Abstract
Higman’s lemma, a specific instance of Kruskal’s theorem, is an interesting result from the area of combinatorics, which has often been used as a test case for theorem provers. We present a constructive proof of Higman’s lemma in the theorem prover Isabelle, based on a paper proof by Coquand and Fridlender. Making use of Isabelle’s newly-introduced infrastructure for program extraction, we show how a program can automatically be extracted from this proof, and analyze its computational behaviour.
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References
Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)
Barras, B., et al.: The Coq proof assistant reference manual – version 7.2. Technical Report 0255, INRIA (February 2002)
Benl, H., Berger, U., Schwichtenberg, H., Seisenberger, M., Zuber, W.: Proof theory at work: Program development in the Minlog system. In: Bibel, W., Schmitt, P. (eds.) Automated Deduction – A Basis for Applications. Systems and Implementation Techniques of Applied Logic Series, vol. II, pp. 41–71. Kluwer Academic Publishers, Dordrecht (1998)
Berghofer, S.: Program Extraction in simply-typed Higher Order Logic. In: Geuvers, H., Wiedijk, F. (eds.) TYPES 2002. LNCS, vol. 2646, pp. 21–38. Springer, Heidelberg (2003)
Constable, R.L., et al.: Implementing Mathematics with the Nuprl Proof Development System. Prentice-Hall, NJ (1986)
Coquand, T., Fridlender, D.: A proof of Higman’s lemma by structural induction. Unpublished draft (November 1993), available at http://www.math.chalmers.se/~frito/Papers/open.ps.gz
Fridlender, D.: Higman’s lemma in type theory. In: Giménez, E. (ed.) TYPES 1996. LNCS, vol. 1512, pp. 112–133. Springer, Heidelberg (1998)
Higman, G.: Ordering by divisibility in abstract algebras. In: Proceedings of the London Mathematical Society, vol. 3(2), pp. 326–336 (1952)
Magnusson, L.: The Implementation of ALF—a Proof Editor Based on Martin- Löf ’s Monomorphic Type Theory with Explicit Substitution. Phd thesis, Dept. of Computing Science, Chalmers Univ. of Technology and Univ. of Göteborg (1994)
Murthy, C.: Extracting Constructive Content from Classical Proofs. PhD thesis, Cornell University (1990)
Murthy, C.R., Russell, J.R.: A constructive proof of Higman’s lemma. In: Mitchell, J.C. (ed.) Proceedings of the 5th Annual IEEE Symposium on Logic in Computer Science, Philadelphia, PA, June 1990, pp. 257–269. IEEE Computer Society Press, Los Alamitos (1990)
Nash-Williams, C.: On well-quasi-ordering finite trees. Proceedings of the Cambridge Philosophical Society 59(4), 833–835 (1963)
Paulin-Mohring, C.: Inductive Definitions in the System Coq - Rules and Properties. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664, pp. 49–92. Springer, Heidelberg (1993)
Seisenberger, M.: Konstruktive Aspekte von Higmans Lemma. Master’s thesis, Fakultät für Mathematik, Ludwig-Maximilians-Universität München (1998)
Seisenberger, M.: On the Constructive Content of Proofs. PhD thesis, Fakultät für Mathematik, Ludwig-Maximilians-Universität München (2003)
Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)
Wenzel, M.: Isabelle/Isar – a versatile environment for human-readable formal proof documents. PhD thesis, Institut für Informatik, TU München (2002), http://tumb1.biblio.tu-muenchen.de/publ/diss/in/2002/wenzel.html
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Berghofer, S. (2004). A Constructive Proof of Higman’s Lemma in Isabelle. In: Berardi, S., Coppo, M., Damiani, F. (eds) Types for Proofs and Programs. TYPES 2003. Lecture Notes in Computer Science, vol 3085. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24849-1_5
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DOI: https://doi.org/10.1007/978-3-540-24849-1_5
Publisher Name: Springer, Berlin, Heidelberg
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