In first-order logic, forward search using a complete strategy such as the inverse method can get stuck deriving larger and larger consequence sets when the goal query is unprovable. This is the case even in trivial theories where backward search strategies such as tableaux methods will fail finitely. We propose a general mechanism for bounding the consequence sets by means of finite approximations of infinite types. If the inverse method also implements forward subsumption and globalization, then the search space under this approximation is finite. We therefore obtain a type-directed iterative refinement algorithm for disproving queries. The method has been implemented for intuitionistic first-order logic, and we discuss its performance on a variety of problems.