C. ?. There, R. Such, C. ?. Such, R. ??, and R. ?. , Since P is closed it means that it does not communicate with any context C[·] it is inserted in means that reductions have occurred either in the catalyser C or in P, separately. Let now Since P is lock-free, by definition P ? * P i and for some n ? i, P i ? * P n and P n has both action and co-action on some channels

C. [. , R. Such, C. ?. , and R. ??, In particular, this holds also for the empty catalyser Here we can assume, without any loss of generality (the other cases are trivial) that R is an input or an output process. Furthermore, we know that there existy} R ? for some x and y such that (?xy) is a restriction in Since P is closed, P i is also closed and this means that it does not communicate with any catalyser it is inserted in means that reductions have occurred either in the catalyser C ? or in E[R], separately, Notice that R is part of E[R] ? P i , and since R occurs in the redex E ? [R][R ? ] together with its counterpart R ? , it means that P i ? * P n where P n is a subprocess of E ? [R][R ? ], and the communication occurs over (?xy). We conclude by applying the definition of lock-freedom. It follows as a corollary from Theorems 14.3.1 and 14.3.2 that the lockfreedom and progress properties coincide for closed terms

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