The paper presents the results of a comprehensive case study which examined to what extent the semantics of particle verbs with auf can be understood as composed from the semantics of their parts. We give substance to our leading hypothesis that the semantics of particle constructions can be reconstructed as rule-based by means of semantics construction algorithms in a DRT-based framework. The compositional process is displayed through merging the semantic representations of the verbal root and the particle. Composition, as it is made operative in the DRT-framework, is akin to processes of presupposition justification as familiar from Dynamic Semantics. This method is shown appropriate for modelling the wide spectrum of compositional mechanisms which have to be taken into account in the formal semantics of particle verbs. The ambiguity of the particle auf can be reconstructed as restricted to only a few core meanings, some of which are familiar from the literature. We present lexical entries for the core meanings, apply them in the semantics construction and discuss sub-cases as well as metaphorical extensions.