@. and ?. Cn, ? k ) such that ? k (q).status = In and ?p ? P F ree (? k ) ? CN t, p.p / ? {q} ? CN q

. Proof, Let e = (? j ) j?0 be an execution of LRA ? T C and let i ? 0 such that T C has stabilized in ? i . Let t ? V be the unique tokenholder in ? i Assume that R(e, i, 6) exists

. Assume, Then, by hypothesis, t.req = ? holds in all configurations between ? R(e,i,2) and ? R(e,i,6) . Moreover, RsT -action is disabled at t in all configurations between ? R(e,i,2) and ? R(e,i,6) , by Claim 1. Hence, t sets t.status to Wait by R-action within two rounds from

L. @bullet-t-c, Let i ? 0 such that T C has stabilized at ? i Assume that R(e, i, 6n(N tok + 1)) exists and R(e, i, 6n(N tok + 1)) ? M (e, i) Similarly to the proof of Lemma 14, P F ree cannot increase, hence if it is empty at some configuration ? k with k ? {R(e, i, 6n(N tok + 1)), . . . , M (e, i)}, we are done, Proof. Let e =)), . . . , M (e, i)}. Assume P F ree (? k ) = ? and let p ? P F ree, p.6

. Lemma-16, L. Algorithm, and T. @bullet-t-c-meets-the, No Livelock property of strong concurrency: there exists a number of rounds T P C > 0 such that for every execution e = (? i ) i?0 and for every index i ? 0

. Proof and T. P. We-pose, N tok + 1) + 4n ? 4 Let e = (? i ) i?0 be an execution of LRA ? T C and let i ? 0 Assume that R(e, i, T P C ) exists and R(e, i, T P C ) ? M (e, i) After T tok rounds, T C has stabilized, Using Lemma, vol.15

T. After, N tok + 1), P F ree is empty and remains so until M (e, i)

?. For-every-k and . {r, i) ? 1}, P assT oken is not executed in step ? k ? ? k+1 . Note that this implies that P assT oken is not executed during the last 6 rounds by the tokenholder t: this allows to apply Lemma 13: there exists a conflicting neighbor of t, q, such that ?p ? P F ree ? CN t, p.p / ? {q} ? CN q

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