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In this paper, we present robust stability results for constrained discrete-time nonlinear systems using a finite-horizon model predictive control (MPC) algorithm for which we do not require the terminal cost to have any particular properties. We introduce a definition that attempts to characterize the robustness properties of the MPC optimization problem. We assume the systems under consideration satisfy this definition (for which we give sufficient conditions) and make two further assumptions. These are that the value function is bounded by a Kinfin function of a state measure (related to the distance from the state to some target set) and that this measure is detectable from the stage cost used in the MPC algorithm. We show that these assumptions lead to stability that is robust to sufficiently small disturbances. While in general the results are semiglobal and practical, when the detectability and upper bound assumptions are satisfied with linear Kinfin functions, the stability and robustness are either semiglobal or global (with respect to the feasible set). We discuss algorithms employing terminal inequality constraints and also provide a specific example of an algorithm that employs a terminal equality constraint.