On the Stability by Union of Reducibility Candidates

We investigate some aspects of proof methods for the termination of (extensions of) the second-order λ-calculus in presence of union and existential types. We prove that Girard’s reducibility candidates are stable by union iff they are exactly the non-empty sets of terminating terms which are downward-closed w.r.t. a weak observational preorder. We show that this is the case for the Curry-style second-order λcalculus. As a corollary, we obtain that reducibility candidates are exactly the Tait’s saturated sets that are stable by reduction. We then extend the proof to a system with product, co-product and positive iso-recursive types.