During his master internship with Frédéric Blanqui and Bruno Barras, Gabriel Hondet developed a new rewrite engine for Dedukti . The algorithm used in the new rewriting engine is formalised and a correctness proof is provided. This algorithm is based on the pattern matching algorithm by Maranget and used in OCaml. It is extended to rewriting rules, λ terms and non linear patterns. Some interesting implementation details are evinced and then we compare the efficiency of the new engine to a naive matching algorithm and to the rewriting engine of Dedukti. The results show that our implementation handles large rewrite systems better that the naive algorithm, and is always better than Dedukti's.

During her internship with Frédéric Blanqui and Emilio Gallego, Houda Mouzoun developed a Dedukti plugin for the VSCode editor.

During his internship with Frédéric Blanqui and Valentin Blot, Jui-Hsuan Wu implemented a prototype algorithm for deciding whether a function defined by rewriting rules is injective or not , and also a new algorithm proposed by Frédéric Blanqui for checking that user-defined rewrite rules preserve typing.

Bruno Barras has developed a reduction machine implementing a strong call-by-need strategy for $\beta$-reduction and pattern-matching. Higher-order pattern-matching is not yet fully implemented. Regarding efficiency, an exponential speed-up can be observed compared to the current call-by-name implementation on a large class of examples, but a constant slow-down shows on examples where call-by-name is the optimal strategy. With Beniamino Accattoli, he has started studying the correctness of this machine, without pattern-matching. They proved that the machine correctly implements $\beta$-reduction, but have no result yet regarding the strategy or the time complexity.

Dependency pairs are a key concept at the core of modern automated termination provers for first-order term rewriting systems. In , 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 , 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 . 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 .

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 

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

Proof assistants often call automated theorem provers to prove subgoals. However, each prover has its own proof calculus and the proof traces that it produces often lack many details to build a complete proof. Hence these traces are hard to check and reuse in proof assistants. Dedukti is a proof checker whose proofs can be translated to various proof assistants: Coq, HOL, Lean, Matita, PVS. Yacine El Haddad, Guillaume Burel and Frédéric Blanqui implemented a tool Ektraskto that extracts TPTP subproblems from a TSTP file and reconstructs complete proofs in Dedukti using automated provers able to generate Dedukti proofs like ZenonModulo or ArchSAT. This tool is generic: it assumes nothing about the proof calculus of the prover producing the trace, and it can use different provers to produce the Dedukti proof. We applied our tool on traces produced by automated theorem provers on the CNF problems of the TPTP library and we were able to reconstruct a proof for a large proportion of them, significantly increasing the number of Dedukti proofs that could be obtained for those problems.

Zenon Modulo and iProverModulo, two automated theorem provers that can produce Dedukti proofs, have been presented in an article accepted in the Journal of Automated Reasoning .

Agda is a dependently-typed programming language developed at Chalmers University, Gothenburg, Sweden, for 20 years. Thanks to the propositions-as-types correspondence of Curry-Howard, Agda is often used as a proof-assistant. Guillaume Genestier developed with Jesper Cockx a prototypical translator from Agda to Dedukti, which supports well some of the mainly used features of Agda and translates hundreds of definitions of the standard libraries. This implementation led to new encodings of theories in Dedukti, regarding: Universe Polymorphism, Inductive and Record Types, Dependent Pattern Matching, eta convertibility. The implementation of this translator permits to improve both Agda and Dedukti. Indeed, we discovered some bugged (almost not used) functions in Agda and had to extend some existing functions to our purpose. On the Dedukti side, this implementation was the first usage of the newly implemented feature of rewriting modulo associativity and commutativity, which required some minor improvements. Furthermore, our translation of eta-expansion using a defined function led to an improvement in the verification of type preservation of rewriting rules in Dedukti.

Isabelle is a logical framework developed at Technical University of Munich and Cambridge University since the 90s. It implements several logics such as HOL and ZF and is used as part of large verification projects such as seL4 and Flyspeck. Gabriel Hondet developed with Makarius Wenzel (from Augsburg) an export from Isabelle propositions to Dedukti, which was later extended by Michael Färber and Makarius Wenzel to export proofs. This required substantial work on the Isabelle kernel to extend the reconstruction of proof terms based on the work of Stefan Berghofer. Our newly developed proof export allows for an independent verification of a substantial portion of the Isabelle/HOL standard library as well as for the integration of results proved in Isabelle into Logipedia.

Bruno Barras and Rehan Malak have developed further their Dedukti library of presheaves. Using this library, they have built a semi-simplicial model of System F.

Gilles Dowek, Ying Jiang, and Wu Peng, have proposed a proof system for the probabilistic modal logic PCTL. A paper is in preparation.

Gilles Dowek, Ying Jiang, Wu Peng, and Wenhui Zhang have started to study a proof system for CTL*, that mixes constructive and classical aspects.

The article Towards Combining Model Checking and Proof Checking, of Ying Jiang, Jian Liu, Gilles Dowek, and Kailiang Ji, has been published in The Computer Journal .

Gilles Dowek and Alejandro Díaz-Caro have defined a lambda-calculus, the system I, to represent the proofs of a variant minimal propositional logic where isomorphic propositions are identified. Their paper Proof Normalisation in a Logic Identifying Isomorphic Propositions, has been presented at the International Conference on Formal Structures for Computation and Deduction. A second paper The virtues of eta-expansion in System I, showing that the addition of eta-expansion to system I actually simplies the system has been submitted to publication.

The call-by-need evaluation strategy for the $\lambda$-calculus is an evaluation strategy that lazily evaluates arguments only if needed, and if so, shares computations across all places where it is needed. To implement this evaluation strategy, abstract machines require some form of global environment. While abstract machines usually lead to a better understanding of the flow of control during the execution, easing in particular the definition of continuation-passing style translations, the case of machines with global environments turns out to be much more subtle.

In collaboration with Hugo Herbelin, Étienne Miquey introduced $F_{\Upsilon }$, a calculus featuring a data type for typed stores and a mechanism of explicit coercions witnessing store extensions. This calculus defines a generic target of typed continuationand-environment-passing style translations for several calculi with global environment: it is compatible with different evaluation strategy (call-by-need, call-by-name, call-by-value) and different type systems (simple types, system F). On the logical side, these translations broadly amounts to a Kripke forcing-like translation mixed with a negative translation (for the continuation-passing part).

The axiom scheme of comprehension is the cornerstone of second-order arithmetic, a logical theory in which most of mathematics can be formalized. Historically, comprehension was obtained from the negative translation of the axiom of choice, this axiom being interpreted by bar recursion. This led to cluttered and inefficient interpretations of second-order arithmetic.

Valentin Blot simplified this interpretation by proving that the axiom scheme of comprehension has a direct computational interpretation through a variant of bar recursion called update recursion. This new interpretation leads to a more efficient computational interpretation of proofs in second-order arithmetic, and paves the way for a convergence of the two existing interpretations: bar recursion and System F.

Émilie Grienenberger and Gilles Dowek have studied in practice the alignment of logical connectives between proofs systems, a first step towards concept alignement, by the export of the HOL Light standard library using axiomatized connectives to Dedukti. More theoretically, an ecumenical system–where classical and intuitionistic logics coexist–was introduced to act as an exchange platform between proof systems.

The article Two linearities for quantum computing in the lambda calculus, of Alejandro Díaz-Caro, Gilles Dowek, and Juan Pablo Rinaldi, first published in the proceedings of Theory and Practice of Natural Computing 2017, has been published in the journal Biosystems.