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I came across this definition in my book but I'm not sure that I understand it correctly. enter image description here

Isn't this the same as saying: $\langle S, s\rangle \rightarrow s = \langle S, s\rangle \rightarrow s$ implies that $s=s$? Why do they use a different prime symbol here even though they are the same statement $S$ being executed on the same state $s$.

2nd Question: How does this statement by the author above allow us to "uniquely determine a final state s' if (and only if) the execution of S terminates"

I think I'm missing the point here because this definition came after Induction on the Shape of Derivation Trees.

3rd Question: How will this proof help us perform induction on the shape of derivation trees?

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    $\begingroup$ 1st Equality of a state with itself is implicit, the statement would be silly. $\endgroup$
    – user16034
    May 15, 2023 at 6:16
  • $\begingroup$ 2nd This is a definition, there is nothing "allowed". $\endgroup$
    – user16034
    May 15, 2023 at 6:17
  • $\begingroup$ Please heed How to reference material written by others. $\endgroup$
    – greybeard
    May 15, 2023 at 14:39

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I've come back to this as I solved it but forgot to post my reasoning. The statement above simply states that if we execute a statement S on a state s twice, the resulting state should be the same. Hence, we can "determine" the final result unambiguously.

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