arXiv subjects

Cody Roux

Publications and source records attributed to Cody Roux.

6 recordsLinked to original sources

arXiv2025

The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale

We report on the Equational Theories Project (ETP), an online collaborative pilot project to explore new ways to collaborate in mathematics with machine assistance. The project successfully determined all 22 028 942 edges of the implication graph between the 4694 simplest equational laws on magmas, by a combination of human-generated and automated proofs, all validated by the formal proof assistant language Lean. As a result of this project, several new constructions of magmas satisfying specific laws were discovered, and several auxiliary questions were also addressed, such as the effect of restricting attention to finite magmas.

math.RA↗

arXiv2021

Formal Methods for the Informal Engineer: Workshop Recommendations

Formal Methods for the Informal Engineer (FMIE) was a workshop held at the Broad Institute of MIT and Harvard in 2021 to explore the potential role of verified software in the biomedical software ecosystem. The motivation for organizing FMIE was the recognition that the life sciences and medicine are undergoing a transition from being passive consumers of software and AI/ML technologies to fundamental drivers of new platforms, including those which will need to be mission and safety-critical. Drawing on conversations leading up to and during the workshop, we make five concrete recommendations to help software leaders organically incorporate tools, techniques, and perspectives from formal methods into their project planning and development trajectories.

cs.SE↗

arXiv2016

A heuristic prover for real inequalities

We describe a general method for verifying inequalities between real-valued expressions, especially the kinds of straightforward inferences that arise in interactive theorem proving. In contrast to approaches that aim to be complete with respect to a particular language or class of formulas, our method establishes claims that require heterogeneous forms of reasoning, relying on a Nelson-Oppen-style architecture in which special-purpose modules collaborate and share information. The framework is thus modular and extensible. A prototype implementation shows that the method works well on a variety of examples, and complements techniques that are used by contemporary interactive provers.

cs.MS↗

arXiv2015

Elaboration in Dependent Type Theory

To be usable in practice, interactive theorem provers need to provide convenient and efficient means of writing expressions, definitions, and proofs. This involves inferring information that is often left implicit in an ordinary mathematical text, and resolving ambiguities in mathematical expressions. We refer to the process of passing from a quasi-formal and partially-specified expression to a completely precise formal one as elaboration. We describe an elaboration algorithm for dependent type theory that has been implemented in the Lean theorem prover. Lean's elaborator supports higher-order unification, type class inference, ad hoc overloading, insertion of coercions, the use of tactics, and the computational reduction of terms. The interactions between these components are subtle and complex, and the elaboration algorithm has been carefully designed to balance efficiency and usability. We describe the central design goals, and the means by which they are achieved.

cs.LO↗

arXiv2011

Refinement Types as Higher Order Dependency Pairs

Refinement types are a well-studied manner of performing in-depth analysis on functional programs. The dependency pair method is a very powerful method used to prove termination of rewrite systems; however its extension to higher order rewrite systems is still the object of active research. We observe that a variant of refinement types allow us to express a form of higher-order dependency pair criterion that only uses information at the type level, and we prove the correctness of this criterion.

cs.LO↗

arXiv2009

On the relation between size-based termination and semantic labelling

We investigate the relationship between two independently developed termination techniques. On the one hand, sized-types based termination (SBT) uses types annotated with size expressions and Girard's reducibility candidates, and applies on systems using constructor matching only. On the other hand, semantic labelling transforms a rewrite system by annotating each function symbol with the semantics of its arguments, and applies to any rewrite system. First, we introduce a simplified version of SBT for the simply-typed lambda-calculus. Then, we give new proofs of the correctness of SBT using semantic labelling, both in the first and in the higher-order case. As a consequence, we show that SBT can be extended to systems using matching on defined symbols (e.g. associative functions).

cs.LO↗