We present a chase procedure for solving the implication problem of constrained tuple-generating dependencies (ctgds), a general class of database dependencies that is also able to handle data and predicates on interpreted domains. Current chase procedures for ctgds are sound but not complete, in the sense that they are unguaranteed to stop successfully whenever implication is true. The procedure we present is sound and complete, the first to our knowledge. It follows a linear reasoning over constraint domains that have the Independence of Negative Constraints property.We then soundly extend this procedure by a disjunctive reasoning over unrestricted constraint domains. To achieve these results, we used a different approach. Previous chases act like a closure operator, whereas we used a goal-directed design.