# Lecture 18: summary of proof techniques

• Reading: fa15 notes

• Systematic review of proof techniques
• Suggestions for when you get stuck on a proof

• statements to be proven often suggest a proof structure. For example, a proof that $$∀ x, P(x)$$ often starts with "choose an arbitrary $$x$$." See the fa15 notes for a systematic list of proof structures.
Induction principle: $$P(0)$$ and $$∀n, (P(n) \implies P(n+1))$$ together imply $$∀n, P(n)$$.
Breaking this down by the rules linked above, to use this prove $$∀n, P(n)$$, you can first prove ($$P(0)$$ and $$∀n, (P(n) \implies P(n+1))$$. Proving $$P(0)$$ is of course the base case, while proving $$∀n, (P(n)\implies P(n+1))$$ gives the inductive step. How does one prove $$∀n, (P(n) \implies P(n+1))$$? By the table linked above, you start by choosing an arbitrary $$n$$. Then, you are trying to prove $$P(n) \implies P(n+1)$$ for that $$n$$; so you assume $$P(n)$$, and your goal becomes showing that $$P(n+1)$$ holds.