Stop When You Are Almost-Full - Adventures in Constructive Termination

Disjunctive well-foundedness, size-change termination, and well-quasi-orders are examples of techniques that have been successfully applied to program termination. Although these works originate in different communities, they rely on closely related principles and both employ similar arguments from Ramsey theory. At the same time there is a notable absence of these techniques in programming systems based on constructive type theory. In this paper we’d like to highlight the aforementioned connection and make the core ideas widely accessible to theoreticians and Coq programmers, by offering a Coq development which culminates in some novel tools for induction. The benefit is nice composability properties of termination arguments at the cost of intuitive and lightweight user obligations. Inevitably, we have to present some Ramsey-like arguments: Though similar proofs are typically classical, we offer an entirely constructive development standing on the shoulders of Bezem and Veldman, and Richman and Stolzenberg.