Computing with Classical Real Numbers

There are two incompatible Coq libraries that have a theory of the real numbers; the Coq standard library gives an axiomatic treatment of classical real numbers, while the CoRN library from Nijmegen defines constructively valid real numbers. Unfortunately, this means results about one structure cannot easily be used in the other structure. We present a way interfacing these two libraries by showing that their real number structures are isomorphic assuming the classical axioms already present in the standard library reals. This allows us to use O'Connor's decision procedure for solving ground inequalities present in CoRN to solve inequalities about the reals from the Coq standard library, and it allows theorems from the Coq standard library to apply to problem about the CoRN reals

Heights and totally real numbers

1973 Schinzel proved that the standard logarithmic height h on the maximal totally real field extension of the rationals is either zero or bounded from below by a positive constant. In this paper we study this property for canonical heights associated to rational functions and the corresponding dynamical system on the affine line. At the end, we will give a few remarks on the behavior of h on finite extensions of the maximal totally real field.Comment: Major changes regarding the first version. E.g. the last chapter was cancele

The real numbers - a survey of constructions

We present a comprehensive survey of constructions of the real numbers (from either the rationals or the integers) in a unified fashion, thus providing an overview of most (if not all) known constructions ranging from the earliest attempts to recent results, and allowing for a simple comparison-at-a-glance between different constructions