A Pointfree approach to Constructive Analysis in Type Theory

The rst paper in this thesis presents a machine checked formalisation, in Martin-Löf's type theory, of pointfree topology with applications to domain theory. In the other papers pointfree topology is used in an approach to constructive analysis. The continuum is de ned as a formal space from a base of rational intervals. Then the closed rational interval [a; b] is de ned as a formal space, in terms of the continuum, and the Heine-Borel covering theorem is proved constructively. The basic de nitions for a pointfree approach to functional analysis are given in such a way that the linear functionals from a seminormed linear space to the reals are points of a particular formal space, and in this setting the Alaoglu and the Hahn-Banach theorems are proved in an entirely constructive way. The proofs have been carried out in intensional Martin-Löf type theory with one universe and nitary inductive de nitions, and the proofs have also been mechanically checked in an implementation of that system. Acknowledgements First I want to thank my supervisor Jan Smith. A few years ago he suggested to me that I start studying pointfree topology as an approach to constructive mathematics. Since then he has repeatedly encouraged me to continue investigating this area. The suggestions and the response I have got from him during our discussions have been valuable for me. But most importantly, he has always shown enthusiasm in my work. Two more people have in uenced my work considerably. Many ideas that I have used are due to Thierry Coquand. Thierry also invented the logical framework Half that was suitable for my kind of formalisation. The last two years I have also had an inspiring collaboration with Sara Negri, one that I hope to continue. Dan Synek is worthy a great praise for implementing a type-checker and an emacs-interface to Half. He made the system fast enough that my formalisations could be carried through. I am grateful to Giovanni Sambin for generously inviting me to Padova. Giovanni has developed the particular notion of formal space that I have been using. His papers and lectures during his visits in Gothenburg have also contributed to my work. On the invitation of Jan von Plato, I have paid two very pleasant visits to the Department of Philosophy in Helsinki. Both times I had interesting discussions with Jan and his colleagues. The programming methodology group in Gothenburg has o ered a creative atmosphere and I have had fruitful discussions with many of its members. I would in particular like to thank Michael Hedberg, Henrik Persson, Mary Sheeran and Martin Weichert for interesting discussions and useful comments.