Proving properties on programs accessing data structures such as arrays often requires universally quantified invariants, e.g., "all elements below index i are nonzero". In this article, we propose a general data abstraction scheme operating on Horn formulas, into which we recast previously published abstractions. We show that our instantiation scheme is relatively complete: the generated purely scalar Horn clauses have a solution (inductive invariants) if and only if the original problem has one expressible by the abstraction.