Direct Proof

A rigorous guide to direct proofs built from definitions, hypotheses, and valid forward implications.

Claim → evidence → conclusion. A claim or task is translated into assumptions or inputs and a target or output, supported by a finite chain of valid steps, and closed by reconnecting the result to the original goal.

Precise definition

A direct proof of an implication $P\Rightarrow Q$ assumes an arbitrary object satisfying $P$ and derives $Q$ through definitions, established results, and valid algebra or logic. Because the object is arbitrary within the stated domain, the argument covers every permitted case rather than selected examples.

Notation and mathematical language

Write $P(x)$ for the hypothesis and $Q(x)$ for the conclusion. Definitions translate words into usable form: an integer $n$ is even iff $n=2k$ for some $k\in\mathbb Z$, and $a\mid b$ iff $b=ak$ for some integer $k$. Existential witnesses must belong to the required domain.

Conceptual picture

A direct proof is a forward chain: unpack the hypothesis, preserve true statements, and reshape them until the conclusion's definition is visible. Planning often works backward from the conclusion—asking what witness would suffice—but the written proof should make the forward justification clear.

Fully worked example

Interpretation and application

Direct proof is the default language of theorem verification in algebra, number theory, and discrete structures. It also improves program specifications: if input conditions imply an invariant and the invariant plus termination implies the postcondition, the same forward logic establishes correctness.