## Section: New Results

### Confluence

Jean-Pierre Jouannaud, in collaboration with Jiaxiang Liu, has started a program in order to enable confluence proofs in $\lambda \Pi $ modulo, investigating several open confluence problems for non-terminating relations. In [27] , together with Mizuhito Ogawa, they introduced the new class of layered rewrite systems, and showed that their confluence can be reduced to that of their critical pairs computed by using unification over infinite rational terms when they do not increase the layer-depth of terms. This shows why an old example of non-terminating, left non-linear, critical pair free rewrite system due to Klop was non-confluent: it indeed had a critical pair in infinite rational trees. In the same paper, they also give an example of a non-confluent, layer-depth increasing system which has no critical pairs, hence showing that layer-depth plays a key role.