Advanced Techniques Reference

Intro to Proofs · Axiom Academy

FORMULA SHEET Advanced Proof Techniques Reference for well-ordering, extremal arguments, invariants, and infinite descent Assume property P fails for some n By WOP, S has a smallest element m Derive a contradiction from the minimality of m Identify the operations / transformations Find a quantity Q that stays constant Show the initial and target states have different Q Conclude the target is unreachable Assume a solution exists in positive integers Choose one minimal in some measure (smallest sum, smallest max element) From this minimal solution, construct a strictly smaller one This contradicts minimality no solution exists

This is the written version of the interactive lesson above. See the full Intro to Proofs course.