Section: New Results

$\lambda \Pi$-calculus modulo theory

Gilles Dowek, Jean-Pierre Jouannaud and Jiaxiang Liu have started a program for developing new techniques for proving confluence of dependently typed theories, which do not rely on termination. These results have been presented at Types 2016, and will be submitted to a Journal early 2019. Target applications for these techniques are encodings of the Calculus of inductive constructions with polymorphic universes in the $\lambda \Pi$-calculus modulo theory.

Frédéric Blanqui has published in the Journal of Functional Programming a long article synthesizing his work on the use of size annotations for proving termination [12]. This paper provides a general and modular criterion for the termination of simply-typed $\lambda$-calculus extended with function symbols defined by user-defined rewrite rules. Following a work of Hughes, Pareto and Sabry, for functions defined with a fixpoint operator and pattern-matching, several criteria use typing rules for bounding the height of arguments in function calls. In this paper, we extend this approach to rewriting-based function definitions and more general user-defined notions of size.

Size-change termination is a technique introduced for first-order functional programs. In [16], Frédéric Blanqui and Guillaume Genestier show how it can be used to study the termination of higher-order rewriting in the $\lambda \Pi$-calculus modulo theory.

Dependency pairs are a key concept at the core of modern automated termination provers for first-order term rewrite systems. In [22], Frédéric Blanqui, Guillaume Genestier and Olivier Hermant introduced an extension of this technique for a large class of dependently-typed higher-order rewrite systems. This improves previous results by Wahlstedt on one hand and Frédéric Blanqui on the other hand to strong normalization and non-orthogonal rewrite systems. This new criterion has been implemented in the type-checker Dedukti .