Abstract
This paper describes a method of representing algebraic structures in the theorem prover Isabelle. We use Isabelle’s higher order logic extended with set theoretic constructions. Dependent types, constructed as HOL sets, are used to represent modular structures by semantical embedding. The modules remain first class citizen of the logic. Hence, they enable adequate formalization of abstract algebraic structures and a natural proof style. Application examples drawn from abstract algebra and lattice theory — the full version of Tarski’s fixpoint theorem — validate the concept.
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References
R. Burstall. Computer Assisted Proof for Mathematics: an Introduction using the LEGO Proof System. Technical Report ECS-LFCS-91-132, University of Edinburgh, 1990. 131
A. Church. A Formulation of the Simple Theory of Types. Journal of Symbolic Logic, pages 56–68, 1940. 125
W. M. Farmer, J. D. Guttman, and F. J. Thayer. imps: an Interactive Mathematical Proof System. Journal of Automated Reasoning, 11:213–248, 1993. 121, 122, 123
John V. Guttag and James J. Horning, editors. Larch: Languages and Tools for Formal Specification. Texts and Monographs in Computer Science. Springer-Verlag, 1993. With Stephen J. Garland, Kevin D. Jones, Andrés Modet, and Jeannette M. Wing. 121, 123
A. Heyting. Intuitionism: An Introduction. North Holland, Amsterdam, 1956. 125
F. Kammüller and L. C. Paulson. A Formal Proof of Sylow’s First Theorem-An Experiment in Abstract Algebra with Isabelle HOL. Journal of Automated Reasoning, 1999. To appear. 121
F. Kammüller and M. Wenzel. Locales-a Sectioning Concept for Isabelle. Technical Report 449, University of Cambridge, Computer Laboratory, 1998. 131
Z. Luo and R. Pollack. Lego proof development system: User’s manual. Technical Report ECS-LFCS-92-211, University of Edinburgh, 1992. 131
Z. Luo. A Higher-order Calculus and Theory Abstraction. Information and Computation, 90(1), 1990. 131
Z. Luo. An Extended Calculus of Constructions. PhD thesis, University of Edinburgh, 1990. Also as Report CST-65-90. 131
D. B. MacQueen. Using Dependant Types to Express Modular Structures. In Proc. 13th ACM Symp. Principles Programming Languages. ACM Press, 1986. 131
B. Nordström, K. Petersson, and J. M. Smith. Programming in Martin-Löf’ s Type Theory — An Introduction. Oxford Science Publications. Clarendon Press, Oxford, 1990. 125
W. Naraschewski and M. Wenzel. Object-oriented Verification based on Record Subtyping in Higher-Order Logic. In 11th International Conference on Theorem Proving in Higher Order Logics, volume 1479 of LNCS, ANU, Canberra, Australia, 1998. Springer-Verlag. 121, 123
S. Owre, N. Shankar, and J. M. Rushby. The PVS Specification Language (Beta Release). Technical report, SRI International, 1993. 121, 123
R. Pollack. The Tarski Fixpoint Theorem. http://e-mail to: proof-sci@se.chalmers.cs, 1990. 131
R. Pollack. Theories in Type Theory. Slides, available on the Web as http://www.brics.dk/pollack, unknown year. 132
D. T. Sannella and R. M. Burstall. Structured Theories in LCF. In CAAP’83: Trees in Algebra and Programming, volume 159 of LNCS, pages 377–91. Springer-Verlag, 1983. 131
Alfred Tarski. A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, 5:285–309, 1955. 130
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Kammüller, F. (1999). Modular Structures as Dependent Types in Isabelle. In: Altenkirch, T., Reus, B., Naraschewski, W. (eds) Types for Proofs and Programs. TYPES 1998. Lecture Notes in Computer Science, vol 1657. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48167-2_9
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DOI: https://doi.org/10.1007/3-540-48167-2_9
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