On the Stability by Union of Reducibility Candidates

Abstract

Abstract. 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 λ-calculus. 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.

Mixed Inductive/Coinductive Types and Strong Normalization

“…The last property is the advantage of saturated-sets semantics, it does not always hold for candidates of reducibility or biorthogonals, and even when it holds the proof is non-trivial [16].…”

Union of Reducibility Candidates for Orthogonal Constructor Rewriting

“…In [18], we have given a necessary and sufficient condition for reducibility candidates to be stable by union; and in [19], we have given a necessary and sufficient condition for the closure by union of Biorthogonals [11,16,8] to be reducibility candidates. The second condition is strictly more general than the first one.…”

“…We now recall the condition established in [18] for the stability by union of reducibility candidates, and then show that it is met with orthogonal constructor rewriting.…”

“…The study of stability by union of reducibility candidates of [18] carries on with our generalization of neutral terms and elimination contexts. The key observation is a characterization of the membership of a term to a candidate using a weak observational preorder.…”

“…By instantiating the condition of [18], we show that reducibility candidates are stable by union for orthogonal constructor rewriting. The proof uses a result of [13] on the existence of external redexes for orthogonal CCERSs.…”