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## Section: New Results

### Termination

F. Blanqui revised his paper on “size-based termination of higher-order rewrite systems” submitted to the Journal of Functional Programming [19]. This paper provides a general and modular criterion for the termination of simply-typed λ-calculus extended with function symbols defined by user-defined rewrite rules. Following a work of Hughes, Pareto and Sabry for functions defined with a fixpoint operator and pattern-matching [33], several criteria use typing rules for bounding the height of arguments in function calls. In this paper, we extend this approach to rewriting-based function definitions and more general user-defined notions of size.

R. Lepigre worked on his paper “Practical Subtyping for System F with Sized (Co-)Induction” [39] (joint work with C. Raffalli), which was submitted to the journal Transactions on Programming Languages and Systems (TOPLAS) and is now under revision. This paper proposes a practical type system for a rich, normalizing, extension of (Curry-style) System F. The termination of recursive programs is established using a new mechanism based on circular proofs, which is also used to deal with (sized) inductive and coinductive types (in subtyping). The idea is to build (possibly ill-formed) infinite, circular typing (resp. subtyping) derivations, and to check for their well-foundedness a posteriori. The normalization proof then follows using standard realizability (or reducibility) techniques, the main point being that the adequacy lemma can still be proved by (well-founded) induction on the structure of the “circular” typing (resp. subtyping) derivations.