Diophantus' 20th Problem and Fermat's Last Theorem for n=4: Formalization of Fermat's Proofs in the Coq Proof Assistant

We present the proof of Diophantus' 20th problem (book VI of Diophantus' Arithmetica), which consists in wondering if there exist right triangles whose sides may be measured as integers and whose surface may be a square. This problem was negatively solved by Fermat in the 17th century, who used the "wonderful" method (ipse dixit Fermat) of infinite descent. This method, which is, historically, the first use of induction, consists in producing smaller and smaller non-negative integer solutions assuming that one exists; this naturally leads to a reductio ad absurdum reasoning because we are bounded by zero. We describe the formalization of this proof which has been carried out in the Coq proof assistant. Moreover, as a direct and no less historical application, we also provide the proof (by Fermat) of Fermat's last theorem for n=4, as well as the corresponding formalization made in Coq.Comment: 16 page

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

The relationship between andragogical and pedagogical orientation and the implications for adult learning

Current literature suggests that the relationship between andragogy and pedagogy is based on a continuum. This study found that the relationship of andragogical and pedagogical orientations, measured by the Students' Orientation Questionnaire, is more correctly represented as being orthogonal or at right angles to each other. Such an orthogonal relationship reflects the complexities involved in adult learning. This paper discusses the implications for both the learning process and for future research

Integrating Simplex with Tableaux

International audienceWe propose an extension of a tableau-based calculus to deal with linear arithmetic. This extension consists of a smooth integration of arithmetic deductive rules to the basic tableau rules, so that there is a natural interleaving between arithmetic and regular analytic rules. The arithmetic rules rely on the general simplex algorithm to compute solutions for systems over rationals, as well as on the branch and bound method to deal with integer systems. We also describe our implementation in the framework of Zenon, an automated theorem prover that is able to deal with first order logic with equality. This implementation has been provided with a backend verifier that relies on the Coq proof assistant , and which can verify the validity of the generated arithmetic proofs. Finally, we present some experimental results over the arithmetic category of the TPTP library, and problems of program verification coming from the benchmark provided by the BWare project

Distributed Trajectory Flexibility Preservation for Traffic Complexity Mitigation

The growing demand for air travel is increasing the need for mitigation of air traffic congestion and complexity problems, which are already at high levels. At the same time new information and automation technologies are enabling the distribution of tasks and decisions from the service providers to the users of the air traffic system, with potential capacity and cost benefits. This distribution of tasks and decisions raises the concern that independent user actions will decrease the predictability and increase the complexity of the traffic system, hence inhibiting and possibly reversing any potential benefits. In answer to this concern, the authors propose the introduction of decision-making metrics for preserving user trajectory flexibility. The hypothesis is that such metrics will make user actions naturally mitigate traffic complexity. In this paper, the impact of using these metrics on traffic complexity is investigated. The scenarios analyzed include aircraft in en route airspace with each aircraft meeting a required time of arrival in a one-hour time horizon while mitigating the risk of loss of separation with the other aircraft, thus preserving its trajectory flexibility. The experiments showed promising results in that the individual trajectory flexibility preservation induced self-separation and self-organization effects in the overall traffic situation. The effects were quantified using traffic complexity metrics based on Lyapunov exponents and traffic proximity

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

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

Proof Certification in Zenon Modulo: When Achilles Uses Deduction Modulo to Outrun the Tortoise with Shorter Steps

International audienceWe present the certifying part of the Zenon Modulo automated theorem prover, which is an extension of the Zenon tableau-based first order automated theorem prover to deduction modulo. The theory of deduction modulo is an extension of predicate calculus, which allows us to rewrite terms as well as propositions, and which is well suited for proof search in axiomatic theories, as it turns axioms into rewrite rules. In addition, deduction modulo allows Zenon Modulo to compress proofs by making computations implicit in proofs. To certify these proofs, we use Dedukti, an external proof checker for the λΠ-calculus modulo, which can deal natively with proofs in deduction modulo. To assess our approach, we rely on some experimental results obtained on the benchmarks provided by the TPTP library