Abstract
This paper is part of a research project where we are exploring methods to extend the computational content of various systems of typed λ-calculus adding new reductions. Our previous study had its focus on isomorphisms of simple inductive types and related extensions of term rewriting. In this paper we present some new results concerning representation of finite sets as inductive types and related algebraic structures.
The work was supported partly by the grant N 01-05 of the French-Russian Liapunov Institute
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References
Barendregt, H., Barnedsen, E.: Autarkic computations in formal proofs. Journal of Automated Reasoning 28, 321–336 (2002)
Di Cosmo, R.: Isomorphisms of Types: From λ-Calculus to Information Retrieval and Language Design. Progress in Theoretical Computer Science. Birkhäuser, Boston, MA (1995)
Flegontov, A., Soloviev, S.: Type theory in differential equations. In: VIII International Workshop on Advenced Computing and Analysis Techniques in Physics Research (ACA 2002), Moscow (2002) (abstract)
Soloviev, S.: The category of finite sets and Cartesian closed categories. In: Theoretical Applications of Methods of Mathematical Logic III, Nauka, Leningrad. Zapiski Nauchnykh Seminarov LOMI, vol. 105, pp. 174–194 (1981); English translation in Journal of Soviet Mathematics 22(3), 1387–1400 (1983)
Chemouil, D.: Isomorphisms of Simple Inductive Types through Extensional Rewriting. Accepted for publication in Mathematical Structures in Computer Science, 42
Chemouil, D., Soloviev, S.: Remarks on isomorphisms of simple inductive types. In: Geuvers, H., Kamareddine, F. (eds.) Electronic Notes in Theoretical Computer Science, vol. 85, Elsevier, Amsterdam (2003)
Schwichtenberg, H.: Minimal Logic for Computable Functionals. Mathematisches Institut der Universität München, Available on (2003), http://www.minlog-system.de
Blanqui, F., Jouannaud, J.P., Okada, M.: Inductive-data-type systems. Theoretical Computer Science 272, 41–68 (2002)
Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, New York (1998)
Geser, A.: Relative Termination. PhD thesis, Universität Passau, Passau, Germany (1990)
Di Cosmo, R.: On the power of simple diagrams. In: Ganzinger, H. (ed.) RTA 1996. LNCS, vol. 1103, pp. 200–214. Springer, Heidelberg (1996)
Akama, Y.: On Mints’ reductions for ccc-calculus. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664, pp. 1–12. Springer, Heidelberg (1993)
Di Cosmo, R., Kesner, D.: Combining algebraic rewriting, extensional lambda calculi, and fixpoints. Theoretical Computer Science 169, 201–220 (1996)
Girard, J.Y., Lafont, Y., Taylor, P.: Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, Cambridge (1988)
Matthes, R.: Extensions of System F by Iteration and Primitive Recursion on Monotone Inductive Types. PhD thesis, Fachbereich Mathematik, Ludwig- Maximilians-Universität München (1998)
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. 328–345. Springer, Heidelberg (1993)
Mayr, R., Nipkow, T.: Higher-order rewrite systems and their confluence. Technical Report Sep26-1, Technical University of Munich (1995)
Di Cosmo, R., Kesner, D.: Rewriting with extensional polymorphic lambdacalculus. In: Kleine Büning, H. (ed.) CSL 1995. LNCS, vol. 1092, Springer, Heidelberg (1996)
Bachmair, L., Dershowitz, N.: Commutation, transformation, and termination. In: Siekmann, J.H. (ed.) CADE 1986. LNCS, vol. 230, pp. 5–20. Springer, Heidelberg (1986)
Doornbos, H., von Karger, B.: On the union of well-founded relations. Logic Journal of the IGPL 6, 195–201 (1998)
Delahaye, D.: Information Retrieval in a Coq Proof Library using Type Isomorphisms. In: Coquand, T., Nordström, B., Dybjer, P., Smith, J. (eds.) TYPES 1999. LNCS, vol. 1956, pp. 131–147. Springer, Heidelberg (2000)
Blanqui, F., Jouannaud, J.P., Okada, M.: The calculus of algebraic constructions. In: Narendran, P., Rusinowitch, M. (eds.) RTA 1999. LNCS, vol. 1631, pp. 301–316. Springer, Heidelberg (1999)
Blanqui, F.: Théorie des types et récriture. PhD thesis, Université Paris xi (2001)
Fiore, M., Cosmo, R.D., Balat, V.: Remarks on isomorphisms in typed lambda calculi with empty and sum types. In: Logic in Computer Science, Los Alamitos, CA, USA, pp. 147–156. IEEE Computer Society, Los Alamitos (2002)
Balat, V.: Une étude des sommes fortes: isomorphismes et formes normales. PhD thesis, Université Paris 7 (2002)
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Soloviev, S., Chemouil, D. (2004). Some Algebraic Structures in Lambda-Calculus with Inductive Types. 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_22
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