CIRC is an automated theorem prover based on the circular coinduction principle. The tool is used for the verification of programs, behavioral equivalence checking, and proving properties over infinite data structures. In this paper we present two extensions of CIRC that handle the case when the prover indicates an infinite execution for a certain goal. The first extension involves goal simplification rules and a procedure for checking that the new execution is indeed a proof, while the second one refers to finding and proving a generalization of the goal. Each of the extensions is presented based on a case study: Binary Process Algebra (BPA) for checking the proof correctness and Streams for using generalization.