  Building on \'Esik and Kuich's completeness result for finitely weighted
Kleene algebra, we establish relational and language completeness results for
finitely weighted Kleene algebra with tests. Similarly as \'Esik and Kuich, we
assume that the finite semiring of weights is commutative, partially ordered
and zero-bounded, but we also assume that it is integral. We argue that
finitely weighted Kleene algebra with tests is a natural framework for
equational reasoning about weighted programs in cases where an upper bound on
admissible weights is assumed.
