This paper proposes a new approach to writing and verifying divide-and-conquer programs in Coq. Extending the rich line of previous work on algebraic approaches to recursion schemes, we present an algebraic approach to divide-and-conquer recursion: recursions are represented as a form of algebra, and from outer recursions, one may initiate inner recursions that can construct data upon which the outer recursions may legally recurse. Termination is enforced entirely by the typing discipline of our recursion schemes. Despite this, our approach requires little from the underlying type system, and can be implemented in System F ω plus a limited form of positive-recursive types. Our implementation of the method in Coq does not rely on structural recursion or on dependent types. The method is demonstrated on several examples, including mergesort, quicksort, Harper’s regular-expression matcher, and others. An indexed version is also derived, implementing a form of divide-and-conquer induction that can be used to reason about functions defined via our method.

Payment card fraud causes multibillion dollar losses for banks and merchants worldwide, often fueling complex criminal activities. To address this, many real-time fraud detection systems use tree-based models, demanding complex feature engineering systems to efficiently enrich transactions with historical data while complying with millisecond-level latencies. In this work, we do not require those expensive features by using recurrent neural networks and treating payments as an interleaved sequence, where the history of each card is an unbounded, irregular sub-sequence. We present a complete RNN framework to detect fraud in real-time, proposing an efficient ML pipeline from preprocessing to deployment. We show that these feature-free, multi-sequence RNNs outperform state-of-the-art models saving millions of dollars in fraud detection and using fewer computational resources.