## Section: New Results

### Proof theory

The conservativity of the embedding of pure type systems in the $\lambda \Pi $-calculus modulo was proved by Ali Assaf. This result extends those of Cousineau and Dowek [46] and further justifies the use of the $\lambda \Pi $-calculus modulo as a logical framework. This embedding is the basis for the automated translation tools Holide and Coqine.

Frédéric Blanqui, Jean-Pierre Jouannaud (Univ. Paris
11) and Albert Rubio (Technical University of Catalonia)
have developed a method
aiming at carrying out termination proofs for
higher-order calculi. CPO appears to be the ultimate improvement of
the higher-order recursive path ordering (HORPO)
[45] in the sense that this definition captures the
essence of computability arguments *à la* Tait and Girard,
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.

Frédéric Blanqui worked on the formalization in the Coq proof assistant of various definitions of the notion of $\alpha $-equivalence on pure $\lambda $-terms. In particular, he formalized and formally proved equivalent the definitions of Church (1932), Curry and Feys (1958), Krivine (1993), and Gabbay and Pitts (1999). This work is freely available from the CoLoR library released on December 13th.

Alejandro Díaz-Caro and Gilles Dowek have introduced an extension of $\lambda $-calculus with pairs where isomorphic types are equated. Identifying some types requires to also identify some terms via an equivalence relation on terms, leading to an interesting calculus, which is related to several known non-deterministic and probabilistic calculi. A preliminary version of this work has been published on [24] . A complete version in simple types, with its proof of normalisation, is currently under review.

Together with Ying Jiang, Gilles Dowek has started to investigate the links between model-checking and proof-checking. This has materialized by an encoding of CTL for a finite model in predicate logic and by the definition of a proof-system for CTL.

Olivier Hermant has studied optimized versions of double-negation translations, that allow to switch between classical and intuitionistic logics. Such an algorithm has been implemented in Zenon's backend to Dedukti by Frédéric Gilbert. Gilles Dowek has given new version of Gödel's translation of classical logic into constructive logic. This translation is homomorphic, hence it can be seen as a mere definition of the classical connectives from the constructive ones.