Section: New Results

Embeddings in the $\lambda \Pi$-calculus modulo

A new version of Coqine has been developed by Ali Assaf. This version is designed using a Coq plugin architecture, which allows for a smoother integration with Coq's code base and alleviates problems of maintainability that affected the previous version.

The implementation of Holide has been improved, by Ali Assaf. This improved version incorporates sharing at the level of terms and types. This optimization allows to reduce the type-checking time of the OpenTheory standard library from more than 30 minutes to less than 1 minute.

Catherine Dubois and Raphaël Cauderlier have studied a translation in the $\lambda \Pi$-calculus modulo of features coming from object oriented programming languages, such as inheritance and late binding. This compilation scheme has been applied to produce a new back-end for FoCaLize called Focalide, through a compilation to Dedukti. This translation can benefit from the flexibility of Dedukti to deal with more dynamic object-oriented languages than FoCaLiZe; they are currently working on generalizing this translation using $\zeta$-calculus as a theoretical foundation for objects.

Resolution and superposition are proof-search methods that are used in state-of-the-art first-order automated theorem provers such as iProver, Vampire, E or SPASS. A shallow embedding of resolution and superposition proofs in the $\lambda \Pi$-calculus modulo has been proposed by Guillaume Burel, thus offering a way to check these proofs in a trusted setting, and to combine them with other proofs. This embedding has been implemented in particular as a backend of iProver Modulo, therefore allowing to check proofs found by iProver Modulo using Dedukti [20] .

A shallow embedding in Dedukti of the tableaux proofs generated by Zenon modulo has been designed and implemented by Frédéric Gilbert [22] , [23] . The embedding is based on a refined version of previous double-negation translations, introducing as less as possible double negations. This optimization has shown that more than half of the proofs found by Zenon modulo are not using the excluded-middle law, therefore being purely intuitionistic.