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 proving 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 is its soundness proof.

We present formalisations of these results in the Coq proof assistant. In particular we have developed support for coinductive proofs that is comparable to that provided by Coq for inductive proofs. This may be of interest to a broader class of Coq users.

可到达性逻辑是一种形式主义，除其他外，可用于表达过渡系统的部分正确性。在本文中，我们为这种形式主义提出了三个证明系统，所有这些系统都是健全而完整的，并继承了逻辑的共性。但是，证明系统在几个方面有所不同。首先，他们以不同的比例使用归纳法和共归法。第二个方面涉及组成性，广义上讲是指它们有能力在较小的系统上证明较简单的公式，并将这些公式用作引理，以在较大的系统上证明更复杂的公式。第三方面是其稳健性证明的难度。

我们表明，证明系统使用的归纳方法越多，并且其对共归的使用（就我们的问题领域而言）越专业，证明系统的组成越多，但其健全性证明就越困难。

我们在Coq证明助手中提供了这些结果的形式化描述。特别是，我们开发了对共归证明的支持，该支持与Coq为归纳证明提供的支持相当。广大的Coq用户可能对此很感兴趣。