Motivated by satisfiability of constraints with function symbols, we consider numerical inequalities on non-negative integers. The constraints we consider are a conjunction of a linear system Ax = b and a conjunction of non-convex constraints of the form xi ≤ x n j . We show that the satisfiability of these constraints is in NP. As a consequence, we obtain NP completeness for an extension of quantifier-free constraints on sets with cardinalities (QFBAPA) with function images S = f [P n ]. We also present related hardness results and consequences of complexity for dual, convex, constraints.

We show that the satisfiability problem for the quantifierfree theory of product structures with the equicardinality relation is in NP. As an application, we extend the combinatory array logic fragment to handle cardinality constraints. The resulting fragment is independent of the base element and index set theories.

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