Existence of a 0- and 1-valent configurations in the proof of FLP impossibility result

Question

In the known paper Impossibility of Distributed Consensus with one Faulty Process (JACM85), FLP (Fisher, Lynch and Paterson) proved the surprising result that no completely asyn

Answer 1

Define a mapping f such that f(C) = 0, if e(C) is 0-valent, otherwise, f(C) = 1, if e(C) is 1-valent.

Because e(C) could not be bivalent, if we assume that D has no bivalent configuration, f(C) could only be either 0 or 1.

Arrange accessible configurations from the initial bivalent configuration in a tree, there must be two neighbors C0, C1 in the tree that f(C0) != f(C1). Because, if not, all f(C) are the same, which means that D has only either all 0-valent configurations or all 1-valent configurations.