Section: New Results

Theory of $\lambda \Pi$-calculus modulo rewriting

Dependency pairs are a key concept at the core of modern automated termination provers for first-order term rewriting systems. In [14][15], Frédéric Blanqui, Guillaume Genestier and Olivier Hermant introduced an extension of this technique for a large class of dependently-typed higher-order rewriting systems. This improves previous results by Wahlstedt on the one hand and the first author on the other hand to strong normalization and non-orthogonal rewriting systems. This new result has been implemented in the termination-checker SizeChangeTool [17], which participated in the Termination Competition and is used by Dedukti.

During his internship with Frédéric Blanqui and Valentin Blot, Jui-Hsuan Wu designed an algorithm for deciding whether a function defined by rewriting rules is injective or not [23]. This allows to improve the unification algorithm used in Dedukti for inferring types and missing arguments.

The expressiveness of dependent type theory can be extended by identifying types modulo some additional computation rules. But, for preserving the decidability of type-checking or the logical consistency of the system, one must make sure that those user-defined rewriting rules preserve typing. Frédéric Blanqui has developed a new method to check that property using Knuth-Bendix completion. A prototype implementation by Jui-Hsuan Wu is available in Dedukti.

Confluence is a crucial property of rewriting. Gaspard Férey and Jean-Pierre Jouannaud formalized the higher-order rewriting relation on untyped terms implemented in Dedukti and studied various criteria to obtain confluence of higher-order rewrite systems considered together with beta. In particular Von Oostrom's decreasing diagrams technique is applied to multi-steps extensions of simple term rewriting to achieve confluence criteria based on the decidable computation of (orthogonal) higher-order critical pairs. This work assumes left-linearity of rules for now but current work aims at extending these techniques to prove confluence of non-left-linear rule restricted to subsets of terms [20].

Fran cois Thiré has worked on a criterion that would help proving metatheoretical results on Cumulative Type Systems, such as expansion postponment and the equivalence between typed and untyped presentations of conversion. This has been published andpresented at LFMTP'19 [19]

Frédéric Gilbert has written a preprint about the definition of proof certificates for predicative subtyping [21].