Type classes for efficient exact real arithmetic in Coq

Floating point operations are fast, but require continuous effort on the part of the user in order to ensure that the results are correct. This burden can be shifted away from the user by providing a library of exact analysis in which the computer handles the error estimates. Previously, we [Krebbers/Spitters 2011] provided a fast implementation of the exact real numbers in the Coq proof assistant. Our implementation improved on an earlier implementation by O'Connor by using type classes to describe an abstract specification of the underlying dense set from which the real numbers are built. In particular, we used dyadic rationals built from Coq's machine integers to obtain a 100 times speed up of the basic operations already. This article is a substantially expanded version of [Krebbers/Spitters 2011] in which the implementation is extended in the various ways. First, we implement and verify the sine and cosine function. Secondly, we create an additional implementation of the dense set based on Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order on undecidable structures, while it was limited to decidable structures before. This hierarchy, based on type classes, allows us to share theory on the naturals, integers, rationals, dyadics, and reals in a convenient way. Finally, we obtain another dramatic speed-up by avoiding evaluation of termination proofs at runtime.Comment: arXiv admin note: text overlap with arXiv:1105.275

Integrals and Valuations

We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of (valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and tightly connected to the Riesz space structure.Comment: Submitted for publication 15/05/0

Constructive Theory of Banach algebras

We present a way to organize a constructive development of the theory of Banach algebras, inspired by works of Cohen, de Bruijn and Bishop. We illustrate this by giving elementary proofs of Wiener's result on the inverse of Fourier series and Wiener's Tauberian Theorem, in a sequel to this paper we show how this can be used in a localic, or point-free, description of the spectrum of a Banach algebra

Sets in homotopy type theory

Homotopy Type Theory may be seen as an internal language for the $\infty$-category of weak $\infty$-groupoids which in particular models the univalence axiom. Voevodsky proposes this language for weak $\infty$-groupoids as a new foundation for mathematics called the Univalent Foundations of Mathematics. It includes the sets as weak $\infty$-groupoids with contractible connected components, and thereby it includes (much of) the traditional set theoretical foundations as a special case. We thus wonder whether those `discrete' groupoids do in fact form a (predicative) topos. More generally, homotopy type theory is conjectured to be the internal language of `elementary' $\infty$-toposes. We prove that sets in homotopy type theory form a $\Pi W$-pretopos. This is similar to the fact that the $0$-truncation of an $\infty$-topos is a topos. We show that both a subobject classifier and a $0$-object classifier are available for the type theoretical universe of sets. However, both of these are large and moreover, the $0$-object classifier for sets is a function between $1$-types (i.e. groupoids) rather than between sets. Assuming an impredicative propositional resizing rule we may render the subobject classifier small and then we actually obtain a topos of sets

A constructive proof of Simpson's Rule

For most purposes, one can replace the use of Rolle's theorem and the mean value theorem, which are not constructively valid, by the law of bounded change. The proof of two basic results in numerical analysis, the error term for Lagrange interpolation and Simpson's rule, however seem to require the full strength of the classical Rolle's Theorem. The goal of this note is to justify these two results constructively, using ideas going back to Amp\`ere and Genocchi

Almost periodic functions, constructively

The almost periodic functions form a natural example of a non-separable normed space. As such, it has been a challenge for constructive mathematicians to find a natural treatment of them. Here we present a simple proof of Bohr's fundamental theorem for almost periodic functions which we then generalize to almost periodic functions on general topological groups

Modalities in homotopy type theory

Univalent homotopy type theory (HoTT) may be seen as a language for the category of $\infty$-groupoids. It is being developed as a new foundation for mathematics and as an internal language for (elementary) higher toposes. We develop the theory of factorization systems, reflective subuniverses, and modalities in homotopy type theory, including their construction using a "localization" higher inductive type. This produces in particular the ($n$-connected, $n$-truncated) factorization system as well as internal presentations of subtoposes, through lex modalities. We also develop the semantics of these constructions

The implementation of care pathways to improve children’s allergy services

National reviews demonstrate quality deficiencies in UK health services for allergic children. In response, the RCPCH developed care pathways to improve the organisation and delivery of allergy care. However, care pathway interventions have variable success. Consequently, a literature review and four ethnographic case studies were conducted to theorise care pathway implementation and inform practical recommendations for health service improvements. This process followed the four stages of MELD dialectic: First Moment (1M) of (non-)being, Second Edge (2E) of becoming, Third Level (3L) of integrated totality, and Fourth Dimension (4D) of agency. The 1M results define the mechanisms that cause allergy health service deficiencies and how to address them. Deficiencies were found to emerge from causal mechanisms interacting at physiological, inter-personal, and systemic levels. Care pathway interventions that address these mechanisms could improve the quality of children’s allergy services. However, healthcare improvements also depend on the implementation process and contextual influences. The 2E results describe the unfolding journeys of four teams improving their local health services for allergic children. The 3L results integrate the 1M theoretical understanding and 2E empirical experiences into an advanced understanding of allergy care pathway implementation. The improvement context totality captures contextual factors that manifest differently depending on geohistorical contingencies. This creates unique local challenges and opportunities. The child allergy intervention totality integrates care pathway interventions implemented across case studies. What intervention elements are implemented and to what degree is negotiated based on local challenges and opportunities. The improvement process totality captures the quality improvement methods used to develop interventions and overcome contextual challenges. Children’s allergy health service improvements emerge from the three totalities operating as a dynamic whole. The 4D results discuss the resolution of dialectical contradictions related to the improvement of children’s allergy services. They integrate contradictory theoretical insights identified across academic disciplines by grounding them in the 2L empirical case study experiences and relating them to the 3L structural understanding. These theoretical developments are subsequently translated into practical strategies to navigate and optimise local efforts to improve children’s allergy services. This study used MELD to make theoretical contributions in clinical allergy and improvement science with the aim to support the improvement of children’s allergy services in practice.Open Acces