## Section: New Results

### Theories

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 [28]. 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 [2] 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 [30].