## Section: New Results

### Termination

Frédéric Blanqui, together with Jean-Pierre Jouannaud (Univ. Paris
11) and Albert Rubio (Technical University of Catalonia), have
finished their work on a new version of the higher-order recursive
path ordering (HORPO) [44] , [43] , a
decidable monotone well-founded relation that can be used for
proving the termination of higher-order rewrite systems by checking
that rules are included in it. This new version, called the
computability path ordering (CPO), appears to be the ultimate
improvement of HORPO in the sense that this definition captures the
essence of computability arguments *à la* Tait and Girard
[37] , therefore explaining the name of the
improved ordering. It has been shown that CPO allows to consider
higher-order rewrite rules in a simple type discipline with
inductive types, that most of the guards present in the recursive
calls of its core definition cannot be relaxed in any natural way
without losing well-foundedness, and that the precedence on function
symbols cannot be made more liberal anymore. This new result is
described in a 41-pages papers available on Frédéric Blanqui's web
page which has been submitted to a journal for publication. A Prolog
implementation of CPO is also available on Albert Rubio's web page.

Frédéric Blanqui revised his work on the compatibility of Tait and Girard's notion of computability for proving the termination of higher-order rewrite systems when matching is done modulo $\beta \eta $-equivalence. In particular, he showed that computability is preserved by leaf-$\beta $-expansion, a key property for dealing with higher-order pattern-matching. This work is described in a 46-pages paper available on his web page which has been submitted to a journal for publication.

Frédéric Blanqui did some historical investigations on fixpoint theorems in posets used for instance for defining the semantics of non-basic inductive types (i.e. types with constructors taking functions as arguments) and the termination of functions defined by induction on such non-basic inductive types. These theorems assume the function either extensive or monotone. However, as shown by Salinas in [48] , these two conditions can be subsumed by a more general one. Frédéric Blanqui slightly improved this condition further by using results by Hartogs, Rubin and Rubin, and Abian and Brown. This work is described in a 10-pages note available on his web page [20] .

Kim Quyen Ly finished the development of a new version faster, safer (proved correct in Coq) and standalone version of Rainbow, based on Coq extraction mechanism. She defended her PhD thesis [11] on the automated verification of termination certificates in October.