Proof automation and type synthesis for set theory in the context of TLA+. (Automatisation de preuves et synthèse de types pour la théorie des ensembles dans le contexte de TLA+)

This thesis presents effective techniques for discharging TLA+ proofobligations to automated theorem provers based on unsorted and many-sortedfirst-order logic. TLA+ is a formal language for specifying and verifyingconcurrent and distributed systems. Its non-temporal fragment is based on avariant of Zermelo-Fraenkel set theory for specifying the data structures. TheTLA+ Proof System TLAPS is an interactive proof environment in which users candeductively verify safety properties of TLA+ specifications. While TLAPS is aproof assistant that relies on users for guiding the proof effort, itgenerates proof obligations and passes them to backend verifiers to achieve asatisfactory level of automation.We developed a new back-end prover that soundly integrates into TLAPS externalautomated provers, specifically, ATP systems and SMT solvers. Two maincomponents provide the formal basis for implementing this new back-end. Thefirst is a generic translation framework that allows to plug to TLAPS anyautomated prover supporting the standard input formats TPTP/FOF orSMT-LIB/AUFLIA. In order to encode higher-order expressions, such as sets bycomprehension or total functions with domains, the translation to first-orderlogic relies on term-rewriting techniques coupled with an abstraction method.Sorted theories such as linear integer arithmetic are homomorphically embeddedinto many-sorted logic. The second component is a type synthesis algorithm for(untyped) TLA+ formulas. The algorithm, which is based on constraint solving,implements one type system for elementary types, similar to those ofmany-sorted logic, and an expansion with dependent and refinement types. Theobtained type information is then implicitly exploited to improve thetranslation. Empirical evaluation validates our approach: the ATP/SMT backendsignificantly boosts the proof development in TLAPS.