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A Pointfree approach to Constructive Analysis in Type Theory
, 1997
"... The first paper in this thesis presents a machine checked formalisation, in MartinLöf'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 defined as a formal space from ..."
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The first paper in this thesis presents a machine checked formalisation, in MartinLöf'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 defined as a formal space from a base of rational intervals. Then the closed rational interval [a, b] is defined as a formal space, in terms of the continuum, and the HeineBorel covering theorem is proved constructively. The basic definitions 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 HahnBanach theorems are proved in an entirely constructive way. The proofs have been carried out in intensional MartinLöf type theory with one universe and finitary inductive definitions, and the proofs have also been mechanically checked in an implementation of that system. ...
The HahnBanach Theorem in Type Theory
, 1997
"... We give the basic deønitions for pointfree functional analysis and present constructive proofs of the Alaoglu and HahnBanach theorems in the setting of formal topology. 1 Introduction We present the basic concepts and deønitions needed in a pointfree approach to functional analysis via formal topol ..."
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We give the basic deønitions for pointfree functional analysis and present constructive proofs of the Alaoglu and HahnBanach theorems in the setting of formal topology. 1 Introduction We present the basic concepts and deønitions needed in a pointfree approach to functional analysis via formal topology. Our main results are the constructive proofs of localic formulations of the Alaoglu and HellyHahnBanach 1 theorems. Earlier pointfree formulations of the HahnBanach theorem, in a topostheoretic setting, were presented by Mulvey and Pelletier (1987,1991) and by Vermeulen (1986). A constructive proof based on points was given by Bishop (1967). In the formulation of his proof, the norm of the linear functional is preserved to an arbitrary degree by the extension and a counterexample shows that the norm, in general, is not preserved exactly. As usual in pointfree topology, our guideline is to deøne the objects under analysis as formal points of a suitable formal space. After this has...
A Machine Assisted Proof of the HahnBanach Theorem
, 1997
"... We describe an implementation of a pointfree proof of the Alaoglu and the HahnBanach theorems in Type Theory. The proofs described here are formalisations of the proofs presented in "The HahnBanach Theorem in Type Theory" [4]. The implementation was partially developed simultaneously with ..."
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We describe an implementation of a pointfree proof of the Alaoglu and the HahnBanach theorems in Type Theory. The proofs described here are formalisations of the proofs presented in "The HahnBanach Theorem in Type Theory" [4]. The implementation was partially developed simultaneously with [4] and it was a help in the development of the informal proofs. 1 Introduction We present a machine assisted formalisation of pointfree topology in MartinLof's type theory. The continuum and the basic definitions needed in a pointfree approach to functional analysis are given and in this setting we describe implementations of localic formulations of the Alaoglu and the HahnBanach theorems. The classical HahnBanach theorem says that, if M is a subspace of a normed linear space A and f is a bounded linear functional on M , then f can be extended to a linear functional F on A so that kFk = kfk. (In our proof we use the equivalent formulation: if kfk 1 then f can be extended to F so that kFk 1.) A...
A Pointfree approach to Constructive Analysis in Type Theory
"... 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 h ..."
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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 influenced 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 typechecker and an emacsinterface 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.