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Uniqueness of normal proofs of minimal formulas

Published online by Cambridge University Press:  12 March 2014

Makoto Tatsuta
Makoto Tatsuta*
Affiliation:
Research Institute of Electrical Communication, Tohoku University, 2-1-1 Katahira, Sendai 980, Japan, E-mail: tatsuta@riec.tohoku.ac.jp
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Abstract

A minimal formula is a formula which is minimal in provable formulas with respect to the substitution relation. This paper shows the following: (1) A β-normal proof of a minimal formula of depth 2 is unique in NJ. (2) There exists a minimal formula of depth 3 whose βη-normal proof is not unique in NJ. (3) There exists a minimal formula of depth 3 whose βη-normal proof is not unique in NK.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 3 , September 1993 , pp. 789 - 799
Copyright
Copyright © Association for Symbolic Logic 1993

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References

REFERENCES

[1]Barendregt, H. P., The lambda calculus: Its syntax and semantics, North-Holland, Amsterdam, 1984.Google Scholar
[2]Hindley, J. R. and Seldin, J. P., Introduction to combinators and λ-calculus, Cambridge University Press, Cambridge, 1986.Google Scholar
[3]Hirokawa, S., Principal types of BCK-lambda terms, Theoretical Computer Science, vol. 107(1993), pp. 253–276.CrossRefGoogle Scholar
[4]Howard, W. A., The formulae-as-types notion of construction, To H. B. Curry: Essays on combinatory logic, lambda calculus and formalism, Academic Press, London, 1980.Google Scholar
[5]Komori, Y., BCK algebras and lambda calculus, Proceedings of the 10th symposium on semigroups, Josai University, Sakado, 1987, pp. 5–11.Google Scholar
[6]Komori, Y. and Hirokawa, S., The number of proofs for a BCK-formula, this Journal, vol. 58 (1993), pp. 626–628.Google Scholar
[7]Solov’ev, S. V., Preservation of equivalence of derivations under reduction of depth of formulas, Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya matematicheskogo Instituta im. V. A. Steklova AN SSSR, vol. 88 (1979), pp. 197–207; English translation, Journal of Soviet Mathematics, vol. 20 (1982), pp. 2370–2376.Google Scholar
[8]Troelstra, A., Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin and New York, 1973.CrossRefGoogle Scholar