In this paper, we push forward the approach proposed in [1] aiming at studying semantic interpretation criteria for the purpose of ensuring safety and complexity properties of programs working on streams. The paper improves the previous results by considering global and local upper bounds properties of both theoretical and practical interests guaranteeing that the size of each output stream element is bounded by a function in the maximal size of the input stream elements. Moreover, in contrast to previous studies, these properties also apply to a wide class of stream definitions, that is functions that do not have streams in the input but produce an output stream.