Automatically proving equivalence by type-safe reflection

We are also grateful for the support of the Scottish Informatics and Computer Science Alliance (SICSA) and EPSRC grant EP/N024222/1.One difficulty with reasoning and programming with dependent types is that proof obligations arise naturally once programs become even moderately sized. For example, implementing an adder for binary numbers indexed over their natural number equivalents naturally leads to proof obligations for equalities of expressions over natural numbers. The need for these equality proofs comes, in intensional type theories, from the fact that the propositional equality enables us to prove as equal terms that are not judgementally equal, which means that the typechecker can’t always obtain equalities by reduction. As far as possible, we would like to solve such proof obligations automatically. In this paper, we show one way to automate these proofs by reflection in the dependently typed programming language Idris. We show how defining reflected terms indexed by the original Idris expression allows us to construct and manipulate proofs. We build a hierarchy of tactics for proving equivalences in semi-groups, monoids, commutative monoids, groups, commutative groups, semi-rings and rings. We also show how each tactic reuses those from simpler structures, thus avoiding duplication of code and proofs.Postprin

Adding an Abstraction Barrier to ZF Set Theory

Much mathematical writing exists that is, explicitly or implicitly, based on set theory, often Zermelo-Fraenkel set theory (ZF) or one of its variants. In ZF, the domain of discourse contains only sets, and hence every mathematical object must be a set. Consequently, in ZF, with the usual encoding of an ordered pair ${\langle a, b\rangle}$, formulas like ${\{a\} \in \langle a, b \rangle}$ have truth values, and operations like ${\mathcal P (\langle a, b\rangle)}$ have results that are sets. Such 'accidental theorems' do not match how people think about the mathematics and also cause practical difficulties when using set theory in machine-assisted theorem proving. In contrast, in a number of proof assistants, mathematical objects and concepts can be built of type-theoretic stuff so that many mathematical objects can be, in essence, terms of an extended typed ${\lambda}$-calculus. However, dilemmas and frustration arise when formalizing mathematics in type theory. Motivated by problems of formalizing mathematics with (1) purely set-theoretic and (2) type-theoretic approaches, we explore an option with much of the flexibility of set theory and some of the useful features of type theory. We present ZFP: a modification of ZF that has ordered pairs as primitive, non-set objects. ZFP has a more natural and abstract axiomatic definition of ordered pairs free of any notion of representation. This paper presents axioms for ZFP, and a proof in ZF (machine-checked in Isabelle/ZF) of the existence of a model for ZFP, which implies that ZFP is consistent if ZF is. We discuss the approach used to add this abstraction barrier to ZF

Automating Change of Representation for Proofs in Discrete Mathematics (Extended Version)

Representation determines how we can reason about a specific problem. Sometimes one representation helps us find a proof more easily than others. Most current automated reasoning tools focus on reasoning within one representation. There is, therefore, a need for the development of better tools to mechanise and automate formal and logically sound changes of representation. In this paper we look at examples of representational transformations in discrete mathematics, and show how we have used Isabelle's Transfer tool to automate the use of these transformations in proofs. We give a brief overview of a general theory of transformations that we consider appropriate for thinking about the matter, and we explain how it relates to the Transfer package. We show our progress towards developing a general tactic that incorporates the automatic search for representation within the proving process

Designing an automated clinical decision support system to match clinical practice guidelines for opioid therapy for chronic pain

Abstract Background Opioid prescribing for chronic pain is common and controversial, but recommended clinical practices are followed inconsistently in many clinical settings. Strategies for increasing adherence to clinical practice guideline recommendations are needed to increase effectiveness and reduce negative consequences of opioid prescribing in chronic pain patients. Methods Here we describe the process and outcomes of a project to operationalize the 2003 VA/DOD Clinical Practice Guideline for Opioid Therapy for Chronic Non-Cancer Pain into a computerized decision support system (DSS) to encourage good opioid prescribing practices during primary care visits. We based the DSS on the existing ATHENA-DSS. We used an iterative process of design, testing, and revision of the DSS by a diverse team including guideline authors, medical informatics experts, clinical content experts, and end-users to convert the written clinical practice guideline into a computable algorithm to generate patient-specific recommendations for care based upon existing information in the electronic medical record (EMR), and a set of clinical tools. Results The iterative revision process identified numerous and varied problems with the initially designed system despite diverse expert participation in the design process. The process of operationalizing the guideline identified areas in which the guideline was vague, left decisions to clinical judgment, or required clarification of detail to insure safe clinical implementation. The revisions led to workable solutions to problems, defined the limits of the DSS and its utility in clinical practice, improved integration into clinical workflow, and improved the clarity and accuracy of system recommendations and tools. Conclusions Use of this iterative process led to development of a multifunctional DSS that met the approval of the clinical practice guideline authors, content experts, and clinicians involved in testing. The process and experiences described provide a model for development of other DSSs that translate written guidelines into actionable, real-time clinical recommendations.http://deepblue.lib.umich.edu/bitstream/2027.42/78267/1/1748-5908-5-26.xmlhttp://deepblue.lib.umich.edu/bitstream/2027.42/78267/2/1748-5908-5-26.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/78267/3/1748-5908-5-26-S3.TIFFhttp://deepblue.lib.umich.edu/bitstream/2027.42/78267/4/1748-5908-5-26-S2.TIFFhttp://deepblue.lib.umich.edu/bitstream/2027.42/78267/5/1748-5908-5-26-S1.TIFFPeer Reviewe

Generalized Theorems for Nonlinear State Space Reconstruction

Takens' theorem (1981) shows how lagged variables of a single time series can be used as proxy variables to reconstruct an attractor for an underlying dynamic process. State space reconstruction (SSR) from single time series has been a powerful approach for the analysis of the complex, non-linear systems that appear ubiquitous in the natural and human world. The main shortcoming of these methods is the phenomenological nature of attractor reconstructions. Moreover, applied studies show that these single time series reconstructions can often be improved ad hoc by including multiple dynamically coupled time series in the reconstructions, to provide a more mechanistic model. Here we provide three analytical proofs that add to the growing literature to generalize Takens' work and that demonstrate how multiple time series can be used in attractor reconstructions. These expanded results (Takens' theorem is a special case) apply to a wide variety of natural systems having parallel time series observations for variables believed to be related to the same dynamic manifold. The potential information leverage provided by multiple embeddings created from different combinations of variables (and their lags) can pave the way for new applied techniques to exploit the time-limited, but parallel observations of natural systems, such as coupled ecological systems, geophysical systems, and financial systems. This paper aims to justify and help open this potential growth area for SSR applications in the natural sciences

Social Networks and Friendships at School: Comparing Children With and Without ASD

Self, peer and teacher reports of social relationships were examined for 60 high-functioning children with ASD. Compared to a matched sample of typical children in the same classroom, children with ASD were more often on the periphery of their social networks, reported poorer quality friendships and had fewer reciprocal friendships. On the playground, children with ASD were mostly unengaged but playground engagement was not associated with peer, self, or teacher reports of social behavior. Twenty percent of children with ASD had a reciprocated friendship and also high social network status. Thus, while the majority of high functioning children with ASD struggle with peer relationships in general education classrooms, a small percentage of them appear to have social success

High TWIST1 mRNA expression is associated with poor prognosis in lymph node-negative and estrogen receptor-positive human breast cancer and is co-expressed with stromal as well as ECM related genes

Introduction: The TWIST homolog 1 (TWIST1) is a transcription factor that induces epithelial to mesenchymal transition (EMT), a key process in metastasis. The purpose of this study was to investigate whether TWIST1 expression predicts disease progression in a large breast cancer cohort with long-term clinical follow-up, and to reveal the biology related to TWIST1 mediated disease progression.Methods: TWIST1 mRNA expression level was analyzed by quantitative real-time reverse polymerase chain reaction (RT-PCR) in 1,427 primary breast cancers. In uni- and multivariate analysis using Cox regression, TWIST1 mRNA expression level was associated with metastasis-free survival (MFS), disease-free survival (DFS) and overall survival (OS). Separate analyses in lymph node-negative patients (LNN, n = 778) who did not receive adjuvant systemic therapy, before and after stratification into estrogen receptor (ER)-positive (n = 552) and ER-negative (n = 226) disease, were also performed. The association of TWIST1 mRNA with survival endpoints was assessed using Kaplan-Meier analysis. Using gene expression arrays, genes showing a significant Spearman rank correlation with TWIST1 were used to identify overrepresented Gene Ontology (GO) terms and Kyoto Encyclopedia of Genes and Genomes (KEGG)-annotated biological pathways.Results: Increased mRNA expression level of TWIST1 analyzed as a continuous variable in both uni- and multivariate analysis was associated with shorter MFS in all patients (hazard ratio (HR): 1.17, 95% confidence interval, (95% CI):1.09 to 1.26; and HR: 1.17, 95% CI: 1.08 to 1.26; respectively), in LNN patients (HR: 1.22, 95% CI: 1.09 to 1.36; and HR: 1.21, 95% CI: 1.07 to 1.36; respectively) and in the ER-positive subgroup of LNN patients (HR: 1.34, 95% CI: 1.17 to 1.53; and HR: 1.32, 95% CI: 1.14 to 1.53; respectively). Similarly, high TWIST1 expression was associated with shorter DFS and OS in all patients and in the LNN/ER-positive subgroup. In contrast, no association of TWIST1 mRNA expression with MFS, DFS or OS was observed in ER-negative patients. Genes h