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The distribution of a limited rate of high-pressure gas to gas-lifted wells, while respecting injection bounds and activation precedence constraints, consists of a mixed-integer nonlinear programming problem. This paper proposes a mixed-integer linear formulation obtained by piecewise-linearizing the nonlinear functions, thereby allowing the use of integer programming algorithms. Valid inequalities for the convex hull of feasible solutions are derived from knapsack covers, for which exact and approximate lifting procedures yield stronger inequalities. Numerical results show that these cover-based cuts reduce the number of nodes explored in a branch-and-bound search.