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An executed purpose, in short, is a transaction in which the time and energy spent on the execution are balanced against the resulting assets, and the ideal case is one in which the former approximates to zero and the latter to infinity.

I think since it is the ideal case, it should use the one instead of one, is that so?

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It depends on the intent behind choosing the "ideal case".

the ideal case is one in which the former approximates to zero

indicates the ideal case is one of several possible cases, without any implied preference. There could be more than one case meeting your criteria.

the ideal case is the one in which the former approximates to zero

indicates that you are specifically pointing out this case as the ideal one. Also it would be the only case meeting your criteria.

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