Gradual Certified Programming in Coq

Expressive static typing disciplines are a powerful way to achieve high-quality software. However, the adoption cost of such techniques should not be under-estimated. Just like gradual typing allows for a smooth transition from dynamically-typed to statically-typed programs, it seems desirable to support a gradual path to certified programming. We explore gradual certified programming in Coq, providing the possibility to postpone the proofs of selected properties, and to check "at runtime" whether the properties actually hold. Casts can be integrated with the implicit coercion mechanism of Coq to support implicit cast insertion a la gradual typing. Additionally, when extracting Coq functions to mainstream languages, our encoding of casts supports lifting assumed properties into runtime checks. Much to our surprise, it is not necessary to extend Coq in any way to support gradual certified programming. A simple mix of type classes and axioms makes it possible to bring gradual certified programming to Coq in a straightforward manner.Comment: DLS'15 final version, Proceedings of the ACM Dynamic Languages Symposium (DLS 2015

Formal Proofs for Nonlinear Optimization

We present a formally verified global optimization framework. Given a semialgebraic or transcendental function $f$ and a compact semialgebraic domain $K$, we use the nonlinear maxplus template approximation algorithm to provide a certified lower bound of $f$ over $K$. This method allows to bound in a modular way some of the constituents of $f$ by suprema of quadratic forms with a well chosen curvature. Thus, we reduce the initial goal to a hierarchy of semialgebraic optimization problems, solved by sums of squares relaxations. Our implementation tool interleaves semialgebraic approximations with sums of squares witnesses to form certificates. It is interfaced with Coq and thus benefits from the trusted arithmetic available inside the proof assistant. This feature is used to produce, from the certificates, both valid underestimators and lower bounds for each approximated constituent. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of multivariate transcendental inequalities. We illustrate the performance of our formal framework on some of these inequalities as well as on examples from the global optimization literature.Comment: 24 pages, 2 figures, 3 table

Tactics for Reasoning modulo AC in Coq

We present a set of tools for rewriting modulo associativity and commutativity (AC) in Coq, solving a long-standing practical problem. We use two building blocks: first, an extensible reflexive decision procedure for equality modulo AC; second, an OCaml plug-in for pattern matching modulo AC. We handle associative only operations, neutral elements, uninterpreted function symbols, and user-defined equivalence relations. By relying on type-classes for the reification phase, we can infer these properties automatically, so that end-users do not need to specify which operation is A or AC, or which constant is a neutral element.Comment: 16

CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates

Termination is an important property of programs; notably required for programs formulated in proof assistants. It is a very active subject of research in the Turing-complete formalism of term rewriting systems, where many methods and tools have been developed over the years to address this problem. Ensuring reliability of those tools is therefore an important issue. In this paper we present a library formalizing important results of the theory of well-founded (rewrite) relations in the proof assistant Coq. We also present its application to the automated verification of termination certificates, as produced by termination tools

Confirming the Factors of Professional Readiness in Athletic Training

Background: Healthcare professionals such as athletic trainers must be prepared for autonomous practice immediately after graduation. Although certified, new athletic trainers have been shown to have clinical areas of strength and weakness. To better assess professional readiness and improve the preparedness of new athletic trainers, the factors of athletic training professional readiness must be defined. However, limited research exists defining the holistic aspects of professional readiness needed for athletic trainers. Confirming the factors of professional readiness in athletic training could enhance the professional preparation of athletic trainers and result in more highly prepared new professionals. Therefore, the objective of this study was to further explore and confirm the factors of professional readiness in athletic training. Methods: We used a qualitative design based in grounded theory. Participants included athletic trainers with greater than 24 months of experience from a variety of work settings from each district of the National Athletic Trainer’s Association. Participants took the demographic questionnaire electronically using Qualtrics Survey Software (Prove UT). After completing the demographic questionnaire, we selected 20 participants to complete one-on-one interviews using GoToMeeting audiovisual web conferencing software. IMB Statistical Package for the Social Sciences (SPSS, v. 21.0) was used to calculate descriptive statistics for participant demographics. The researcher transcribed all interviews verbatim and a utilized a grounded theory approach during qualitative data analysis. Data were analyzed using a constant comparative analysis as well as open and axial coding. We established trustworthiness by using reflexivity, member checks, and peer reviews. Results: Analysis revealed four overarching themes including management, interpersonal relations, clinical decision-making, and confidence. Conclusion: Athletic trainers should be well-rounded. They must possess communication and organizational skills, the ability to collaborate, value self-reflection and continuing education, and have clinical expertise. Future research should be conducted to finalize a comprehensive model of professional readiness for athletic training, to develop a holistic assessment instrument for athletic training professional readiness, and to explore the preparedness of new athletic trainers as athletic training education transitions to the professional masters degree

Implementing and reasoning about hash-consed data structures in Coq

We report on four different approaches to implementing hash-consing in Coq programs. The use cases include execution inside Coq, or execution of the extracted OCaml code. We explore the different trade-offs between faithful use of pristine extracted code, and code that is fine-tuned to make use of OCaml programming constructs not available in Coq. We discuss the possible consequences in terms of performances and guarantees. We use the running example of binary decision diagrams and then demonstrate the generality of our solutions by applying them to other examples of hash-consed data structures