An anonymous credential system allows a user to convince a service provider anonymously that he/she owns certified attributes. Previously, a system to prove AND and OR relations simultaneously by CNF formulas was proposed. To achieve a constant-size proof of the formula, this system adopts an accumulator that compresses multiple attributes into a single value. However, this system has a problem: the proof generation requires a large computational time in case of lots of OR literals in the formula. One of the example formulas consists of lots of birthdate attributes to prove age. This greatly increases the public parameters correspondent to attributes, which causes a large delay in the accumulator computation due to multiplications of lots of parameters. In this paper, we propose an anonymous credential system with constant-size proofs for monotone formulas on attributes, in order to obtain more efficiency in the proof generation. The monotone formula is a logic formula that contains any combination of AND and OR relations. Our approach to prove the monotone formula is that the accumulator is extended to be adapted to the tree expressing the monotone formula. Since the use of monotone formulas increases the expression capability of the attribute proof, the number of public parameters multiplied in the accumulator is greatly decreased, which impacts the reduction of the proof generation time.