Automated Deduction in the B Set Theory using Typed Proof Search and Deduction Modulo

International audienceWe introduce an encoding of the set theory of the B method using polymorphic types and deduction modulo, which is used for the automated verication of proof obligations in the framework of theBWare project. Deduction modulo is an extension of predicate calculus with rewriting both on terms and propositions. It is well suited for proof search in theories because it turns many axioms into rewrite rules. We also present the associated automated theorem prover Zenon Modulo, an extension of Zenon to polymorphic types and deduction modulo, along with its backend to the Dedukti universal proof checker, which also relies on types and deduction modulo, and which allows us to verify the proofs produced by Zenon Modulo. Finally, we assess our approach over the proof obligation benchmark of BWare

Checking Zenon Modulo Proofs in Dedukti

Dedukti has been proposed as a universal proof checker. It is a logical framework based on the lambda Pi calculus modulo that is used as a backend to verify proofs coming from theorem provers, especially those implementing some form of rewriting. We present a shallow embedding into Dedukti of proofs produced by Zenon Modulo, an extension of the tableau-based first-order theorem prover Zenon to deduction modulo and typing. Zenon Modulo is applied to the verification of programs in both academic and industrial projects. The purpose of our embedding is to increase the confidence in automatically generated proofs by separating untrusted proof search from trusted proof verification.Comment: In Proceedings PxTP 2015, arXiv:1507.0837

Deduction modulo theory

This paper is a survey on Deduction modulo theor

Tableaux Modulo Theories Using Superdeduction

We propose a method that allows us to develop tableaux modulo theories using the principles of superdeduction, among which the theory is used to enrich the deduction system with new deduction rules. This method is presented in the framework of the Zenon automated theorem prover, and is applied to the set theory of the B method. This allows us to provide another prover to Atelier B, which can be used to verify B proof rules in particular. We also propose some benchmarks, in which this prover is able to automatically verify a part of the rules coming from the database maintained by Siemens IC-MOL. Finally, we describe another extension of Zenon with superdeduction, which is able to deal with any first order theory, and provide a benchmark coming from the TPTP library, which contains a large set of first order problems.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0117

Semantic A-translation and Super-consistency entail Classical Cut Elimination

We show that if a theory R defined by a rewrite system is super-consistent, the classical sequent calculus modulo R enjoys the cut elimination property, which was an open question. For such theories it was already known that proofs strongly normalize in natural deduction modulo R, and that cut elimination holds in the intuitionistic sequent calculus modulo R. We first define a syntactic and a semantic version of Friedman's A-translation, showing that it preserves the structure of pseudo-Heyting algebra, our semantic framework. Then we relate the interpretation of a theory in the A-translated algebra and its A-translation in the original algebra. This allows to show the stability of the super-consistency criterion and the cut elimination theorem

SMT Solving Modulo Tableau and Rewriting Theories

International audienceWe propose an automated theorem prover that combines an SMT solver with tableau calculus and rewriting. Tableau inference rules are used to unfold propositional content into clauses while atomic formulas are handled using satisfiability decision procedures as in traditional SMT solvers. To deal with quantified first order formulas, we use metavariables and perform rigid unification modulo equalities and rewriting, for which we introduce an algorithm based on superposition, but where all clauses contain a single atomic formula. Rewriting is introduced along the lines of deduction modulo theory, where axioms are turned into rewrite rules over both terms and propositions. Finally, we assess our approach over a benchmark of problems in the set theory of the B method

Dedukti: a Logical Framework based on the λ\lambdaΠ\Pi-Calculus Modulo Theory

Dedukti is a Logical Framework based on the $\lambda$$\Pi$-Calculus Modulo Theory. We show that many theories can be expressed in Dedukti: constructive and classical predicate logic, Simple type theory, programming languages, Pure type systems, the Calculus of inductive constructions with universes, etc. and that permits to used it to check large libraries of proofs developed in other proof systems: Zenon, iProver, FoCaLiZe, HOL Light, and Matita

Automated theorem proving in first-order logic modulo: on the difference between type theory and set theory

Resolution modulo is a first-order theorem proving method that can be applied both to first-order presentations of simple type theory (also called higher-order logic) and to set theory. When it is applied to some first-order presentations of type theory, it simulates exactly higherorder resolution. In this note, we compare how it behaves on type theory and on set theory