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 . 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 , 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 , 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.

Frédéric Blanqui and Guillaume Genestier have formally defined the operational semantics of Dedukti 2.5, showing some problems with non left-linear rewrite rules.

Rodolphe Lepigre, Frédéric Blanqui and Franck Slama developed a new version of Dedukti, available on https://github.com/Deducteam/lambdapi, with meta-variables and a small set of tactics in order to be able to build Dedukti proofs interactively.

Aristomenis-Dionysios Papadopoulos has added a rewrite tactic in the style of Ssreflect .

Emilio Gallego added an LSP server for communicating with editors.

Ismail Lachheb has developed a plugin for Dedukti based on the LSP protocol into the Atom editor .

Guillaume Burel added support for polarized Deduction modulo theory in Dedukti.

Quentin Ye has developed an algorithm to compare $\lambda$-terms. The main point was to take sharing into account, so as to relate the complexity with the space used to represent the term, rather than with the size of the term. He has implemented this algorithm in the Dedukti codebase. He has also run his algorithm on examples that show an exponential speed-up compared to the naive algorithm .

Gaspard Férey and François Thiré defined a new encoding for Cumulative type systems (CTS) in the $\lambda \Pi$-calculus modulo theory, extending the work of Ali Assaf's PhD  . This encoding relies on explicit subtyping which requires additional computational rules. It provides a way to encode a larger class of CTS, which sheds a new light on the computational content of explicit subtyping. This encoding should be extendable to express more advanced features such as universe polymorphism in the Calculus of Inductive Construction, a first step to have a faithful encoding of the Coq system. The encoding has been proven correct under the hypothesis that the computational rules are confluent.

François Thiré redesigned the tool Universo, so that it can be used for a larger class of CTS. The specification for Universo can be given by rewrite rules which makes Universo much easier to use. This tool is a first step to have an automatic chain of translations to translate proofs in the encoding of Matita to STT${\forall }_{\beta \delta }$.which would make these proofs interoperable with 5 different systems.

François Thiré changed the encoding provided by Krajono to integrate some ideas of the encoding discussed above. This encoding is compatible with the tool Universo.

Gaspard Férey updated the CoqInE software to translate Coq's 8.8 version. In this version, the standard library relies on universe polymorphism so partial support for the translation of this feature was integrated. Since encodings of the many features of Coq (inductive constructions, floating universes, several kinds of universe polymorphisms, etc) are a current work in progress, the software was made parameterizable to allow experimentations of multiple encodings of these features.

Gaspard Férey showcased an encoding of the Calculus of Inductive Constructions (CiC) relying on associative-commutative (AC) rewriting on the arithmetic library translated from Matita. This practical experiment shows the limitations of AC-rewriting (as implemented in Dedukti) in terms of performance and the need for special care when defining encodings relying on this feature.

Guillaume Burel began to write a tool translating SAT proof traces in LRAT format into Dedukti proofs. The main issue was that steps in LRAT traces are not logical consequences of previous clauses but only preserve provability.

Mohamed Yacine El Haddad developed a tool to extract TPTP problems from a TSTP trace (generated by automated theorem provers) and reconstruct the proof of the trace in Dedukti format.

Bruno Barras has started to develop a model of Homotopy Type Theory (HoTT) in Dedukti. This is basically a presheaf model, where the choice of the base category leads either to the simplicial sets model or to the cubical model of HoTT. This construction generalizes the setoid model construction  to an arbitrary dimension. Since this involves encoding notions of category theory, the rewriting feature of Dedukti is intensively used to represent, among others, the associativity of morphism composition, or the naturality conditions.

Guillaume Bury has proposed an automation-friendly set theory for the B method. This theory is expressed using first order logic extended to polymorphic types and rewriting. Rewriting is introduced along the lines of deduction modulo theory, where axioms are turned into rewrite rules over both propositions and terms. This work has been published in  .

François Thiré has defined in Dedukti a constructive version of simple type theory with prenex polymorphism: STT${\forall }_{\beta \delta }$. This work has been published at the LFMTP workshop in . STT${\forall }_{\beta \delta }$ has been used to encode an arithmetic library able to prove little Fermat's theorem. Then these proofs has been exported to different systems that are: Coq, Matita, Lean and OpenTheory. Gilles Dowek, César Muñoz, and François Thiré have developed a translation of STT${\forall }_{\beta \delta }$ to PVS.

Then, Walid Moustaoui and François Thiré have built a website called Logipedia which allows the user to inspect this arithmetic library and the user can download the proof of this theorem to one of the systems mentioned above.

Shared and cyclic structures are very common in both programming and proving, which requires generalizing term rewriting techniques to graphs. Jean-Pierre Jouannaud and Nachum Dershowitz have introduced a very general class of multigraphs, called drags, equipped with a composition operator $\otimes$ which provides with a rich categorical structure. Rewriting a drag $D$ can then be defined in a very simple way, by writing $D$ as the composition of a left-hand side of rules $L$ and a context $C$, and then replacing $L$ by $R$, the right-hand side of the rule, which yields the rewritten drag $R\otimes C$. The fundamental aspects of the algebra of drags have been presented at TERMGRAPH'2018 and have also been submitted to a special issue of TCS. Termination of drag rewriting in investigated in .

Gilles Dowek, Liu Jian, and Ying Jiang have reworked the presentation of CTL in sequent calculus proposed by Gilles Dowek and Ying Jiang in 2012 and provided an implementation of it. This work has been published in .