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Theses

Interoperability between proof systems using the logical framework Dedukti

François Thiré 1 
1 DEDUCTEAM - Deduction modulo, interopérabilité et démonstration automatique
Inria Saclay - Ile de France, LMF - Laboratoire Méthodes Formelles
Abstract : There is today a large family of proof systems based upon variouslogics: The Calculus of Inductive Constructions, Higher-Order logic orSet theory, etc. The diversity of proof systems has the negativeconsequence that theorems are formalized many times. One way toovercome this issue would be to make proof systems interoperable. Inthis thesis, we have tackled the interoperability problem for proofsystems both on the theoretical and the practical side using theDedukti logical framework.We begin our journey by looking at Cumulative Type Systems (CTS), afamily of type systems which extends that of Pure Type Systemswith a subtyping relation. CTS provides a common skeleton to manylogics used today. The logic behind Coq, HOL-Light, Lean, Matita orPVS can be seen as an extension of CTS with various features(inductive types, proof irrelevance, predicate subtyping, …). Wedefine a new notion of embedding between CTS. We also provide a soundbut incomplete algorithm to decide whether a proof in one CTS can betranslated into another CTS. This algorithm can also be seen as anextension of Coq's algorithm to check that the floating universeconstraints are consistent. Then, we propose a new embedding of CTSinto Dedukti and give a soundness proof of this embedding. Theseresults show that Dedukti is suitable for studying interoperability onthe theoretical side.We continue our journey on a case of study: The proof of Fermat'slittle theorem written in Matita. We show how we were able totranslate this proof to various proof systems through Dedukti. Thistranslation mainly relies on two tools created for this purpose:— Dkmeta, a tool which proposes to use rewriting as a way to writeproofs transformation programs. One advantage of this tool is that itreuses the syntax of Dedukti itself.— Universo, a tool which implements the aforementioned algorithm whichallows to translate a proof in one CTS (written in Dedukti) toanother.This semi-automatic translation allows to translate the proof ofFermat's little theorem into a weak but expressive logic calledSTTforall. STTforall is a constructive version of Simple Type Theorywith prenex polymorphism. As a consequence, a proof in STTforall canbe exported easily to many proof systems. This case of study showsthat Dedukti is also suitable for interoperability on the practicalside.The tools used for these transformations could be reused also forproofs coming from other proof systems for which an encoding inDedukti is known (such as Coq or Agda).The journey ends with the exportation of the proof of Fermat's littletheorem encoded in STTforall towards 5 different proof systems: Coq,Lean, Matita, OpenTheory (a member of the HOL-family proof systems)and PVS. We have implemented a user interface for that via a websitecalled Logipedia.Logipedia was designed with the goal of containing many more proofs thatcould be shared between proof systems and as such is intended to be anencyclopedia of formal proofs.
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Submitted on : Thursday, July 15, 2021 - 11:37:04 AM
Last modification on : Friday, August 5, 2022 - 2:58:08 PM

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François Thiré. Interoperability between proof systems using the logical framework Dedukti. Logic in Computer Science [cs.LO]. Université Paris-Saclay, 2020. English. ⟨NNT : 2020UPASG053⟩. ⟨tel-03224039v3⟩

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