Abstract
Inductive type theories, as used in systems like Coq or Lego [11,14,4], provide a systematic approach to program recursive functions over inductive data-structures and to reason about these functions. Recursive computation is described by reduction rules, included in the type system under the name ι-reduction. If t is an element of a recursive type, f is a recursive function over that type and v is the value of f(t), then the equality f(t) = v is a simple tautology, because f(t) and v are equal modulo ι-reduction.
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
Peter Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic and the Foundations of Mathematics, 1977.
Robert Constable, S. F. Allen, H. M. Bromley, W. R. Cleaveland, J. F. Cremer, R. W. Harber, D. J. Howe, T. B. Knoblock, N. P. Mendler, P. Panangaden, J. T. Sasaki, and S. F. Smith. Implementing mathematics with the Nuprl proof development system. Prentice-Hall, 1986.
Robert L. Constable and Nax P. Mendler. Recursive definitions in type theory. In Rohit Parikh, editor, Logics of Programs, volume 193 of LNCS, pages 61–78. Springer Verlag, June 1985.
Thierry Coquand and Christine Paulin-Mohring. Inductively defined types. In P. Martin-Löf and G. Mints, editors, Proceedings of Colog’88, volume 417 of Lecture Notes in Computer Science. Springer-Verlag, 1990.
Thierry Coquand and Henrik Persson. Gröbner bases in type theory. In Types’98, volume 1658 of Lecture Notes in Computer Science. Springer-Verlag, 1998.
Michael J. C. Gordon and Thomas F. Melham. Introduction to HOL: a theorem proving environment for higher-order logic. Cambridge University Press, 1993.
John Harrison. Inductive definitions: Automation and application. In P. J. Windley, T. Schubert, and J. Alves-Foss, editors, Higher Order Logic Theorem Provoing and Its Applications: Proceedings of the 8th International Workshop, volume 971 of Lecture Notes in Computer Sciences. Springer-Verlag, 1995.
Thomas F. Melham. A package for inductive relation definitions in HOL. In Proceedings of the 1991 International Workshop on the HOL Theorem Proving system and its Applications, pages 350–357. IEEE Computer Society Press, 1992.
Bengt Nordström. Terminating general recursion. BIT, 28, 1988.
Catherine Parent. Synthesizing proofs from programs in the Calculus of Inductive Constructions. In Mathematics of Program Construction, volume 947 of Lecture Notes in Computer Science. Springer-Verlag, July 1995.
Christine Paulin-Mohring. Inductive Definitions in the System Coq-Rules and Properties. In M. Bezem and J.-F. Groote, editors, Proceedings of the conference Typed Lambda Calculi and Applications, number 664 in Lecture Notes in Computer Science, 1993. LIP research report 92-49.
Lawrence C. Paulson. Constructing recursion operators in intuitionistic type theory. Technical report 57, University of Cambridge, Computer Laboratory, October 1984.
Lawrence C. Paulson and Tobias Nipkow. Isabelle: a generic theorem prover, volume 828 of Lecture Notes in Computer Science. Springer-Verlag, 1994.
Frank Pfenning and Christine Paulin-Mohring. Inductively defined types in the Calculus of Constructions. In Proceedings of Mathematical Foundations of Programming Semantics, volume 442 of Lecture Notes in Computer Science. Springer-Verlag, 1990. technical report CMU-CS-89-209.
Konrad Slind. Function definition in higher order logic. In Theorem Proving in Higher Order Logics, volume 1125 of Lecture Notes in Computer Science. Sprinter Verlag, August 1996.
Laurent Théry. A certified version of Buchberger’s algorithm. In Automated Deduction (CADE-15), volume 1421 of Lecture Notes in Artificial Intelligence. Springer-Verlag, July 1998.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Balaa, A., Bertot, Y. (2000). Fix-Point Equations for Well-Founded Recursion in Type Theory. In: Aagaard, M., Harrison, J. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2000. Lecture Notes in Computer Science, vol 1869. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44659-1_1
Download citation
DOI: https://doi.org/10.1007/3-540-44659-1_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67863-2
Online ISBN: 978-3-540-44659-0
eBook Packages: Springer Book Archive