CMP.1:4.4 - Follow the direction of the consequence

For an established reduction from A to B:

• A suitable B-solver, together with the reduction, gives a suitable A-solver.
• If no such A-solver can exist under the stated model and guarantee, no B-solver with the assumed capability can exist.

Write the constructed A-procedure before using the second conclusion. It shows what the assumed B-solver would enable and where the contradiction arises.

Keep the scope of a limit. An impossibility for a total decision procedure on an unrestricted input class leaves other questions open: positive-case recognition, bounded execution, a restricted class, or a different computational model. Choose an alternative only when it supplies a useful answer for the work. The finite-state return in :5.2 shows such a change.

For a complexity consequence, use a reduction with the required resource bound. An unboundedly expensive input conversion supplies no efficient A-algorithm merely because B has one.