Abstract. Type-based termination is a semantically intuitive method that ensures termination of recursive definitions by tracking the size of datatype elements, and by checking that recursive calls operate on smaller arguments. However, many systems using type-based termination rely on a semantical anomaly to guarantee strong normalization; namely, they impose that non-recursive elements of a datatype, e.g. the empty list, have size 1 instead of 0. This semantical anomaly also prevents functions such as quicksort to be given a precise typing. The main contribution of this paper is a type system that remedies this anomaly, and still ensures termination. In addition, our type system features prenex stage polymorphism, a weakening of existential quantification over stages, and is precise enough to type quicksort as a non-size increasing function. Moreover, our system accomodate stage addition with all positive inductive types.

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.

We discuss a complete axiomatization of Monadic Second-Order Logic (MSO) on infinite words.By using model-theoretic methods, we give an alternative proof of D. Siefkes' result that a fragment with full comprehension and induction of second-order Peano's arithmetic is complete w.r.t. the validity of MSO-formulas on infinite words. We rely on Feferman-Vaught Theorems and the Ehrenfeucht-Fraïssé method for Henkin models of MSO. Our main technical contribution is an infinitary Feferman-Vaught Fusion of such models. We show it using Ramseyan factorizations similar to those for standard infinite words.

When enriching the λ-calculus with rewriting, union types may be needed to type all strongly normalizing terms. However, with rewriting, the elimination rule (∨ E) of union types may also allow to type non normalizing terms (in which case we say that (∨ E) is unsafe). This occurs in particular with non-determinism, but also with some confluent systems. It appears that studying the safety of (∨ E) amounts to the characterization, in a term, of safe interactions between some of its subterms.In this paper, we study the safety of (∨ E) for an extension of the λ-calculus with simple rewrite rules. We prove that the union and intersection type discipline without (∨ E) is complete w.r.t. strong normalization. This allows to show that (∨ E) is safe if and only if an interpretation of types based on biorthogonals is sound for it. We also discuss two sufficient conditions for the safety of (∨ E), and study an alternative biorthogonality relation, based on the observation of the least reducibility candidate.