Next we need to show that,

um, starting from a bivalent configuration,

there is always another bivalent configuration that is reachable.

Notice that this proof doesn't say

that you can never reach consensus ever.

It says that there is always some way in which

you can be prevented from reaching consensus.

Let the red configuration be a bivalent configuration,

and let, uh, the event e,

which consists of the process p receiving a message m,

that is the global message buffer in the red configuration,

the sum event that is applicable to the initial configuration,

so m is in the global message buffer in the red configuration.

Now let's put our hand on e and prevent e from being applied.

This means that you might be able to still apply

some other events on the red configuration,

and there are some configurations

that you might be able to reach,

starting from this red configuration.

We call that set to be C, okay?

Those are the blue configurations

that are shown in the triangle.

These are the configurations

that are reached without applying the special event e.

why are we not applying e?

You'll see in a moment, there is a reason for it.

Now, if you take any one of these

blue or the red configurations in the-in the triangle,

and you apply the single event e to it,

the special event e to it,

you will reach another dark blue event.

Let that set of events, the dark blue set of events,

be called as D, 'kay?

Once again, D, any event in,

any configuration D is reached by applying the special event e

on any one of the configurations in the triangle.

Now, this is the summary of what we have discussed.

You have the initial bivalent configuration, the red one.

You don't apply the event e to it, and you reach,

and all the possible states, uh,

that are reachable are in the triangle.

You take one of the configurations in the triangle

and you apply the event e to it,

you'll reach a state that is,

or a configuration that is in the set D.

Okay, so we claim that

the set D contains a bivalent configuration, okay,

and, again, the proof here is by contradiction.

If you can show that the state D contains

a bivalent configuration,

then you can show that

there is a sequence that consists of at least one event

that starts from a bivalent configuration, the red one,

that also leads to another bivalent configuration.

Let's assume the contradiction.

Suppose that D only has 0 and 1-valent contradiction, uh,

configurations and no, uh, bivalent ones.

Okay, So there are these states in D,

are going to be tagged with a 0 or a 1,

and because each stated D has a parent in, uh, C from which,

on which e was applied to obtain that D state,

we also, uh, tag its parent with the corresponding 0 or 1.

Now what you have is you have a C of mixing 0's and 1's here,

and therefore, just by the same argument that we use before,

where we showed that

there has to be at least one bivalent configuration

because at least one 0-valence state and one 1-valence state

are adjacent to each other,

you can show here as well, that in this triangle

there's going to be at least two configurations,

that are adjacent to each other,

because all of them are linked by other events other than e

so that one is tagged with a 0, the other is tagged with a 1.

In other words, it has to be the case that there are states

or configurations D0 and D1, both in D,

and C0 and C1 both in C

such that D0 is 0-valent, D1 is 1-valent.

D0 is obtained from C0 by applying e,

D1 is obtained from

C1 by applying e, and C1 is adjacent to C0,

which means that C0 is, on C0 if you apply some event e'

a special event e', you will obtain C1.

So given this, there are two possibilities.

First, that the process, receiving the message p'

in the special event e' is the same as p,

which is the process receiving the message in event e,

and the second case is that p' is, uh, not the same as p,

'Kay, so let's consider the first case,

p' is not the-the same as p.

from the previous slides,

C0, when you apply e' to it, you get C1.

C0, when you apply e to it, you get D0.

C1, when e is applied to it you get D1.

Since e and e'

have different sets of receiving processes,

these are disjoint sequences,

and so flipping the order in which they are applied,

e' first, followed by e, or e followed by e',

will give you the same final configuration.

In other words, you can draw this red arrow here

and show that you can reach from D0 you can reach D1 as well.

But this is a contradiction, because we said D0 was 0-valent,

but we have just showed that

from D0 you can reach a 1-valent state as well,

which means that in fact D0 is bivalent,

and that is a contradiction.