Compositional proofs about systems of many components require circular reasoning principles in which properties of other components need to be assumed in proving the properties of each individual component. A number of such circular assume-guarantee rules have been proposed for different concurrency models and different forms of property specifications. In this paper, we provide a framework that unifies and extends these results. We define an assume-guarantee semantics for properties expressible as least or greatest fixed points, and a circular compositional rule that is sound with respect to this semantics. We demonstrate the utility of this general rule by applying it to trace semantics with linear temporal logic specifications, and trace tree semantics with automata refinement specifications. For traces, we derive a new assume-guarantee rule for the weakly until operator of linear temporal logic and show that previously proposed assume-guarantee rules can be seen as special instances of our rule. For trace trees, we derive a rule for parallel composition of Moore machines, and show that the rule of  is a special instance thus yielding an alternate proof of the results in .