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Formalising Overlap Algebras in Matita†

Published online by Cambridge University Press:  01 July 2011

CLAUDIO SACERDOTI COEN  and
ENRICO TASSI
CLAUDIO SACERDOTI COEN
Affiliation:
Dipartimento di Scienze dell'Informazione, Università di Bologna, via Mura Anteo Zamboni 7, 40127, Bologna, Italy Email: sacerdot@cs.unibo.it
ENRICO TASSI
Affiliation:
Microsoft Research-INRIA Joint Center, Parc Orsay Université, 28, rue Jean Rostand, 91893, Orsay, France Email: enrico.tassi@inria.fr
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Abstract

We describe some formal topological results, formalised in Matita 1/2, presented in predicative intuitionistic logic and in terms of Overlap Algebras.

Overlap Algebras are new algebraic structures designed to ease reasoning about subsets in an algebraic way within intuitionistic logic. We find that they also ease the formalisation of formal topological results in an interactive theorem prover.

Our main result is the existence of a functor between two categories of ‘generalised topological spaces’, one with points (Basic Pairs) and the other point-free (Basic Topologies). This formalisation is part of a wider scientific collaboration with the inventor of the theory, Giovanni Sambin. His goal is to verify in what sense his theory is ‘implementable’, and to discover what problems may arise in the process. We check that all intermediate constructions respect the stringent size requirements imposed by predicative logic. The formalisation is quite unusual, since it has to make explicit size information that is often hidden.

We found that the version of Matita used for the formalisation was largely inappropriate. The formalisation drove several major improvements of Matita that will be integrated in the next major release (Matita 1.0). We show some motivating examples, taken directly from the formalisation, for these improvements. We also describe a possibly sub-optimal solution in Matita 1/2, which is exploitable in other similar systems. We briefly discuss a better solution available in Matita 1.0.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 21 , Special Issue 4: Interactive Theorem Proving and the Formalisation of Mathematics , August 2011 , pp. 763 - 793
Copyright
Copyright © Cambridge University Press 2011

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References

Asperti, A. and Armentano, C. (2008) A page in number theory. Journal of Formalized Reasoning 1 1–23.Google Scholar
Asperti, A. and Tassi, E (2010) Smart matching. In: Autexier, S. et al. (eds.) Intelligent Computer Mathematics. Springer-Verlag Lecture Notes in Computer Science 6167263–277.CrossRefGoogle Scholar
Asperti, A., Sacerdoti Coen, C., Tassi, E. and Zacchiroli, S. (2007) User interaction with the Matita proof assistant. Journal of Automated Reasoning 39 (2)109–139.CrossRefGoogle Scholar
Asperti, A., Ricciotti, W., Sacerdoti Coen, C. and Tassi, E. (2009a) A compact kernel for the Calculus of Inductive Constructions. Sadhana 34 (1)71–144.CrossRefGoogle Scholar
Asperti, A., Ricciotti, W., Sacerdoti Coen, C. and Tassi, E. (2009b) Hints in unification. In: Berghofer, S., Nipkow, T.Urban, C. and Wenzel, M. (eds.) Theorem Proving in Higher Order Logics, 22nd International Conference (TPHOLs 2009). Springer-Verlag Lecture Notes in Computer Science 567484–98.CrossRefGoogle Scholar
Ciraulo, F. and Sambin, G. (2010) The overlap algebra of regular opens. Journal of Pure and Applied Algebra 214 1988–1995.CrossRefGoogle Scholar
Coquand, T. and Huet, G. P. (1988) The calculus of constructions. Information and Computation 76 (2/3), 95–120.CrossRefGoogle Scholar
Coquand, T., Sambin, G., Smith, J. M. and Valentini, S. (2003) Inductively generated formal topologies. Annals of Pure and Applied Logic 124 (1–3)71–106.CrossRefGoogle Scholar
Courant, J. (2002) Explicit universes for the calculus of constructions. In: Carreño, V. A.Muñoz, C. and Tahar, S. (eds.) Theorem Proving in Higher Order Logics, 15th International Conference (TPHOLs 2002). Springer-Verlag Lecture Notes in Computer Science 2410115–130.CrossRefGoogle Scholar
Cruz-Filipe, L., Geuvers, H. and Wiedijk, F. (2004) C-CoRN, the constructive Coq repository at Nijmegen. Proceedings of MKM2004. Springer-Verlag Lecture Notes in Computer Science 311988–103.Google Scholar
Cubric, D., Dybier, P. and Scott, P. (1997) Normalisation and the Yoneda embedding. Mathematical Structures in Computer Science 8 (2)153–192.CrossRefGoogle Scholar
Johnstone, P. T. (1983) The point of pointless topology. Bulletin of the American Mathematical Society 8 41–53.CrossRefGoogle Scholar
Luo, Z. (1989) ECC, an extended calculus of constructions. Proceedings of LICS1989, IEEE Press 386–395Google Scholar
Maietti, M. E. and Sambin, G. (2005) From sets and types to topology and analysis, toward a minimalist foundation for constructive mathematics, Chapter 6, Oxford University Press.Google Scholar
Padovani, L. and Zacchiroli, S. (2006). From notation to semantics: There and back again. In: Borwein, J. M. and Farmer, W. M. (eds.) Proceedings Mathematical Knowledge Management: 5th International Conference, MKM 2006. Springer-Verlag Lecture Notes in Computer Science 4108194–207.CrossRefGoogle Scholar
Paulin-Mohring, C. (1996) Définitions inductives en théorie des types d'ordre supŕieur, Habilitation à diriger les recherches, Université Claude Bernard Lyon I.Google Scholar
Sacerdoti Coen, C. (2006) A semi-reflexive tactic for (sub-)equational reasoning. In: Fillitre, J.-C., Paulin-Mohring, C. and Werner, B. (eds.) Types for Proofs and Programs, Proceedings of TYPES2004. Springer-Verlag Lecture Notes in Computer Science 383998–114.CrossRefGoogle Scholar
Sacerdoti Coen, C. and Tassi, E. (2011) Nonuniform coercions via unification hints. In: Hirschowitz, T. (ed.) Proceedings Types for Proofs and Programs (TYPES 2009), Revised Selected Papers. EPTCS 53 16–29.CrossRefGoogle Scholar
Sacerdoti Coen, C. and Tassi, E. (2008) A constructive and formal proof of Lebesgue's dominated convergence theorem in the interactive theorem prover Matita. Journal of Formalized Reasoning 1 51–89.Google Scholar
Sacerdoti Coen, C. and Zacchiroli, S. (2008) Spurious disambiguation errors and how to get rid of them. Journal of Mathematics in Computer Science 2 355–378.Google Scholar
Sambin, G. (2011) The Basic Picture: a structural basis for constructive topology, Oxford University Press.Google Scholar
Sozeau, M. (2009) A new look at generalized rewriting in type theory. Journal of Formalized Reasoning 2 (1)41–62.Google Scholar
Sozeau, M. and Oury, N. (2008) First-class type classes. In: Mohamed, O. A., Muñoz, C. and Tahar, S. (eds.) Proceedings of the 21st International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2008). Springer-Verlag Lecture Notes in Computer Science 5170278–293.Google Scholar
Valentini, S. (2005) The problem of the formalization of constructive topology. Archive for Mathematical Logic 44 (1)115–129.CrossRefGoogle Scholar
Werner, B. (1997) Sets in types, types in sets. In: Proceedings of TACS97. Springer-Verlag Lecture Notes in Computer Science 1281530–546.Google Scholar