Section: New Results

Encodings into Dedukti and interoperability

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 [32] proved that Cousineau and Dowek's embedding of functional pure type systems [41] 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 [39] . This translation, which is based on the translation of pure type systems by Cousineau and Dowek [41] , 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 [18] . This translation, which is based on the translation of pure type systems by Cousineau and Dowek [41] , 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 [34] .

In [26] , 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 [33] , 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.