In , Frédéric Blanqui showed how to extend the notion of reducibility introduced by Girard for proving the termination of β-reduction in the polymorphic λ-calculus, to prove the termination of various kinds of rewrite relations on λ-terms, including rewriting modulo some equational theory and rewriting with matching modulo βη, by using the notion of computability closure. This provides a powerful termination criterion for various higher-order rewriting frameworks, including Klop's Combinatory Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems.

In , Frédéric Blanqui, together with Jean-Pierre Jouannaud and Albert Rubio, introduced the computability path ordering (CPO), a recursive relation on terms obtained by lifting a precedence on function symbols. A first version, core CPO, is essentially obtained from the higher-order recursive path ordering (HORPO) by eliminating type checks from some recursive calls and by incorporating the treatment of bound variables as in the so-called computability closure. The well-foundedness proof shows that core CPO captures the essence of computability arguments à la Tait and Girard, therefore explaining its name. We further show that no more type check can be eliminated from its recursive calls without loosing well-foundedness, but one for which we found no counterexample yet. Two extensions of core CPO are then introduced which allow one to consider: the first, higher-order inductive types; the second, a precedence in which some function symbols are smaller than application and abstraction.

Another extension of CPO, to dependently typed terms, has been developed by Jean-Pierre Jouannaud and Jianqi Li in  .

Jean-Pierre Jouannaud and Albert Rubio showed in  how to modify recursive path orders for higher-order terms which, like CPO, include $\beta \eta$-reductions, into orders that are compatible with $\beta \eta$-conversion. The result is a powerful order for proving termination of higher-order rewrite rules based on higher-order pattern matching.

Gaëtan Gilbert and Olivier Hermant have introduced a constructive way to perform proof normalization through completeness proofs .

Frédéric Blanqui formalized Ramsey's proof of the (infinite) Ramsey's theorem (see http://color.inria.fr/).

Jean-Pierre Jouannaud, in collaboration with Jiaxiang Liu, has started a program in order to enable confluence proofs in $\lambda \Pi$ modulo, investigating several open confluence problems for non-terminating relations. In , together with Mizuhito Ogawa, they introduced the new class of layered rewrite systems, and showed that their confluence can be reduced to that of their critical pairs computed by using unification over infinite rational terms when they do not increase the layer-depth of terms. This shows why an old example of non-terminating, left non-linear, critical pair free rewrite system due to Klop was non-confluent: it indeed had a critical pair in infinite rational trees. In the same paper, they also give an example of a non-confluent, layer-depth increasing system which has no critical pairs, hence showing that layer-depth plays a key role.

In , Guillaume Bury, Raphaël Cauderlier and Pierre Halmagrand presented the extension of the automated theorem prover Zenon to ML-style polymorphism.

In , Guillaume Bury, David Delahaye, Damien Doligez, Pierre Hamalgrand and Olivier Hermant introduced an encoding of the set theory of the B method using polymorphic types and deduction modulo, used for the automated verification of proof obligations in the framework of the BWare project.

In , Kailiang Ji designed a strategy to translate model-checking problems into proving the satisfiability of a set of first-order formulas. The focus is to give an encoding of temporal properties expressed in CTL as first-order formulas, by translating the logical equivalence between temporal operators into rewrite rules. In this way, proof-search algorithms designed for Deduction Modulo, such as Resolution Modulo or Tableaux Modulo, can be used to verify temporal properties of finite transition systems. This strategy is implemented in iProver Modulo, and the testing results show that Resolution Modulo can be considered as a new way to quickly determine whether a temporal property is violated or not in transition system models.

Gaëtan Gilbert, supervised by Arnaud Spiwack, wrote a prototype of a principle unification and type inference mechanism for Dedukti, based on a monadic API. This prototype separates with an abstraction barrier a unifier kernel which implements correct unification primitives from the unification algorithm and heuristics. The unification algorithm is written in a style which closely mirrors a pen-and-paper deduction rule presentation.

Éric Uzena, supervised by David Delahaye and Arnaud Spiwack, wrote a prototype of an extension of Dedukti with associative and commutative symbols and rewriting modulo associativity and commutativity of these symbols.

Ali Assaf, Guillaume Burel, Raphaël Cauderlier, David Delahaye, Gilles Dowek, Catherine Dubois, Frédéric Gilbert, Pierre Hamalgrand, Olivier Hermant, and Ronan Saillard have written a synthetic paper on the Dedukti system and on the expression of theories in this system. This paper is submitted to publication.

Ali Assaf proved that Cousineau and Dowek's embedding of functional pure type systems is conservative with respect to the original systems, using a new notion of reducibility called relative normalization. Together with Cousineau and Dowek's original result on the preservation of typing, this result justifies the use of the $\lambda \Pi$-calculus modulo as a logical framework.

Ali Assaf's translation of the calculus of inductive constructions to the $\lambda \Pi$-calculus modulo, which was presented at the TYPES conference in 2014, has been published in the postproceedings of TYPES 2014 . This translation, which is based on the translation of pure type systems by Cousineau and Dowek , is implemented in the automated translation tool Coqine.

Ali Assaf and Guillaume Burel presented their translation of HOL to Dedukti at the PxTP 2015 workshop . This translation, which is based on the translation of pure type systems by Cousineau and Dowek , is implemented in the automated translation tool Holide.

Raphaël Cauderlier and Catherine Dubois' translation of object calculus and subtyping to Dedukti, which was presented at the TYPES conference in 2014, has been published in the post-proceedings of TYPES 2014 .

In , Raphaël Cauderlier and Pierre Halmagrand presented a shallow embedding into Dedukti of proofs produced by ZenonModulo, an extension of the tableau-based first-order theorem prover Zenon to deduction modulo and typing.

In , Ali Assaf and Raphaël Cauderlier have combined simple developments written in Coq and HOL using Dedukti and the existing translation tools Coqine and Holide. This work is a first step towards using Dedukti as a framework for proof interoperability.

Guillaume Burel, Gilles Dowek and Ying Jiang have introduced a general framework to prove the decidability of reachability and provability problems. This framework uses an analogy between the objects recognized by an automaton and cut-free proofs. Various aspects of this work have been published at FroCoS , LPAR , and another paper is in preparation.

Gilles Dowek's paper on the definition of the classical connectives and quantifiers has been published .

Arnaud Spiwack gave a predicative shallow embedding of a weak version of system $U^{-}$ in dependent type theory, for Hurkens's paradox to hold. He also showed that a variety of incarnations of Hurkens's paradox are straightforward instantiations of this encoding, greatly simplifying existing proofs.

Arnaud Spiwack developped a topos-theoretic methodology to reason equationally on circuit languages. Results that hold for combinational circuits are lifted to sequential circuits thanks to a transfer principle. This approach allows, in particular, to simplify reasoning about more complex temporal gates than the unit delay. These results aim at enriching the compiler of the Faust audio signal processing programming language, which features such complex temporal gates.

For the sake of reliability, the kernels of Interactive Theorem Provers (ITPs) are kept relatively small in general. On top of the kernel, additional symbols and inference rules are defined. Some dependency analysis of symbols of HOL Light indicates that the depth of dependency could be reduced by introducing a few more symbols to the kernel. Shuai Wang showed that extending the kernel of HOL Light is a successful attempt to reduce proof size and speed up proof-checking. More specifically, symbols and inference rules of universal quantification and implication were added to the kernel. This approach has been proved to give equivalent proof-checking results with the size of the proof files reduced to 24% on average and a speedup of 38% for proof-checking overall.

Pablo Arrighi and Gilles Dowek have studied the expression of mecanic motions in cellular automata. Part of this work has been published in TPNC and another paper is in preparation.

Arnaud Spiwack developped a variant of Turing machine where the tape is replaced by an unlabeled tree. The additional structure makes combining machines much easier, making it tractable to give explicit descriptions of rather complex machines. The cost model of these machines models that of purely functional programming languages, making it possible to compare mathematically the complexity of imperative algorithms and of purely functional algorithms.