Translating set theoretical proofs into type theoretical programs
We show how to translate proofs of II 2 0 -sentences in the fragment of set theory KPI U + , which is an extension of Kripke-Platek set theory, into proofs of Martin-Löf's Type Theory using a cut elimination argument. This procedure is effective. The method used is an implementation of techniques from traditional proof theoretic analysis in type theory. It follows that KPI U + and ML1 W show the same II 2 0 -sentences and have therefore the same programs. As a side result we get that II 2 0 -sentences provable in intensional and extensional version of ML1W are the same, solving partly a problem posed by M. Hofmann.
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- [Ba75]Barwise, J.: Admissible Sets and Structures. An Approach to Definability Theory, Springer, 1975.Google Scholar
- [Be85]Beeson, M.: Foundations of Constructive Mathematics Springer, Berlin, 1985Google Scholar
- [BS95]Berger, U. and Schwichtenberg, H.: Program Extraction from Classical Proofs. In: D. Leivant (Ed.): Logic and Computational Complexity, LCC '94, Indianapolis, October 1994, Springer Lecture Notes in Computer Science 960, pp. 77–97.Google Scholar
- [Bu91]Buchholz, W.: Notation systems for infinitary derivations. Arch. Math. Logic (1991) 30:277–296.Google Scholar
- [Bu92]Buchholz, W.: A simplified version of local predicativity. In: Aczel, P. et al. (Eds.): Proof Theory. Cambridge University Press, Cambridge, 1992, pp. 115–147.Google Scholar
- [Jä86]Jäger, G.: Theories for Admissible Sets: A Unifying Approach to Proof Theory. Bibliopolis, Naples, 1986.Google Scholar
- [Sch92]Schwichtenberg, H.: Proofs as programs. In:Aczel, P. et al. (Eds.): Proof Theory. Cambridge University Press, Cambridge, 1992, pp. 81–113.Google Scholar
- [Se93]Setzer, A.: Proof theoretical strength of Martin-Löf Type Theory with W-type and one universe. PhD-thesis, Munich, 1993.Google Scholar
- [Se96]Setzer, A.: Extending Martin-Löf Type Theory by one Mahlo-Universe. Submitted.Google Scholar
- [Se97]Setzer, A.: Well-ordering proofs for Martin-Löf Type Theory with W-type and one universe. Submitted.Google Scholar