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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: [email protected]
ENRICO TASSI
Affiliation:
Microsoft Research-INRIA Joint Center, Parc Orsay Université, 28, rue Jean Rostand, 91893, Orsay, France Email: [email protected]
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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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