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Partiality and recursion in interactive theorem provers – an overview

Published online by Cambridge University Press:  10 November 2014

ANA BOVE ,
ALEXANDER KRAUSS  and
MATTHIEU SOZEAU
ANA BOVE
Affiliation:
Department of Computer Science and Engineering, Chalmers University of Technology, 412 96, Gothenburg, Sweden Email: [email protected]
ALEXANDER KRAUSS
Affiliation:
Technische Universität München, Boltzmannstr. 3, 85748 Garching, Germany Email: [email protected]
MATTHIEU SOZEAU
Affiliation:
INRIA Paris, 23 avenue d'Italie, 75013 Paris, France Email: [email protected]
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Abstract

The use of interactive theorem provers to establish the correctness of critical parts of a software development or for formalizing mathematics is becoming more common and feasible in practice. However, most mature theorem provers lack a direct treatment of partial and general recursive functions; overcoming this weakness has been the objective of intensive research during the last decades. In this article, we review several techniques that have been proposed in the literature to simplify the formalization of partial and general recursive functions in interactive theorem provers. Moreover, we classify the techniques according to their theoretical basis and their practical use. This uniform presentation of the different techniques facilitates the comparison and highlights their commonalities and differences, as well as their relative advantages and limitations. We focus on theorem provers based on constructive type theory (in particular, Agda and Coq) and higher-order logic (in particular Isabelle/HOL). Other systems and logics are covered to a certain extent, but not exhaustively. In addition to the description of the techniques, we also demonstrate tools which facilitate working with the problematic functions in particular theorem provers.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 26 , Special Issue 1: Special Issue: Dependently Typed Programming , January 2016 , pp. 38 - 88
Copyright
Copyright © Cambridge University Press 2014 

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References

Abel, A. (1998) Foetus – termination checker for simple functional programs. Programming Lab Report. Available at http://www.tcs.informatik.uni-muenchen.de/abel/foetus/.Google Scholar
Abel, A. (2006) A Polymorphic Lambda-Calculus with Sized Higher-Order Types, Ph.D. thesis, Ludwig-Maximilians-Universität München.Google Scholar
Abel, A. (2008) Semi-continuous sized types and termination. Logical Methods in Computer Science 4 (2) 1–33. CSL'06 special issue.Google Scholar
Abel, A. (2010) MiniAgda: Integrating sized and dependent types. In: Bove et al. (2010) 14–28.Google Scholar
Abel, A. and Altenkirch, T. (2002) A predicative analysis of structural recursion. Journal of Functional Programming 12 1–41.CrossRefGoogle Scholar
Abramsky, S. and Jung, A. (1994) Domain theory. In: Abramsky, S., Gabbay, D. M. and Maibaum, T. S. E. (eds.) Handbook of Logic in Computer Science, volume 3, Oxford University Press 1–168.Google Scholar
Aczel, P. (1977a) An introduction to inductive definitions. In: Barwise, J. (ed.) Handbook of Mathematical Logic, North-Holland Publishing Company 739–782.CrossRefGoogle Scholar
Aczel, P. (1977b) The strength of Martin-Löf's intuitionistic type theory with one universe. In: Miettinen, S. and Vnanen, J. (eds.) Proceedings of the Symposium on Mathematical Logic (Oulu, 1974), Report No. 2, Department of Philosopy, University of Helsinki, Helsinki 1–32.Google Scholar
Agda (2008) Agda wiki. Available at http://wiki.portal.chalmers.se/agda/agda.php.Google Scholar
Ait Mohamed, O., Muñoz, C. and Tahar, S. (eds.) (2008) In: Theorem Proving in Higher Order Logics (TPHOLs 2008), Proceedings of 21st International Conference, Montreal, Canada, August 18–21, 2008. Springer-Verlag Lecture Notes in Computer Science 5170 319 pp.Google Scholar
Audebaud, P. (1991) Partial objects in the calculus of constructions. In: Kahn, G. (ed.) Logic in Computer Science (LICS 1991), IEEE 86–95.Google Scholar
Balaa, A. and Bertot, Y. (2000) Fix-point equations for well-founded recursion in type theory. In: Aagaard, M. and Harrison, J. (eds.) Theorem Proving in Higher Order Logics (TPHOLs 2000). Springer-Verlag Lecture Notes in Computer Science 1869 1–16.CrossRefGoogle Scholar
Balaa, A. and Bertot, Y. (2002) Fonctions récursives générales par itération en théorie des types. Journées Francophones des Langages Applicatifs - JFLA02, INRIA.Google Scholar
Barringer, H., Cheng, J. H. and Jones, C. B. (1984) A logic covering undefinedness in program proofs. Acta Informatica 21 251–269.CrossRefGoogle Scholar
Barthe, G. and Courtieu, P. (2002) Efficient reasoning about executable specifications in Coq. In: Carreno, V. A., Muñoz, C. and Tahar, S. (eds.) Theorem Proving in Higher-Order Logics (TPHOLs 2002). Springer-Verlag Lecture Notes in Computer Science 2410 31–46.CrossRefGoogle Scholar
Barthe, G., Forest, J., Pichardie, D. and Rusu, V. (2006) Defining and reasoning about recursive functions: A practical tool for the Coq proof assistant. In: Hagiya, M. and Wadler, P. (eds.) Functional and Logic Programming (FLOPS 2006). Springer-Verlag Lecture Notes in Computer Science 3945 114–129.CrossRefGoogle Scholar
Barthe, G., Frade, M. J., Giménez, E., Pinto, L. and Uustalu, T. (2004) Type-based termination of recursive definitions. Mathematical Structures in Computer Science 14 (1) 97–141.CrossRefGoogle Scholar
Barthe, G., Grégoire, B. and Pastawski, F. (2006) CIC^: Type-based termination of recursive definitions in the calculus of inductive constructions. In: Hermann, M. and Voronkov, A. (eds.) Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2006). Springer-Verlag Lecture Notes in Computer Science 4246 257–271.CrossRefGoogle Scholar
Barthe, G., Grégoire, B. and Riba, C. (2008) Type-based termination with sized products. In: Kaminski, M. and Martini, S. (eds.) CSL. Springer Lecture Notes in Computer Science 5213 493–507.CrossRefGoogle Scholar
Barthe, G., Grégoire, B. and Riba, C. (2009) A tutorial on type-based termination. In: Bove, A., Barbosa, L. S., Pardo, A. and Pinto, J. S. (eds.) Language Engineering and Rigorous Software Development. Springer-Verlag Lecture Notes in Computer Science 5520 100–152.CrossRefGoogle Scholar
Berghofer, S. and Nipkow, T. (2000) Executing higher order logic. In: Callaghan et al. (2002) 24–40.Google Scholar
Berghofer, S. and Wenzel, M. (1999) Inductive datatypes in HOL – lessons learned in formal-logic engineering. In: Bertot et al. (1999) 19–36.Google Scholar
Bertot, Y., Capretta, V. and Das Barman, K. (2002) Type-theoretic functional semantics. In: Carreno, V. A., Muñoz, C. A. and Tahar, S. (eds.) Theorem Proving in Higher Order Logics: 15th International Conference, TPHOLs 2002. Springer-Verlag Lecture Notes in Computer Science 2410 83–97.CrossRefGoogle Scholar
Bertot, Y. and Castéran, P. (2004) Interactive Theorem Proving and Program Development. Coq'Art: The Calculus of Inductive Constructions, Springer Verlag.CrossRefGoogle Scholar
Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C. and Théry, L. (eds.) (1999) In: Theorem Proving in Higher Order Logics (TPHOLs '99). Springer-Verlag Lecture Notes in Computer Science 1690 358 pp.Google Scholar
Bertot, Y. and Komendantsky, V. (2008) Fixed point semantics and partial recursion in Coq. In: Antoy, S. and Albert, E. (eds.) Principles and Practice of Declarative Programming (PPDP '08), ACM, New York, NY, USA 89–96.Google Scholar
Blanqui, F. (2004) A type-based termination criterion for dependently-typed higher-order rewrite systems. In: van Oostrom, V. (ed.) Rewriting Techniques and Applications, 15th International Conference, RTA 2004, Aachen, Germany, June 3–5, 2004, Proceedings. Springer Lecture Notes in Computer Science 3091 24–39.CrossRefGoogle Scholar
Blanqui, F. (2005) Decidability of type-checking in the calculus of algebraic constructions with size annotations. In: Ong, L. (ed.) Computer Science Logic (CSL 2005). Springer-Verlag Lecture Notes in Computer Science 3634 135–150.CrossRefGoogle Scholar
Bove, A. (2009) Another look at function domains. In: Abramsky, S., Mislove, M. and Palamidessi, C. (eds.) Mathematical Foundations of Programming Semantics (MFPS 2009). Electronic Notes in Theoretical Computer Science 249C 61–74.CrossRefGoogle Scholar
Bove, A. and Capretta, V. (2005a) Modelling general recursion in type theory. Mathematical Structures in Computer Science 15 (4) 671–708.CrossRefGoogle Scholar
Bove, A. and Capretta, V. (2005b) Recursive functions with higher-order domains. In: Urzyczyn, P. (ed.) Typed Lambda Calculi and Applications (TLCA 2005). Springer-Verlag Lecture Notes in Computer Science 3461 116–130.CrossRefGoogle Scholar
Bove, A. and Capretta, V. (2007) Computation by prophecy. In: Rocca, S. R. D. (ed.) Typed Lambda Calculi and Applications (TLCA 2007). Springer-Verlag Lecture Notes in Computer Science 4583 70–83.CrossRefGoogle Scholar
Bove, A. and Capretta, V. (2008) A type of partial recursive functions. In: Ait Mohamed et al. (2008) 102–117.Google Scholar
Bove, A., Dybjer, P. and Sicard-Ramírez, A. (2009) Embedding a logical theory of constructions in Agda, Programming Languages meets Program Verification (PLPV) 2009, ACM Digital Library.CrossRefGoogle Scholar
Bove, A., Dybjer, P. and Sicard-Ramírez, A. (2012) Combining interactive and automatic reasoning in first order theories of functional programs. In: Birkedal, L. (ed.) 15th International Conference on Foundations of Software Science and Computational Structures, FoSSaCS 2012. Lecture Notes in Computer Science 7213 104–118.CrossRefGoogle Scholar
Bove, A., Komendantskaya, E. and Niqui, M. (eds.) (2010) Workshop on Partiality and Recursion in Interative Theorem Provers (PAR 2010), Satellite Workshop of ITP'10 at FLoC 2010. Electronic Proceedings in Theoretical Computer Science 43 93 pp.Google Scholar
Boyer, R. S. and Moore, J. S. (1979) A Computational Logic, Academic Press, New York.Google Scholar
Boyer, R. S. and Moore, J. S. (1996) Mechanized formal reasoning about programs and computing machines. In: Veroff, R. (ed.) Automated Reasoning and Its Applications: Essays in Honor of Larry Wos, MIT Press.Google Scholar
Bulwahn, L., Krauss, A. and Nipkow, T. (2007) Finding lexicographic orders for termination proofs in Isabelle/HOL. In: Schneider, K. and Brandt, J. (eds.) Theorem Proving in Higher Order Logics (TPHOLs 2007). Springer-Verlag Lecture Notes in Computer Science 4732 38–53.CrossRefGoogle Scholar
Callaghan, P., Luo, Z., McKinna, J. and Pollack, R. (eds.) (2002) In: Types for Proofs and Programs, International Workshop, TYPES 2000, Durham, UK, December 8–12, 2000, Selected Papers. Springer-Verlag Lecture Notes in Computer Science 2277 245 pp.Google Scholar
Capretta, V. (2005) General recursion via coinductive types. Logical Methods in Computer Science 1 (2) 1–18.Google Scholar
Charguéraud, A. (2010) The optimal fixed point combinator. In: Kaufmann and Paulson (2010) 195–210.Google Scholar
Cheng, J. H. and Jones, C. B. (1991) On the usability of logics which handle partial functions. In: Morgan, C. and Woodcock, J. C. P. (eds.) 3rd Refinement Workshop, Springer Verlag 51–69.Google Scholar
Constable, R. L. and Mendler, N. P. (1985) Recursive definitions in type theory. In: Parikh, R. (ed.) Logic of Programs. Springer-Verlag Lecture Notes in Computer Science 193 61–78.CrossRefGoogle Scholar
Constable, R. L. and Smith, S. F. (1987) Partial objects in constructive type theory. Logic in Computer Science (LICS 1987), IEEE, Washington, DC 183–193.Google Scholar
development team (2010) Coq 8.3 Reference Manual, INRIA. http://coq.inria.fr/refman/.Google Scholar
Coquand, T. and Huet, G. (1988) The calculus of constructions. Information and Computation 76 (2/3) 95–120.CrossRefGoogle Scholar
Coquand, T. and Paulin, C. (1990) Inductively defined types. In: Martin-Löf, P. and Mints, G. (eds.) Proceedings of Colog '88. Springer-Verlag Lecture Notes in Computer Science 417 50–66.CrossRefGoogle Scholar
Di Gianantonio, P. and Miculan, M. (2003) A unifying approach to recursive and co-recursive definitions. In: Geuvers, H. and Wiedijk, F. (eds.) Types for Proofs and Programs (TYPES 2002). Springer-Verlag Lecture Notes in Computer Science 2646 148–161.CrossRefGoogle Scholar
Dubois, C. and Donzeau-Gouge, V. V. (1998) A step towards the mechanization of partial functions: domains as inductive predicates. CADE-15 Workshop on Mechanization of Partial Functions.Google Scholar
Dybjer, P. (1985) Program verification in a logical theory of constructions. In: Jouannaud, J.-P. (ed.) Functional Programming Languages and Computer Architecture. Lecture Notes in Computer Science 201 334–349.CrossRefGoogle Scholar
Dybjer, P. (1991) Inductive sets and families in Martin-Löf's type theory and their set-theoretic semantics. Logical Frameworks, Cambridge University Press 280–306.CrossRefGoogle Scholar
Dybjer, P. (2000) A general formulation of simultaneous inductive-recursive definitions in type theory. Journal of Symbolic Logic 65 (2) 525–549.CrossRefGoogle Scholar
Farmer, W. M. (1993) A simple type theory with partial functions and subtypes. Annals of Pure and Applied Logic 64 (3) 211–240.CrossRefGoogle Scholar
Farmer, W. M., Guttman, J. D. and Thayer, F. J. (1993) IMPS: An interactive mathematical proof system. Journal of Automated Reasoning 11 653–654.CrossRefGoogle Scholar
Finn, S., Fourman, M. and Longley, J. (1997) Partial functions in a total setting. Journal of Automated Reasoning 18 (1) 85–104.CrossRefGoogle Scholar
Furbach, U. and Shankar, N. (eds.) (2006) In: Automated Reasoning, 3rd International Joint Conference, IJCAR 2006, Seattle, WA, USA, August 17-20, 2006, Proceedings. Springer Verlag Lecture Notes in Artificial Intelligence 4130 680 pp.Google Scholar
Giesl, J. (1997) Termination of nested and mutually recursive algorithms. Journal of Automated Reasoning 19 (1) 1–29.CrossRefGoogle Scholar
Giesl, J. (2001) Induction proofs with partial functions. Journal of Automated Reasoning 26 (1) 1–49.CrossRefGoogle Scholar
Giménez, E. (1995) Codifying guarded definitions with recursive schemes. In: Dybjer, P., Nordström, B. and Smith, J. M. (eds.) Types for Proofs and Programs (TYPES 1994). Springer-Verlag Lecture Notes in Computer Science 996 39–59.CrossRefGoogle Scholar
Goguen, H., McBride, C. and McKinna, J. (2006) Eliminating dependent pattern matching. In: Futatsugi, K., Jouannaud, J.-P. and Meseguer, J. (eds.) Algebra, Meaning, and Computation, Essays Dedicated to Joseph A. Goguen. Springer-Verlag Lecture Notes in Computer Science 4060 521–540.CrossRefGoogle Scholar
Gordon, M. J. C. and Melham, T. F. (eds.) (1993) Introduction to HOL: A Theorem Proving Environment for Higher Order Logic, Cambridge University Press.Google Scholar
Gordon, M. J. C., Milner, R. and Wadsworth, C. P. (1979) Edinburgh LCF: A Mechanised Logic of Computation. Springer-Verlag Lecture Notes in Computer Science 78 162 pp.Google Scholar
Greve, D. (2009) Assuming termination. ACL2 Workshop Proceedings.CrossRefGoogle Scholar
Haftmann, F. and Nipkow, T. (2010) Code generation via higher-order rewrite systems. In: Blume, M., Kobayashi, N. and Vidal, G. (eds.) Functional and Logic Programming (FLOPS 2010). Springer-Verlag Lecture Notes in Computer Science 6009 103–117.CrossRefGoogle Scholar
Harrison, J. (1995) Inductive definitions: Automation and application. In: Schubert et al. (1995) 200–213.Google Scholar
Howard, W. A. (1980) The formulae-as-types notion of construction. In: Seldin, J. P. and Hindley, J. R. (eds.) To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, London 479–490.Google Scholar
Huffman, B. (2008) Reasoning with powerdomains in Isabelle/HOLCF. In: Ait Mohamed, O., Muñoz, C. and Tahar, S. (eds.) TPHOLs 2008: Emerging Trends Proceedings, Department of Electrical and Computer Engineering, Concordia University 45–56.Google Scholar
Huffman, B. (2009) A purely definitional universal domain. In: Berghofer, S., Nipkow, T., Urban, C. and Wenzel, M. (eds.) Theorem Proving in Higher Order Logics (TPHOLs 2009). Springer-Verlag Lecture Notes in Computer Science 5674 260–275.CrossRefGoogle Scholar
Hughes, J., Pareto, L. and Sabry, A. (1996) Proving the correctness of reactive systems using sized types, Principles of Programming Languages (POPL 1996), ACM 410–423.Google Scholar
Jones, C. B. (1990) Systematic Software Development using VDM, Prentice-Hall.Google Scholar
Kaufmann, M. and Paulson, L. C. (eds.) (2010) In: Interactive Theorem Proving, Proceedings of 1st International Conference, ITP 2010, Edinburgh, UK, 11–14 July, 2010. Springer-Verlag Lecture Notes in Computer Science 6172 497 pp.Google Scholar
Krauss, A. (2006) Partial recursive functions in higher-order logic. In: Furbach and Shankar (2006) 589–603.Google Scholar
Krauss, A. (2007) Certified size-change termination. In: Pfenning, F. (ed.) Automated Deduction (CADE-21). Springer-Verlag Lecture Notes in Computer Science 4603 460–476.CrossRefGoogle Scholar
Krauss, A. (2010a) Partial and nested recursive function definitions in higher-order logic. Journal of Automated Reasoning 44 (4) 303–336.CrossRefGoogle Scholar
Krauss, A. (2010b) Recursive definitions of monadic functions. In: Bove et al. (2010) 1–13.Google Scholar
Krstić, S. and Matthews, J. (2003) Inductive invariants for nested recursion. In: Basin, D. A. and Wolff, B. (eds.) Theorem Proving in Higher Order Logics (TPHOLs 2003). Springer-Verlag Lecture Notes in Computer Science 2758 253–269.CrossRefGoogle Scholar
Leroy, X. (2006) Formal certification of a compiler back-end, or: programming a compiler with a proof assistant. In: Morrisett, J. G. and Peyton Jones, S. L. (eds.) Principles of Programming Languages (POPL 2006), ACM Press 42–54.Google Scholar
Manna, Z. and Shamir, A. (1976) The theoretical aspects of the optimal fixed point. SIAM Journal on Computing 5 (3) 414–426.CrossRefGoogle Scholar
Manolios, P. and Moore, J. S. (2003) Partial functions in ACL2. Journal of Automated Reasoning 31 (2) 107–127.CrossRefGoogle Scholar
Martin-Löf, P. (1984) Intuitionistic Type Theory, Bibliopolis, Napoli.Google Scholar
Matthews, J. (1999) Recursive function definition over coinductive types. In: Bertot et al. (1999) 73–90.Google Scholar
McBride, C. (2002) Elimination with a motive. In: Callaghan et al. (2002) 197–216.Google Scholar
McBride, C. (2003) First-order unification by structural recursion. Journal of Functional Programming 13 (6) 1061–1075.CrossRefGoogle Scholar
McBride, C. (2004) Epigram: Practical programming with dependent types. In: Vene, V. and Uustalu, T. (eds.) Advanced Functional Programming (AFP 2004), Springer-Verlag Lecture Notes in Computer Science 3622 130–170.CrossRefGoogle Scholar
McBride, C. and McKinna, J. (2004) The view from the left. Journal of Functional Programming 14 (1) 69–111.CrossRefGoogle Scholar
Megacz, A. (2007) A coinductive monad for Prop-bounded recursion. In: Stump, A. and Xi, H. (eds.) Programming Languages meets Program Verification (PLPV 2007), ACM, New York, NY, USA 11–20.CrossRefGoogle Scholar
Milner, R. (1972) Logic for computable functions: Description of a machine implementation, Technical report, Stanford, CA, USA.CrossRefGoogle Scholar
Moggi, E. (1991) Notions of computation and monads. Information and Computation 93 (1) 55–92.CrossRefGoogle Scholar
Müller, O., Nipkow, T., von Oheimb, D. and Slotosch, O. (1999) HOLCF=HOL+LCF. Journal of Functional Programming 9 (2) 191–223.CrossRefGoogle Scholar
Nipkow, T., Bauer, G. and Schultz, P. (2006) Flyspeck I: Tame graphs. In: Furbach and Shankar (2006) 21–35.Google Scholar
Nipkow, T., Paulson, L. C. and Wenzel, M. (2002) Isabelle/HOL – A Proof Assistant for Higher-Order Logic , Springer Verlag Lecture Notes in Computer Science 2283 215 pp.Google Scholar
Nordström, B. (1988) Terminating general Recursion. BIT 28 (3) 605–619.CrossRefGoogle Scholar
Nordström, B., Petersson, K. and Smith, J. (1990) Programming in Martin-Löf's Type Theory. An Introduction, Oxford University Press.Google Scholar
Norell, U. (2007) Towards a Practical Programming Language Based on Dependent Type Theory, Ph.D. thesis, Chalmers University of Technology.Google Scholar
OCaml (1996) Ocaml web page. Available at http://caml.inria.fr/ocaml/.Google Scholar
Owens, S. and Slind, K. (2008) Adapting functional programs to higher-order logic. Higher-Order and Symbolic Computation 21 (4) 377–409.CrossRefGoogle Scholar
Paulin-Mohring, C. (1993) Inductive definitions in the system Coq - rules and properties. In: Typed Lambda Calculi and Applications (TLCA 1993). Springer-Verlag Lecture Notes in Computer Science 664 328–345.CrossRefGoogle Scholar
Paulin-Mohring, C. (2009) A constructive denotational semantics for Kahn networks in Coq. In: Bertot, Y., Huet, G., Levy, J.-J. and Plotkin, G. (eds.) From Semantics and Computer Science: Essays in Honor of Gilles Kahn, Cambridge University Press.Google Scholar
Paulson, L. C. (1986) Constructing recursion operators in intuitionistic type theory. Journal of Symbolic Computation 2 (4) 325–355.CrossRefGoogle Scholar
Paulson, L. C. (1989) The foundation of a generic theorem prover. Journal of Automated Reasoning 5 (3) 363–397.CrossRefGoogle Scholar
Peyton Jones, S. (ed.) (2003) Haskell 98 Language and Libraries The Revised Report, Cambridge University Press.Google Scholar
Regensburger, F. (1995) HOLCF: Higher order logic of computable functions. In: Schubert et al. (1995) 293–307.Google Scholar
Schubert, E. T., Windley, P. J. and Alves-Foss, J. (eds.) (1995) In: Higher Order Logic Theorem Proving and its Applications, Proceedings of 8th International Workshop, Aspen Grove, UT, USA, 11–14 September, 1995. Springer-Verlag Lecture Notes in Computer Science 971 400 pp.Google Scholar
Scott, D. S. (1993) A type-theoretical alternative to iswim, cuch, owhy. Theoretical Computer Science 121 411–440.CrossRefGoogle Scholar
Setzer, A. (2006) Partial recursive functions in Martin-Löf type theory. In: Beckmann, A., Berger, U., Löwe, B. and Tucker, J. V. (eds.) Logical Approaches to Computational Barriers: Proceedings of 2nd Conference on Computability in Europe, CiE 2006, Swansea, UK, 30 June-5 July, 2006. Springer Lecture Notes in Computer Science 3988 505–515.CrossRefGoogle Scholar
Setzer, A. (2007) A data type of partial recursive functions in Martin-Löf type theory. 35 pp, submitted.CrossRefGoogle Scholar
Slind, K. (1996) Function definition in higher-order logic. In: von Wright, J., Grundy, J. and Harrison, J. (eds.) Theorem Proving in Higher Order Logics (TPHOLs '96). Springer-Verlag Lecture Notes in Computer Science 1125 381–397.CrossRefGoogle Scholar
Slind, K. (1999) Reasoning About Terminating Functional Programs, Ph.D. thesis, Institut für Informatik, Technische Universität München.Google Scholar
Sozeau, M. (2006) Subset coercions in Coq. In: Altenkirch, T. and McBride, C. (eds.) TYPES. Springer-Verlag Lecture Notes in Computer Science 4502 237–252.CrossRefGoogle Scholar
Sozeau, M. (2010) Equations: A dependent pattern matching compiler. In: Kaufmann and Paulson (2010) 419–434.Google Scholar
Sutcliffe, G. (2009) The TPTP problem library and associated infrastructure. The FOT and CNF parts, v.3.5.0. Journal of Automated Reasoning 43 (4) 337–362.CrossRefGoogle Scholar
Wahlstedt, D. (2007) Dependent Type Theory with Parameterized First-Order Data Types and Well-Founded Recursion, Ph.D. thesis, Chalmers University of Technology.Google Scholar
Wenzel, M., Paulson, L. C. and Nipkow, T. (2008) The Isabelle framework. In: Ait Mohamed et al. (2008) 33–38.Google Scholar
Xi, H. (2001) Dependent types for program termination verification. Logic in Computer Science (LICS 2001), IEEE 231–242.Google Scholar