Abstract
This paper introduces the logical system HOL2P that extends classical higher order logic (HOL) with type operator variables and universal types. HOL2P has explicit term operations for type abstraction and type application. The formation of type application terms t [T] is restricted to small types T that do not contain any universal types. This constraint ensures the existence of a set-theoretic model and thus consistency.
The expressiveness of HOL2P allows category-theoretic concepts such as natural transformations and initial algebras to be applied at the level of polymorphic HOL functions. The parameterisation of terms with type operators adds genericity to theorems. Type variable quantification can also be expressed.
A prototype of HOL2P has been implemented on top of HOL-Light. Type inference is semi-automatic, and some type annotations are necessary. Reasoning is supported by appropriate tactics. The implementation has been used to check some sample derivations.
This is a preview of subscription content, log in via an institution to check access.
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
Backhouse, R., Jansson, P., Jeuring, J., Meertens, L.: Generic programming - an introduction. In: Swierstra, S.D., Oliveira, J.N. (eds.) AFP 1998. LNCS, vol. 1608, Springer, Heidelberg (1999)
Bird, R.S., de Moor, O.: Algebra of Programming. Prentice-Hall, Englewood Cliffs (1997)
Coquand, T.: A new paradox in type theory. In: Prawitz, D., Skyrms, B., Westerståhl, D. (eds.) Proceedings 9th Int. Congress of Logic, Methodology and Philosophy of Science, pp. 555–570. North-Holland, Amsterdam (1994)
Girard, J.-Y., Lafont, Y., Taylor, P.: Proofs and Types. Cambridge Tracts in Theoretical Computer Science, vol. 7. Cambridge University Press, Cambridge (1989)
Glimming, J.: Logic and Automation for Algebra of Programming. PhD thesis, MSc Thesis, University of Oxford (2001)
Gordon, M., Wadsworth, C.P., Milner, R.: Edinburgh LCF. LNCS, vol. 78. Springer, Heidelberg (1979)
Gordon, M.J.C., Melham, T.F.: Introduction to HOL. Cambridge University Press, Cambridge (1993)
Gordon, M.J.C., Pitts, A.: The HOL Logic and System. In: Bowen, J. (ed.) Towards Verified Systems. Real-Time Safety Critical Systems Series, vol. 2, Elsevier, Amsterdam (1994)
Harrison, J.: The HOL Light System Reference – Version 2.20 (2006), http://www.cl.cam.ac.uk/~jrh13/hol-light
Harrison, J.: Towards self-verification in HOL-Light. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, Springer, Heidelberg (2006)
Hu, Z., Iwasaki, H., Takeichi, M.: Deriving structural hylomorphisms from recursive definitions. In: ICFP, pp. 73–82 (1996)
Jones, S.P., Vytiniotis, D., Weirich, S., Shields, M.: Practical type inference for arbitrary-rank types. Journal of Functional Programming 17(1) (2007)
Mac Lane, S.: Categories for the Working Mathematician. Springer, Heidelberg (1971)
Melham, T.F.: The HOL logic extended with quantification over type variables. Formal Methods in System Design 3(1–2), 7–24 (1994)
Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)
Pierce, B.C.: Types and Programming Languages. MIT Press, Cambridge (2002)
The Coq Development Team. The Coq Proof Assistant Reference Manual – Version V8.0 (2004), http://coq.inria.fr
Völker, N.: HOL2P Prototype Implementation (2007), http://cswww.essex.ac.uk/staff/norbert/hol2p
Vytiniotis, D., Weirich, S., Jones, S.P.: Boxy types: inference for higher-rank types and impredicativity. ACM SIGPLAN Notices 41(9), 251–262 (2006)
Wadler, P.: Theorems for free? In: Functional Programming Languages and Computer Architecture, Springer, Heidelberg (1989)
Wadler, P.: The essence of functional programming. In: POPL, pp. 1–14 (1992)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Völker, N. (2007). HOL2P - A System of Classical Higher Order Logic with Second Order Polymorphism. 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_25
Download citation
DOI: https://doi.org/10.1007/978-3-540-74591-4_25
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)