Find more details here.

**Supervisor:** Frédéric Blanqui

**Place:** Deducteam, Laboratoire Méthodes Formelles (LMF), ENS Paris-Saclay, 4 avenue des Sciences, Gif-sur-Yvette

**Context:** I propose various projects on formal proofs as digital objects, more specifically the translation of formal proofs between different proof systems. Formal proofs are used in mathematics but also in the industry for certifying the correctness of protocols, software and hardware.

Interoperability is a very important feature in computer science and engineering to avoid useless work duplication and allow more safety. Unfortunately, interoperability between proof systems is not well developed. One important difficulty is that proof systems may have incompatible features: their combination may be inconsistent. Therefore, to translate a proof from one system to the other, we need to analyze the features of the first system used in the proof and check whether they are compatible with the features of the target system.

The λΠ-calculus modulo rewriting, and its implementation Dedukti, is a powerful logical framework allowing users to define their own logic and represent the proofs in those logics. For instance, one can represent in Dedukti first-order logic and its proofs, simple type theory and its proofs, the Isabelle logic and its proofs, the Coq logic and its proofs, the Agda logic and its proofs, etc. In addition, there is a number of tools for transforming those proofs and translate them back to various other systems: HOL-Light, Coq, PVS, Lean, etc.

There exist several tools for checking the correctness of Dedukti files: dkcheck, kontroli and lambdapi. While dkcheck and kontroli are mere checkers taking complete Dedukti files as input, Lambdapi is a proof assistant featuring implicit arguments, type inference, coercions, tactics, the possibility of calling external automated theorem provers, etc. for building Dedukti proofs interactively.