A universal characterization of the closed Euclidean interval

We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basic arithmetic operations and to verify equations between them. We test the notion in categories of interest. In the category of sets, any closed and bounded interval of real numbers is an interval object. In the category of topological spaces, the interval objects are closed and bounded intervals with the Euclidean topology. We also prove that an interval object exists in any elementary topos with natural numbers object.</p

On the Cauchy Completeness of the Constructive Cauchy Reals

It is consistent with constructive set theory (without Countable Choice, clearly) that the Cauchy reals (equivalence classes of Cauchy sequences of rationals) are not Cauchy complete. Related results are also shown, such as that a Cauchy sequence of rationals may not have a modulus of convergence, and that a Cauchy sequence of Cauchy sequences may not converge to a Cauchy sequence, among others

Coinductive Formal Reasoning in Exact Real Arithmetic

In this article we present a method for formally proving the correctness of the lazy algorithms for computing homographic and quadratic transformations -- of which field operations are special cases-- on a representation of real numbers by coinductive streams. The algorithms work on coinductive stream of M\"{o}bius maps and form the basis of the Edalat--Potts exact real arithmetic. We use the machinery of the Coq proof assistant for the coinductive types to present the formalisation. The formalised algorithms are only partially productive, i.e., they do not output provably infinite streams for all possible inputs. We show how to deal with this partiality in the presence of syntactic restrictions posed by the constructive type theory of Coq. Furthermore we show that the type theoretic techniques that we develop are compatible with the semantics of the algorithms as continuous maps on real numbers. The resulting Coq formalisation is available for public download.Comment: 40 page

Exact Real Search: Formalised Optimisation and Regression in Constructive Univalent Mathematics

The real numbers are important in both mathematics and computation theory. Computationally, real numbers can be represented in several ways; most commonly using inexact floating-point data-types, but also using exact arbitrary-precision data-types which satisfy the expected mathematical properties of the reals. This thesis is concerned with formalising properties of certain types for exact real arithmetic, as well as utilising them computationally for the purposes of search, optimisation and regression. We develop, in a constructive and univalent type-theoretic foundation of mathematics, a formalised framework for performing search, optimisation and regression on a wide class of types. This framework utilises Mart\'in Escard\'o's prior work on searchable types, along with a convenient version of ultrametric spaces -- which we call closeness spaces -- in order to consistently search certain infinite types using the functional programming language and proof assistant Agda. We formally define and prove the convergence properties of type-theoretic variants of global optimisation and parametric regression, problems related to search from the literature of analysis. As we work in a constructive setting, these convergence theorems yield computational algorithms for correct optimisation and regression on the types of our framework. Importantly, we can instantiate our framework on data-types from the literature of exact real arithmetic, allowing us to perform our variants of search, optimisation and regression on ternary signed-digit encodings of the real numbers, as well as a simplified version of Hans-J. Boehm's functional encodings of real numbers. Furthermore, we contribute to the extensive work on ternary signed-digits by formally verifying the definition of certain exact real arithmetic operations using the Escard\'o-Simpson interval object specification of compact intervals.Comment: A thesis submitted to the University of Birmingham for the degree of Doctor of Philosophy. 198 pages. Supervised by Dan Ghica and Mart\'in Escard\'