Abstract
We propose a method for realising the proofs of Intuitionistic Zermelo-Fraenkel set theory (IZF) by strongly normalising λ-terms. This method relies on the introduction of a Curry-style type theory extended with specific subtyping principles, which is then used as a low-level language to interpret IZF via a representation of sets as pointed graphs inspired by Aczel’s hyperset theory.
As a consequence, we refine a classical result of Myhill and Friedman by showing how a strongly normalising λ-term that computes a function of type ℕ→ℕ can be extracted from the proof of its existence in IZF.
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References
Aczel, P.: Non well-founded sets. Center for the Study of Language and Information (1988)
Barendregt, H.: Introduction to generalized type systems. Technical Report 90-8, University of Nijmegen, Department of Informatics (May 1990)
Barras, B., Boutin, S., Cornes, C., Courant, J., Filliâtre, J.C., Giménez, E., Herbelin, H., Huet, G., Muñoz, C., Murthy, C., Parent, C., Paulin, C., Saïbi, A., Werner, B.: The Coq Proof Assistant Reference Manual – Version V6.1. Technical Report 0203, INRIA (August 1997)
Friedman, H.: Some applications of Kleene’s methods for intuitionistic systems. In: Cambridge Summer School in Mathematical Logic. Springer Lecture Notes in Mathematics, vol. 337, pp. 113–170. Springer, Heidelberg (1973)
Geuvers, J.H., Nederhof, M.J.: A modular proof of strong normalization for the calculus of constructions. Journal of Functional Programming 1,2, 155–189 (1991)
Krivine, J.-L.: Typed lambda-calculus in classical Zermelo-Fraenkel set theory. Archive for Mathematical Logic 40(3), 189–205 (2001)
Krivine, J.-L.: Dependent choice, ‘quote’ and the clock. Theoretical Computer Science (2003)
Martin-Löf, P.: Intuitionistic Type Theory. Bibliopolis, Napoli (1984)
McCarty, D.: Realizability and Recursive Mathematics. PhD thesis, Ohio State University (1984)
Melliès, P.-A., Werner, B.: A generic normalization proof for pure type systems. In: Giménez, E. (ed.) TYPES 1996. LNCS, vol. 1512, Springer, Heidelberg (1998)
Miquel, A.: Le calcul des constructions implicite: syntaxe et sémantique. PhD thesis, Université Paris VII (2001)
Myhill, J.: Some properties of intuitionistic Zermelo-Fraenkel set theory. In: Cambridge Summer School in Mathematical Logic. Springer Lecture Notes in Mathematics, vol. 337, pp. 206–231. Springer, Heidelberg (1973)
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Miquel, A. (2003). A Strongly Normalising Curry-Howard Correspondence for IZF Set Theory. In: Baaz, M., Makowsky, J.A. (eds) Computer Science Logic. CSL 2003. Lecture Notes in Computer Science, vol 2803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45220-1_35
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DOI: https://doi.org/10.1007/978-3-540-45220-1_35
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