Common use of Struct Clause in Contracts

Struct. in Comp. Science (2017), vol. 27, pp. 1236–1264. ×c Cambridge University Press 2015 doi:10.1017/▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇ First published online 9 December 2015 Enhanced coalgebraic bisimulation ▇▇▇▇▇▇▇▇ ▇▇▇†, ▇▇▇▇▇▇▇ ▇▇▇▇▇ ▇‡, M ▇▇▇▇▇▇▇ ▇▇▇▇▇▇▇▇ E§,¶, DAMIEN P OU S‡, ▇▇▇ ▇▇▇▇▇ N¶,∗ and ▇▇▇▇▇▇▇▇▇ S ILV A†† †Universit´e ▇▇ ▇▇▇▇, CNRS, ENS ▇▇ ▇▇▇▇, UCBL, LIP, 46 All´ee d’Italie, 69364 Lyon, France Email: ▇▇▇▇▇▇▇▇.▇▇▇@▇▇▇-▇▇▇▇.▇▇ ‡CNRS, Plume team, LIP (UMR 5668, ENS ▇▇ ▇▇▇▇, UCBL, Universit´e ▇▇ ▇▇▇▇), 46 All´ee §LIACS - Leiden University, ▇▇▇▇▇ ▇▇▇▇▇▇▇ ▇, ▇▇▇▇▇▇, ▇▇▇ ▇▇▇▇▇▇▇▇▇▇▇ Email: ▇.▇.▇▇▇▇▇▇▇▇▇@▇▇▇▇▇.▇▇▇▇▇▇▇▇▇▇.▇▇ ¶Centrum Wiskunde en Informatica (CWI), Science Park 123, 1098 XG Amsterdam, The ∗Radboud University Nijmegen, Toernooiveld ▇▇▇, ▇▇▇▇ ▇▇ ▇▇▇▇▇▇▇▇, ▇▇▇ ▇▇▇▇▇▇▇▇▇▇▇ ††University College London, ▇▇▇▇▇ Street, London WC1E 6BT, U.K. We present a systematic study of bisimulation-up-to techniques for coalgebras. This enhances the bisimulation proof method for a large class of state based systems, including labelled transition systems but also stream systems and weighted automata. Our approach allows for compositional reasoning about the soundness of enhancements. Applications include the soundness of bisimulation up to bisimilarity, up to equivalence and up to congruence. All in all, this gives a powerful and modular framework for simplified coinductive proofs of equivalence.