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I've found this example of constraint propagation using projection rule enter image description here

We have

C = { x1 ≠ x2, x1 ≥ x2 }

< C; x1 ∈ {1,2,3}, x2 ∈ {1,2,3} >

They say that applying propagation rule, does not give any simplification.

I'm not sure why this is the case. Shouldn't we get?

< C; x1 ∈ {2,3}, x2 ∈ {1,2} >

Other steps in the example, make sense that to me, e.g.

< C; x1 ∈ {2}, x2 ∈ {1,2,3} >

produces

< C; x1 ∈ {2}, x2 ∈ {1} >
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You apply the propagation rule to a single constraint.

The rules are defined in such a way that they can be applied efficiently. Otherwise, by applying the "generalized propagation rule" you could declare that a SAT instance is unsatisfiable in a single step.

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