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, Structured derivations of consensus algorithms for failure detectors Take V ? ? V such that ?v ? V : S(v, G) ? V ? and V ? is minimal According to property (d4) of G, ?k i > 0 such that V (p i , k i ) Since p i is correct it holds that ?k > k i : V (p i , k) ? V (G). Hence, according to property (d3) of G, it holds that ?k > k i : (V (p i, Proceedings of the 17th Annual ACM Symposium on Principles of Distributed Computing (PODC) V (p i , k)) ? E(G). But we showed that ?v ? V ? : (v, V (p i , k i )) ? E(G). By transitivity (d2) of G, we get: ?k ? k i : ?v ? V ? : (v, V (p i , k)) ? E(G), pp.297-306, 1998.
, It follows that: ?k ? k i : ?v ? V : ?x ? S(v, G) : (x, V (p i , k)) ? E(G) This implies according to rule (r3) of Definition D.4 that: ?k ? k i
, Hence (ii) implies that, ). It follows that (ii ? ) ?x ? S(u, Y ) : (x, w) ? E(Y )
, Given S a multiset, let 1 x S denote the multiplicity of x in S. Given u = (p u , stateU ) and v = (p v , stateV ) any two
, X)), for every permutation of processes identities ?, if ?(u), ?(v), )), then (?(u), ?(v)) ? E(M(X))
, Let u = (p u , s u ), v = (p v , s v ) We distinguish between the case in which (i)
, according to rule (r3) of Definition D.4, we have
, Let Y, X two subgraphs of G such that X
, Let X a (closed) subgraph of G. If
, We say that v 2 ? V (X) is U -stable in M(X)