polynomial_time_reductions

To prove NP-hardness using a polynomial-time reduction, define a computable transformation from a known-hard problem like SAT to your target problem, then establish correctness criteria showing YES-instances map correctly.

Yes, you can formalize complexity theory reductions in Lean 4 by using provided templates that offer skeletons for encoding computational problems, defining reductions, and structuring the formal proofs for verification.

A polynomial-time reduction correctness proof requires establishing both forward and backward directions, ensuring that YES-instances of the source problem map correctly to YES-instances of the target problem and vice versa.

To show a computational problem is NP-complete, construct a formal reduction blueprint from a known NP-hard problem to your target problem and verify that the transformation preserves solution existence in both directions.

Yes, defining a computable transformation function from instances of a source problem to instances of a target problem is required, along with proving this transformation executes in polynomial time and preserves answer mappings.