Nick70 wrote:I haven't studied this thoroughly yet because I concentrated on double forcing chains. But of course, a single forcing chain should be considered simpler than double forcing chains?

Well well. All work and no play makes Jack a dull boy.

Let's look again at the double forcing chain above.

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`(3,8)=6 => (8,8)<>6 => (8,6)=6 => (8,6)<>9 => (8,2)=9 `

(3,8)=9 => (1,7)<>9 => (1,6)=9 => (8,6)<>9 => (8,2)=9

There is a very simple thing that I overlooked.

If A => B, then (not B) => (not A).

So we can reverse one of the two chains above, e.g. the first one:

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`(8,2)<>9 => (8,6)=9 => (8,6)<>6 => (8,8)=6 => (3,8)<>6`

And of course, since the first terms of the two chains are conjugate, after the reversal we can join the two chains together:

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`(8,2)<>9 => (8,6)=9 => (8,6)<>6 => (8,8)=6 => (3,8)<>6 =>(3,8)=9 => (1,7)<>9 => (1,6)=9 => (8,6)<>9 => (8,2)=9`

This applies to all "double" forcing chains. So every double forcing chain is just a single forcing chain in disguise.

What about the opposite? Is it true as well that every single forcing chain can be split into a double forcing chain?

Yes, it is. Let's look at a sample single forcing chain, e.g.:

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`(9,4)=3 => (3,4)<>3 => (3,9)=3 => (2,7)<>3 => (9,7)=3 => (9,4)<>3`

All we need to split it is a pair of conjugate possibilities. Note that there is no guarantee that the two possibilities in the middle of the chain ((3,9)=3 and (2,7)=3) are conjugate, because the first one is true, and two possibilities are conjugate only if one being false implies the other being true. But we don't have to split the chain in the middle. We can pick the second and third possibilities, which are conjugates, obtaining this double chain:

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`(3,4)=3 => (9,4)<>3`

(3,9)=3 => (2,7)<>3 => (9,7)=3 => (9,4)<>3

Actually, I didn't say the whole truth. I could have split it in the middle anyway:

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`(3,9)<>3 => (3,4)=3 => (9,4)<>3`

(2,7)<>3 => (9,7)=3 => (9,4)<>3

The two first terms are not conjugate, so there is no guarantee that one of them is true, but we don't care! Because the chains start with them being false, and we can be sure that at least one of them will be false - maybe even both of them.

So a single forcing chain or a double forcing chain are just two ways to see the same thing.

The double chain is probably more pleasing to the eye, however the single chain is probably more formally appropriate.