Abstract
In Mizar, unlike in most other proof assistants, the types are not part of the foundations of the system. Mizar is based on untyped set theory, which means that in Mizar expressions are typed but the values of those expressions are not.
In this paper we present the Mizar type system as a collection of type inference rules. We will interpret Mizar types as soft types, by translating Mizar’s type judgments into sequents of untyped first order predicate logic. We will then prove that the Mizar type system is correct with respect to this translation in the sense that each derivable type judgment translates to a provable sequent.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bancerek, G.: On the structure of Mizar types. Electronic Notes in Theoretical Computer Science, 85 (2003)
Cartwright, R., Fagan, M.: Soft typing. In: Proceedings of the SIGPLAN ’91 Conference on Programming Language Design and Implementation, pp. 278–292 (1991)
Dahn, I.: Interpretation of a Mizar-Like Logic in First-Order Logic. In: Caferra, R., Salzer, G. (eds.) Automated Deduction in Classical and Non-Classical Logics. LNCS (LNAI), vol. 1761, pp. 137–151. Springer, Heidelberg (2000)
Gordon, M.J.C., Melham, T.F. (eds.): Introduction to HOL. Cambridge University Press, Cambridge (1993)
Hurd, J.: Predicate subtyping with predicate sets. In: Boulton, R.J., Jackson, P.B. (eds.) TPHOLs 2001. LNCS, vol. 2152, pp. 265–280. Springer, Heidelberg (2001)
Lee, G., Rudnicki, P.: Alternative Aggregates in Mizar (to be published in Mathematical Knowledge Management, 2007)
Muzalewski, M.: An Outline of PC Mizar. Fondation Philippe le Hodey, Brussels (1993), http://www.cs.ru.nl/~freek/mizar/mizarmanual.ps.gz
Urban, J.: MPTP 0.1: System Description. ENTCS 86(1) (2003)
Urban, J.: Translating Mizar for First Order Theorem Provers. In: Asperti, A., Buchberger, B., Davenport, J.H. (eds.) MKM 2003. LNCS, vol. 2594, pp. 203–215. Springer, Heidelberg (2003)
Urban, J.: MoMM – Fast Interreduction and Retrieval in Large Libraries of Formalized Mathematics. International Journal on Artificial Intelligence Tools 15(1), 109–130 (2006)
Weidenbach, C.: SPASS: Combining superposition, sorts and splitting (1999)
Wiedijk, F. (ed.): The Seventeen Provers of the World. LNCS (LNAI), vol. 3600. Springer, Heidelberg (2006)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wiedijk, F. (2007). Mizar’s Soft Type System. In: Schneider, K., Brandt, J. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2007. Lecture Notes in Computer Science, vol 4732. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74591-4_28
Download citation
DOI: https://doi.org/10.1007/978-3-540-74591-4_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74590-7
Online ISBN: 978-3-540-74591-4
eBook Packages: Computer ScienceComputer Science (R0)