3 research outputs found
A Computer Verified Theory of Compact Sets
Compact sets in constructive mathematics capture our intuition of what
computable subsets of the plane (or any other complete metric space) ought to
be. A good representation of compact sets provides an efficient means of
creating and displaying images with a computer. In this paper, I build upon
existing work about complete metric spaces to define compact sets as the
completion of the space of finite sets under the Hausdorff metric. This
definition allowed me to quickly develop a computer verified theory of compact
sets. I applied this theory to compute provably correct plots of uniformly
continuous functions.Comment: This paper is to be part of the proceedings of the Symbolic
Computation in Software Science Austrian-Japanese Workshop (SCSS 2008
Classical Mathematics for a Constructive World
Interactive theorem provers based on dependent type theory have the
flexibility to support both constructive and classical reasoning. Constructive
reasoning is supported natively by dependent type theory and classical
reasoning is typically supported by adding additional non-constructive axioms.
However, there is another perspective that views constructive logic as an
extension of classical logic. This paper will illustrate how classical
reasoning can be supported in a practical manner inside dependent type theory
without additional axioms. We will see several examples of how classical
results can be applied to constructive mathematics. Finally, we will see how to
extend this perspective from logic to mathematics by representing classical
function spaces using a weak value monad.Comment: v2: Final copy for publicatio