Section: New Results

Type and rewriting theory

Participants : Frédéric Blanqui, Jean-Pierre Jouannaud, Jianqi Li, Qian Wang.

Qian Wang and Bruno Barras have proved the strong normalization property of CoqMTU in presence of strong elimination, a major step towars the full certification of CoqMTU [16] .

Jouannaud and Li have developped a new framework, Normal Abstract Rewriting Systems (NARS), that captures all known Church-Rosser results in presence of a termination assumption allowing to reduce equality of terms to a simpler equality on their normal forms. This result applies to the paticular case of higher-order rewriting for which it solved long-standing open problems [10] .

Jouannaud and Liu have continued their investigation of Church-Rosser properties of non-terminating rewrite systems [10] , showing recently first, that many results found in the litterature could be captured, and generalized, by using van Oostrom's decreasing diagram technique (accepted at Symposium on Algebraic Specifications, Kanazawa, Japan, April 2014). The next step, which has been recently completed, is a powerful result generalizing Knuth and Bendix confluence test to non terminating rewrite system (submitted).

Frédéric Blanqui, Jean-Pierre Jouannaud and Albert Rubio (Technical University of Catalonia) have developed a method aiming at carrying out termination proof for higher-order calculi. CPO appears to be the ultimate improvement of the higher-order recursive path ordering (HORPO) [25] 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. This result is submitted to journal, and has been concurrently generalized to higher-order calculi with dependent types by Jouannaud and Li (submitted).

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.

Frédéric Blanqui worked with John Steinberger (Tsinghua University) on the formal verification in Coq of proofs of theorems on coset arrays and non-negative integer linear combinations.