Proof Techniques

Beyond inference rules, we have proof techniques — patterns for building
a complete argument.

Direct proof. To prove $P \Rightarrow Q$: assume $P$, then derive $Q$
through valid inferences.

Proof by contradiction. To prove $P$: assume $\neg P$, derive a contradiction
(some $R \land \neg R$), conclude $P$.

Proof by contrapositive. To prove $P \Rightarrow Q$: prove $\neg Q \Rightarrow \neg P$
instead.

Proof by cases. When the problem splits naturally — e.g. "for all $x$, $P(x)$"
where $x$ is either even or odd — prove each case separately.

Proof by exhaustion. When the domain is small and finite, check every element.

Proof by counterexample. To DISPROVE $\forall x. P(x)$, find one $x$ with $\neg P(x)$.