We study proof techniques for bisimilarity based on unique solution of equations. We draw inspiration from a result by Roscoe in the denotational setting of CSP and for failure semantics, essentially stating that an equation (or a system of equations) whose infinite unfolding never produces a divergence has the unique-solution property. We transport this result onto the operational setting of CCS and for bisimilarity. We then exploit the operational approach to: refine the theorem, distinguishing between different forms of divergence; derive an abstract formulation of the theorems, on generic LTSs; adapt the theorems to other equivalences such as trace equivalence, and to preorders such as trace inclusion. We compare the resulting techniques to enhancements of the bisimulation proof method (the `up-to techniques). Finally, we study the theorems in name-passing calculi such as the asynchronous $pi$-calculus, and use them to revisit the completeness part of the proof of full abstraction of Milners encoding of the $lambda$-calculus into the $pi$-calculus for Levy-Longo Trees.
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