The typechecking problem for transformations of relational data into tree data is the following: given a TreeQL program P (called transformation), and a DTD d (called output type), decide whether for every database instance D the result of the transformation P of D is of a type consistent with d (see [2]). TreeQL programs with projectionfree conjunctive queries and DTDs with arbitrary regular expressions are considered here. A non-elementary upper bound for the typechecking problem is given in

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... 2] (although in a more general setting, where equality and negation in projection-free conjunctive queries and additional universal integrity constraints are allowed). In this paper we show that the typechecking problem is in coNEXP-TIME. As an intermediate step we consider the following problem, which can be formulated in a language independent of XML notions. Given a set of triples of the form (ϕ, k, j), where ϕ is a projection-free conjunctive query and k, j are natural numbers, decide whether there exists a database D such that for each triple (ϕ, k, j) in the set, there exists a natural number α, such that there are exactly k + j * α tuples satisfying the query ϕ in D. Our main technical contribution consists of a NEXPTIME algorithm for the last problem.