Abstract
The standard versions of HOL only support disjoint sums over finite families of types. This paper introduces disjoint sums over type classes containing possibly a countably infinite number of monomorphic types. The result is a monomorphic sum type together with an overloaded function which represents the family of injections. Model-theoretic reasoning shows the soundness of the construction.
In order to axiomatize the disjoint sums in HOL, datatypes are introduced which mirror the syntactic structure of type classes. The association of a type with its image in the sum type is represented by a HOL function carrier. This allows a translation of the set-theoretic axiomatization of disjoint sums to HOL.
As an application, a sum type U is presented which contains isomorphic copies of many familiar HOL types. Finally, a Z universe is constructed which can server as the basis of a HOL model of the Z schema calculus.
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References
J.-Y. Girard. The system F of variable types, fifteen years later. Theoretical Computer Science, 45:159–192, 1986.
M.J.C. Gordon. Set Theory, Higher Order Logic or Both? In Gunter and Felty A. Felty, editors. Proceedings of the 10th International Conference on Theorem Proving in Higher Order Logics (TPHOLs’97), LNCS 1275. Springer-Verlag, 1997 [4], pages 191–201.
M.J.C. Gordon and T.F. Melham. Introduction to HOL. Cambridge University Press, 1993.
E. L. Gunter and A. Felty, editors. Proceedings of the 10th International Conference on Theorem Proving in Higher Order Logics (TPHOLs’97), LNCS 1275. Springer-Verlag, 1997.
M.C. Henson and S. Reeves. A logic for the schema calculus. In J. P. Bowen, A. Fett, and M. G. Hinchey, editors, ZUM’98: The Z Formal Specification Notation, LNCS 1493, pages 172–191. Springer-Verlag, 1998.
S.P. Jones and J. Hughes, editors. HASKELL98: A Non-strict, Purely Functional Language, February 1999. On-line available via http://haskell.org/.
J.P. Bowen and M.J.C. Gordon. Z and HOL. In J.P. Bowen and J.A. Hall, editors, Z User Workshop, Cambridge 1994, Workshops in Computing, pages 141–167. Springer-Verlag, 1994.
Kolyang, T. Santen, and B. Wolff. A Structure Preserving Encoding of Z in Isabelle/HOL. In J. von Wright, J. Grundy, and J. Harrison, editors, Theorem Proving in Higher Order Logics — 9th International Conference, LNCS 1125, pages 283–298. Springer Verlag, 1996.
L. Lamport and L.C. Paulson. Should your specification language be typed? Technical Report 425, Computer Laboratory, University of Cambridge, May 1997.
Zhaohui Luo. A higher-order calculus and theory abstraction. Information and Computation, 90(1):107–137, 1991.
L.C. Paulson. A fixedpoint approach to implementing (co)inductive definitions. In A. Bundy, editor, Automated Deduction — CADE-12, LNAI 814, pages 148–161. Springer-Verlag, 1994.
L.C. Paulson. Isabelle: A Generic Theorem Prover. LNCS 828. Springer-Verlag, 1994.
F. Regensburger. HOLCF: Eine konservative Erweiterung von HOL um LCF. Dissertation an der Technischen Universität München, 1994.
J.M. Spivey. Understanding Z: a specification language and its formal semantics. Cambridge Tracts in Theoretical Computer Science 3. Cambridge University Press, 1988.
R. Turner. Sets, types and typechecking. Journal of Logic and Computation, To Appear.
M. Wenzel. Type classes and overloading in higher-order logic. In Gunter and Felty A. Felty, editors. Proceedings of the 10th International Conference on Theorem Proving in Higher Order Logics (TPHOLs’97), LNCS 1275. Springer-Verlag, 1997 [4], pages 307–322.
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Völker, N. (1999). Disjoint Sums over Type Classes in HOL. In: Bertot, Y., Dowek, G., Théry, L., Hirschowitz, A., Paulin, C. (eds) Theorem Proving in Higher Order Logics. TPHOLs 1999. Lecture Notes in Computer Science, vol 1690. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48256-3_2
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DOI: https://doi.org/10.1007/3-540-48256-3_2
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