SQL is by far the most widely used and implemented query language. Yet, on some key features, such as correlated queries and NULL value semantics, many implementations diverge or contain bugs. We leverage recent advances in the formalization of SQL and query compilers to develop DBCert, the first mechanically verified compiler from SQL queries written in a canonical form to imperative code. Building DBCert required several new contributions which are described in this paper. First, we specify and mechanize a complete translation from SQL to the Nested Relational Algebra which can be used for query optimization. Second, we define Imp, a small imperative language sufficient to express SQL and which can target several execution languages including JavaScript. Finally, we develop a mechanized translation from the nested relational algebra to Imp, using the nested relational calculus as an intermediate step.

In multiple domains, large amounts of data are daily generated and combined to be analyzed. The interpretation of these analyses requires to track back the provenance of combined data with respect to initial, raw data. The correctness of the provenance is crucial in many critical domains, such as medicine to prescribe treatments. In this article, we propose the first provenance-aware extended relational algebra formalized in a proof assistant (Coq), for a non trivial subset of database queries: queries containing aggregates, null values, and correlated sub-queries. The formalization is validated by an adequacy proof with respect to standard evaluation of queries. This development is a first step towards a posteriori certification of provenance for data manipulation, with strong guaranties.

In this article, we provide a Coq mechanised, executable, formal semantics for a realistic fragment of SQL consisting of "select [distinct] from where group by having" queries with null values, functions, aggregates, quantifiers and nested potentially correlated sub-queries. Relying on the Coq extraction mechanism to Ocaml, we further produce a Coq certified semantic analyser for a SQL compiler. We then relate this fragment to a Coq formalised (extended) relational algebra that enjoys a bag semantics hence recovering all well-known algebraic equivalences upon which are based most of compilation optimisations. By doing so, we provide the first formally mechanised proof of the equivalence of SQL and extended relational algebra.

We present the rewriting toolkit CiME3. Amongst other original features, this version enjoys two kinds of engines: to handle and discover proofs of various properties of rewriting systems, and to generate Coq scripts from proof traces given in certification problem format in order to certify them with a skeptical proof assistant like Coq. Thus, these features open the way for using CiME3 to add automation to proofs of termination or confluence in a formal development in the Coq proof assistant.

Software engineering, automated reasoning, rule-based programming or specifications often use rewriting systems for which termination, among other properties, may have to be ensured.This paper presents the approach developed in Project A3PAT to discover and moreover certify, with full automation, termination proofs for term rewriting systems.

A novel AC-unification algorithm is presented. A combination technique for regular collapse-free theories is provided along the line developed by A. Boudet et al. (1989). The number of calls to the diophantine equations solver is bounded by the number of AC symbols times the number of shared variables. The rest of the algorithm being linear, this gives a much better idea of how the complexity of AC unification is related to the complexity of solving linear diophantine equations. The termination proof is surprisingly easy. Finally, systems of constraint linear diophantine equations can be solved, rather than one equation at a time, using an algorithm which extends Fortenbacher's algorithm to an arbitrary dimension. This allows a much more efficient use of the constraints than in the standard case.>