  We present sound and complete environmental bisimilarities for a variant of
Dybvig et al.'s calculus of multi-prompted delimited-control operators with
dynamic prompt generation. The reasoning principles that we obtain generalize
and advance the existing techniques for establishing program equivalence in
calculi with single-prompted delimited control. The basic theory that we
develop is presented using Madiot et al.'s framework that allows for smooth
integration and composition of up-to techniques facilitating bisimulation
proofs. We also generalize the framework in order to express environmental
bisimulations that support equivalence proofs of evaluation contexts
representing continuations. This change leads to a novel and powerful up-to
technique enhancing bisimulation proofs in the presence of control operators.
