Abstract
We present a Coq formalization of constructive ω-cpos (extending earlier work by Paulin-Mohring) up to and including the inverse-limit construction of solutions to mixed-variance recursive domain equations, and the existence of invariant relations on those solutions. We then define operational and denotational semantics for both a simply-typed CBV language with recursion and an untyped CBV language, and establish soundness and adequacy results in each case.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adams, R.: Formalized metatheory with terms represented by an indexed family of types. In: Filliâtre, J.-C., Paulin-Mohring, C., Werner, B. (eds.) TYPES 2004. LNCS, vol. 3839, pp. 1–16. Springer, Heidelberg (2006)
Agerholm, S.: Domain theory in HOL. In: Joyce, J.J., Seger, C.-J.H. (eds.) HUG 1993. LNCS, vol. 780. Springer, Heidelberg (1994)
Agerholm, S.: Formalizing a model of the lambda calculus in HOL-ST. Technical Report 354, University of Cambridge Computer Laboratory (1994)
Agerholm, S.: LCF examples in HOL. The Computer Journal 38(2) (1995)
Altenkirch, T., Reus, B.: Monadic presentations of lambda terms using generalized inductive types. In: Flum, J., Rodríguez-Artalejo, M. (eds.) CSL 1999. LNCS, vol. 1683, pp. 453–468. Springer, Heidelberg (1999)
Audebaud, P., Paulin-Mohring, C.: Proofs of randomized algorithms in Coq. In: Uustalu, T. (ed.) MPC 2006. LNCS, vol. 4014, pp. 49–68. Springer, Heidelberg (2006)
Bartels, F., Dold, A., Pfeifer, H., Von Henke, F.W., Rueß, H.: Formalizing fixed-point theory in PVS. Technical report, Universität Ulm (1996)
Benton, N., Hur, C.-K.: Biorthogonality, step-indexing and compiler correctness. In: ACM International Conference on Functional Programming (2009)
Capretta, V.: General recursion via coinductive types. Logical Methods in Computer Science 1 (2005)
Coquand, T.: Infinite objects in type theory. In: Barendregt, H., Nipkow, T. (eds.) TYPES 1993, vol. 806. Springer, Heidelberg (1994)
Freyd, P.: Recursive types reduced to inductive types. In: IEEE Symposium on Logic in Computer Science (1990)
Freyd, P.: Remarks on algebraically compact categories. In: Applications of Categories in Computer Science. LMS Lecture Notes, vol. 177 (1992)
Joyal, A., Street, R.: The geometry of tensor calculus. Adv. in Math. 88 (1991)
Kahn, G.: Elements of domain theory. In: The Coq users’ contributions library (1993)
McBride, C.: Type-preserving renaming and substitution (unpublished draft)
Milner, R.: Logic for computable functions: Description of a machine implementation. Technical Report STAN-CS-72-288, Stanford University (1972)
Moggi, E.: Notions of computation and monads. Inf. Comput. 93(1), 55–92 (1991)
Müller, O., Nipkow, T., von Oheimb, D., Slotosch, O.: HOLCF = HOL + LCF. J. Functional Programming 9 (1999)
Nipkow, T.: Winskel is (almost) right: Towards a mechanized semantics textbook. Formal Aspects of Computing 10 (1998)
Paulin-Mohring, C.: A constructive denotational semantics for Kahn networks in Coq. In: From Semantics to Computer Science. Essays in Honour of G Kahn (2009)
Petersen, K.D.: Graph model of LAMBDA in higher order logic. In: Joyce, J.J., Seger, C.-J.H. (eds.) HUG 1993. LNCS, vol. 780. Springer, Heidelberg (1994)
Pitts, A.M.: Computational adequacy via ‘mixed’ inductive definitions. In: Main, M.G., Melton, A.C., Mislove, M.W., Schmidt, D., Brookes, S.D. (eds.) MFPS 1993. LNCS, vol. 802. Springer, Heidelberg (1994)
Pitts, A.M.: Relational properties of domains. Inf. Comput. 127 (1996)
Regensburger, F.: HOLCF: Higher order logic of computable functions. In: Schubert, E.T., Alves-Foss, J., Windley, P. (eds.) HUG 1995. LNCS, vol. 971. Springer, Heidelberg (1995)
Reus, B.: Formalizing a variant of synthetic domain theory. J. Automated Reasoning 23 (1999)
Varming, C., Birkedal, L.: Higher-order separation logic in Isabelle/HOLCF. In: Mathematical Foundations of Programming Semantics (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Benton, N., Kennedy, A., Varming, C. (2009). Some Domain Theory and Denotational Semantics in Coq. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2009. Lecture Notes in Computer Science, vol 5674. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03359-9_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-03359-9_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03358-2
Online ISBN: 978-3-642-03359-9
eBook Packages: Computer ScienceComputer Science (R0)