised April 1987). Unpublished.Google Scholar
  7. 7.
    Corella, Francisco, ‘The double nature of type theory’, Technical Report RC 15473, IBM Research (January 1990).Google Scholar
  8. 8.
    Corella, Francisco, ‘ZF/HOL, a formal theory for the mechanization of mathematics’, RC 16027, IBM Research (1990).Google Scholar
  9. 9.
    Gordon, Michael J. C., ‘A proof generating system for higher-order logic’, in: G. Birtwistle and P. A. Subrahmanyam (Eds.),VLSI Specificatin, Verification and Synthesis, Kluwer (1987).Google Scholar
  10. 10.
    Henkin, Leon, ‘A theory of propositional types’,Fundamenta Mathematicae 52, 323–344 (1963.Google Scholar
  11. 11.
    Ketonen, Jussi and Weening, Joseph S., ‘EKL — An interactive proof checker user's reference manual’, Technical Report STAN-CS-84-1006, Stanford University, Department of Computer Science, Stanford, CA 94305 (1984).Google Scholar
  12. 12.
    Monk, L. G., ‘Inference rules using local contexts’,J. Automated Reasoning 4, 445–462 (1988).Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • F. Corella
    • 1
  1. 1.IBM ResearchYorktown HeightsU.S.A.

Personalised recommendations