Abstract. We describe two methods of extracting constructive content from classical proofs, focusing on theorems involving infinite sequences and nonconstructive choice principles. The first method removes any reference to infinite sequences and transforms the theorem into a system of inductive definitions, the other applies a combination of Gödel’s negativeand Friedman’s A-translation. Both approaches are explained by means of a case study on Higman’s Lemma and its well-known classical proof due to Nash-Williams. We also discuss some proof-theoretic optimizations that were crucial for the formalization and implementation of this work in the interactive proof system Minlog. 1

Disjunctive well-foundedness (used in Terminator), size-change termination, and well-quasi-orders (used in supercompilation and term-rewrite systems) are examples of techniques that have been successfully applied to automatic proofs of program termination and online termination testing, respectively. Although these works originate in different communities, there is an intimate connection between them – 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 performing 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 Veldman and Bezem, and Richman and Stolzenberg. 1.