Section: New Results

Property proofs for adversary rewrite systems

Participant : Isabelle Gnaedig.

We have continued to work on rewriting property proofs in the adversary context. Our inductive proof technique, initially developed for proving termination of rewriting for systems that do not enjoy the strong termination property, was first proposed to establish termination proofs under particular strategies: the innermost, outermost, local strategies  [58] .

We then have tackled the proof problem of weak properties i.e., properties that do hold only on certain derivation branches. Weak property proofs are still marginal in the domain of rewriting, probably because classical proof techniques, especially for termination, work on the rules, so that the phenomenons arising in the induced rewriting relation are hidden. Our technique, developing proof trees simulating rewriting trees by abstraction and narrowing, explicitly describes the behavior of the studied property on derivation branches, allowing to establish it on good branches. In addition, it is constructive, which is very useful in the programming context: the good branches are identified at compile time, when the proof is established. At run time, derivations are computed only on a good derivation branch, which avoids using the costly breadth-first strategy.

We then have proposed a procedure, based on our inductive principle, for weak termination and $C$-reducibility, which can be seen as a weak notion of sufficient completeness. The procedure principle is generic and can be instantiated by specific mechanisms related to both properties [20] .