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[tlaplus] Symmetry sets

Hello, TLA enthusiasts,

I'm having trouble understanding when it is safe to use symmetry sets.

This documentation at https://tla.msr-inria.inria.fr/tlatoolbox/doc/model/model-values.html says:

> The expression {{v1, v2}, {v1, v3}, {v2, v3}} is symmetric for the set {v1, v2, v3} -- for example, interchanging v1 and v3 in this expression produces {{v3, v2}, {v3, v1}, {v2, v1}}, which is equal to the original expression.

then it says:

> `CHOOSE x \in {v1, v2, v3} : TRUE` is not symmetric for {v1, v2, v3}. 

I cannot reconcile these two statements.  Suppose the original CHOOSE expression gives v1.  After the same substitution as in the first quote, the CHOOSE expression gives v3.  Is that not "equal to the original expression", in the same way that the tuples above are equal?

What is unique about CHOOSE that it is "the only TLA+ operator that can produce a non-symmetric expression”?

Thanks for any insights.


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