= Proof by minimal counterexample
Assume that a proposed statement fails for some <natural number>, and choose the least failing index using the <well-ordering principle>. Prove the necessary base cases and reduce the least failure to a smaller index that must satisfy the statement. If that smaller case implies the alleged failure also satisfies it, there is a <contradiction>. Reductions by a step larger than one require enough consecutive base cases to cover every possible residue class.
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