C.29.1:4.4 - Determine whether a merged representation defines the answer

Suppose the source question has answer q(x), and you want a receiving function g with:

q(x) = g(F(x)).

Such a g is well-defined on F(X) exactly when all allowed source cases having the same representation give the same answer:

F(x₁) = F(x₂)  implies  q(x₁) = q(x₂).

If g exists, both source answers equal g of the same receiving value. Conversely, if the source answer is the same for every case represented by y, define g(y) to be that common answer. Choosing another representative then changes nothing.

Use this as a construction, not only as a test. First try to rewrite q in terms of the retained variables. If the unwanted variables disappear, derive the resulting g. If they remain, look for two permitted source cases with the same representation and different q values. That pair proves that the requested answer cannot be recovered from that representation alone. One such pair is enough to refute the proposed function. Failure to find a pair is not a proof of independence.

For an update, ask whether cases sharing the current summary have the same required next summary. For a summary that is also to decide availability, compare the operation conditions across those cases. For a prediction after several steps, compare what the permitted continuations can do. The relevant comparison follows the requested result; it does not require preserving every property of the original object.

Distinguish defining a value on a merged class from changing the question. “The cost of this route” requires the chosen route’s cost. “The least cost among these routes” deliberately combines several costs by taking a minimum. The second is a new, potentially useful function; it does not make the first independent of its representative.