Coinduction up-to in a fibrational setting

Abstract

Bisimulation up-to enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain up-to techniques not only for bisimilarity but for a large class of coinductive predicates modelled as coalgebras. By tuning the parameters of our framework, we obtain novel techniques for unary predicates and nominal automata, a variant of the GSOS rule format for similarity, and a new categorical treatment of weak bisimilarity.

Proving language inclusion and equivalence by coinduction

“…Recently the theory of bisimulation-up-to was generalized from labelled transition systems to a large class of coalgebras [2,28], yielding enhancements of the bisimulation proof method for many different kinds of state-based systems. In fact, the soundness theorems of bisimulation-up-to can be derived from this general theory.…”

“…In order to show this formally, one would have to show that behavioural differential equations can be represented as so-called abstract GSOS specifications, which are a way of defining operational semantics [34]. Then one obtains by the theory of [2,28] that bisimulation up to congruence for all of these cases is sound, meaning that any bisimulation-up-to can be extended to a bisimulation. For a more abstract view on simulation, we refer to [2], which is a framework for up-to techniques for more general coinductive predicates, including simulation.…”

“…Then one obtains by the theory of [2,28] that bisimulation up to congruence for all of these cases is sound, meaning that any bisimulation-up-to can be extended to a bisimulation. For a more abstract view on simulation, we refer to [2], which is a framework for up-to techniques for more general coinductive predicates, including simulation.…”

A general account of coinduction up-to

“…Previous work This paper is an extended version of [10] and [11]. We extended the previous works with careful explanations and detailed proofs, three motivating examples (Sect.…”

The Proof Technique of Unique Solutions of Contractions

“…'Up-to context' techniques have been formalised in a coalgebraic setting, and adapted to languages whose LTS semantics adheres to the GSOS format [5]; see for instance [6], which uses lambda-bialgebras, a generalisation of GSOS to the categorical framework.…”