ME.25:4.3 - Perform the transformation and derive its scope

Apply the mathematical construction with its conditions. A transformation may preserve an answer, establish a one-way refinement, give a bounded approximation or construct a new family of operations. State which of these conclusions is obtained.

For a preservation claim, compare the original and transformed constructions over the admitted inputs. If R extracts the required result from a construction’s outcome, the condition may be:

R(new(x)) = R(old(x)) for every admitted x.

The equality concerns that result. Expand R or the compared behavior when intermediate responses, errors or resource interactions also matter. For a bounded approximation, derive the bound and the inputs on which it holds. For a new capability, establish what the new construction can produce and which old requirements it still satisfies.

Derive the gain separately. A reduction in the number of operations can imply less work under a supplied cost model. It may leave elapsed time unchanged, or increase storage and communication. Use the existing characterization and comparison methods when those trade-offs affect the choice.

A familiar algebraic rule is usable when its assumptions fit. If a proposed rewrite fails, return the failed condition; do not present the transformed expression as a viable candidate. Consider another construction when a promising alternative remains.