(Co)inductive Proof Systems for Compositional Proofs in Reachability Logic

TL;DR

This paper introduces three coinductive proof systems for Reachability Logic, emphasizing their compositionality, and explores the trade-offs between induction, coinduction, and proof complexity.

Contribution

It presents three novel proof systems for Reachability Logic, analyzing their properties, compositionality, and the complexity of their soundness proofs, with mechanisations in Isabelle/HOL and Coq.

Findings

More induction increases compositionality

Specialised coinduction enhances proof reuse

Complexity of soundness proofs varies with proof system design

Abstract

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.