  Reachability Logic is a formalism that can be used, among others, for
expressing partial-correctness properties of transition systems. In this paper
we present three proof systems for this formalism, all of which are sound and
complete and inherit the coinductive nature of the logic. The proof systems
differ, however, in several aspects. First, they use induction and coinduction
in different proportions. The second aspect regards compositionality, broadly
meaning their ability to prove simpler formulas on smaller systems, and to
reuse those formulas as lemmas for more complex formulas on larger systems. The
third aspect is the difficulty of their soundness proofs. We show that the more
induction a proof system uses, and the more specialised is its use of
coinduction (with respect to our problem domain), the more compositional the
proof system is, but the more difficult its soundness proof becomes. We also
briefly present mechanisations of these results in the Isabelle/HOL and Coq
proof assistants.
