Abstract
It is well known that mathematical proofs often contain (abstract) algorithms, but although these algorithms can be understood by a human, it still takes a lot of time and effort to implement these algorithms on a computer; moreover, one runs the risk of making mistakes in the process.
From a fully formalized constructive proof one can automatically obtain a computer implementation of such an algorithm together with a proof that the program is correct. As an example we consider the fundamental theorem of algebra which states that every non-constant polynomial has a root. This theorem has been fully formalized in the Coq proof assistant. Unfortunately, when we first tried to extract a program, the computer ran out of resources. We will discuss how we used logical techniques to make it possible to extract a feasible program. This example is used as a motivation for a broader perspective on how the formalization of mathematics should be done with program extraction in mind.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Barendregt, H.P.: Lambda calculi with types. In: Handbook of logic in computer science, vol. 2, pp. 117–309. Oxford Univ. Press, New York (1992)
Bishop, E.: Foundations of Constructive Analysis. McGraw-Hill Book Company, New York (1967)
Bishop, E.: Mathematics as a numerical language. In: Intuitionism and Proof Theory (Proceedings of the summer Conference at Buffalo, N.Y., 1968), pp. 53–71. North-Holland, Amsterdam (1970)
Capretta, V.: Abstraction and Computation. PhD thesis, University of Nijmegen (2002)
Carlström, J.: Subsets, quotients and partial functions in martin-löf’s type theory. In: Geuvers, H., Wiedijk, F. (eds.) TYPES 2002. LNCS, vol. 2646, pp. 78–94. Springer, Heidelberg (2003) (to appear)
Ciaffaglione, A., Gianantonio, P.D.: A co-inductive approach to real numbers. In: Coquand, T., Nordström, B., Dybjer, P., Smith, J. (eds.) TYPES 1999. LNCS, vol. 1956, pp. 114–130. Springer, Heidelberg (2000)
Courant, J.: MC2: A module calculus for pure type systems. Technical Report 1292, LRI (September 2001)
Fernández, M., Mackie, I., Severi, P., Szasz, N.: A uniform approach to program extraction: Pure type systems with ultra σ-types, http://www.cmat.edu.uy/~severi/publications.html
Geuvers, H.: Inconsistency of classical logic in type theory, http://www.cs.kun.nl/~herman/note.ps.gz
Geuvers, H., Pollack, R., Wiedijk, F., Zwanenburg, J.: The algebraic hierarchy of the FTA Project. In: Linton, S.S. (ed.) Journal of Symbolic Computation, Special Issue on the Integration of Automated Reasoning and Computer Algebra Systems, pp. 271–286. Elsevier, Amsterdam (2002)
Geuvers, H., Wiedijk, F., Zwanenburg, J.: A constructive proof of the Fundamental Theorem of Algebra without using the rationals. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R. (eds.) TYPES 2000. LNCS, vol. 2277, pp. 96–111. Springer, Heidelberg (2002)
Grégoire, B., Leroy, X.: A compiled implementation of strong reduction. In: Proceedings ICFP 2002 (2002)
Letouzey, P.: A new extraction for Coq. In: Geuvers, H., Wiedijk, F. (eds.) TYPES 2002. LNCS, vol. 2646, pp. 200–219. Springer, Heidelberg (2003)
Martin-Löf, P.: Constructive mathematics and computer science. In: Logic, Methodology and the Philosophy of Science VI, pp. 153–175. North-Holland, Amsterdam (1982)
Niqui, M.: Exact arithmetic on Stern-Brocot tree, (2003) (submitted)
Paulin-Mohring, C.: Extracting Fω’s programs from proofs in the Calulus of Constructions. In: Sixteenth Annual ACM Symposium on Principles of Programming Languages, Austin. ACM, New York (1989)
Pottier, L.: Quotients dans le CCI. Technical Report RR-4053, INRIA (November 2000), http://www-sop.inria.fr/rapports/sophia/RR-4053.html
Prost, F.: Marking techniques for extraction. Technical Report 95-47, Laboratoire de l’informatique du Parallélisme, Ecole Normale Supérieure de Lyon (1995)
Schwichtenberg, H.: Minimal logic for computable functionals. Technical report, Mathematisches Institut der Universität München (2002)
The Coq Development Team. The Coq Proof Assistant Reference Manual Version 7.3. INRIA-Rocquencourt (2002)
Troelstra, A.S.: Realizability. Handbook of Proof Theory, pp. 407–473. North-Holland, Amsterdam (1998)
Troelstra, A.S., van Dalen, D.: Constructivism in mathematics. An introduction. Studies in Logic and the Foundations of Mathematics, vol. 123. North-Holland, Amsterdam (1988)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cruz-Filipe, L., Spitters, B. (2003). Program Extraction from Large Proof Developments. In: Basin, D., Wolff, B. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2003. Lecture Notes in Computer Science, vol 2758. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10930755_14
Download citation
DOI: https://doi.org/10.1007/10930755_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40664-8
Online ISBN: 978-3-540-45130-3
eBook Packages: Springer Book Archive