An equational theory for weak bisimulation via generalized parameterized coinduction

Abstract

Coinductive reasoning about infinitary structures such as streams is widely applicable. However, practical frameworks for developing coinductive proofs and finding reasoning principles that help structure such proofs remain a challenge, especially in the context of machine-checked formalization.This paper gives a novel presentation of an equational theory for reasoning about structures up to weak bisimulation. The theory is both compositional, making it suitable for defining general-purpose lemmas, and also incremental, meaning that the bisimulation can be created interactively.To prove the theory's soundness, this paper also introduces generalized parameterized coinduction, which addresses expressivity problems of earlier works and provides a practical framework for coinductive reasoning. The paper presents the resulting equational theory for streams, but the technique applies to other structures too.All of the results in this paper have been proved in Coq, and the generalized parameterized coinduction framework is available as a Coq library.CCS Concepts • Software and its engineering → Formal software verification; • Theory of computation → Program verification; Logic and verification; Equational logic and rewriting.

Leapfrog: certified equivalence for protocol parsers

“…GPaco [33,57] is a framework for modular coinductive reasoning in Coq, which supports "up-to" bisimilarity techniques. It is designed for interactive use and focuses on automating low-level proof steps.…”

Formally Verified Simulations of State-Rich Processes using Interaction Trees in Isabelle/HOL

“…ITrees [39], and their mechanisation in Coq, have been applied in various projects as a way of defining abstract yet executable semantics [21,40,23,41,42,22,33]. They have been used to verify C programs [21] and a HTTP key-value server [22].…”

Non-well-founded Deduction for Induction and Coinduction

“…Automated provers, to the best of our knowledge, currently do not offer any support for coinduction, and while coinductive data types have been implemented in interactive theorem provers (a.k.a. proof assistants) such as Coq [11,47,83], Nuprl [30], Isabelle [13,81,12,38], Agda [1], Lean [4], and Dafny [54], the treatment of these forms of data is often partial. These formalizations, as well as other formal frameworks that support the combination of induction and coinduction, e.g., [80,61,6,46], generally rely on making (co)invariants explicit within proofs.…”

“…Some notable cyclic systems that do support coinduction in various settings include [67,58,72,36,2]. Another related framework is that of Coq's parameterized coinduction [47,83], which offers a different, but highly related, implicit nature of proofs (based on patterns within parameters, rather than within proof sequents). This paper reviews the general method of non-well-founded proof theory, focusing on its use in capturing both implicit inductive and coinductive reasoning.…”