Same Variable in P and Q — Substitution and Algebra Is Usually the Whole Proof

Many introductory direct proofs look intimidating on first read and turn out to be three lines of algebra once you see the trick. The trick is a pattern, and once you learn to spot it the proof almost writes itself.

The pattern is this: when the hypothesis and the conclusion reference the same variable, the proof is almost always "unpack the hypothesis into an equation for that variable, substitute into the conclusion, simplify." No clever rearrangement, no case split, no change of technique. The structure is mechanical.

Learning to recognise this shape at a glance — in two seconds, before you start writing — is the difference between proofs that take you twenty minutes and proofs that take you ninety seconds.

What the shape looks like

Hold a candidate claim up against this mould.

If P(n), then Q(n) — where both P and Q are statements about the same n.

Concrete examples:

If n is even, then n^2 is even. (Both sides about n.)
If n is divisible by 6, then n is divisible by 3. (Both sides about n.)
If x > 0, then x^2 > 0. (Both sides about x.)
If a \mid b, then a \mid 5b. (Both sides about the pair (a, b), but the pair plays the role of "the same variable.")

Contrast with claims where the variables are different across sides:

If a \mid b and b \mid c, then a \mid c. (Three variables, chain of relations.)
If f is continuous and g is continuous, then f + g is continuous. (Three objects: f, g, and the new object f+g.)

These non-matching cases still have direct proofs, but they need a slightly different workflow: combine multiple hypotheses, track the relations between objects, and build the conclusion. The matching-variable case is the simpler one, and it deserves its own two-second recognition move.

The three-step recipe

For a claim of shape "if P(n), then Q(n)":

Step 1 — Unpack P(n) using a definition. "n is even" becomes n = 2k. "n is divisible by 6" becomes n = 6m. Your hypothesis is now an equation you can substitute.

Step 2 — Substitute the equation into Q(n). Wherever the conclusion mentions n, replace it with the right-hand side of the hypothesis equation.

Step 3 — Simplify and repack using the definition needed by Q(n). The algebra should reveal the form Q(n) demands — a factor of 2 for even, a factor of 3 for divisibility by 3, a square for "is a perfect square," and so on.

Why this works: the hypothesis equation n = 2k (or whatever) is a substitution rule — it says "anywhere you see n, you can write 2k instead." Step 2 is just this rule in action. Step 3 is algebra plus the conclusion's definition, which converts the rearranged expression back into the form the conclusion asks for. There is no creative leap; the definitions and the substitution do all the work.

The pattern, visualised

The pipeline that handles the majority of same-variable direct proofs. Step 1 converts the hypothesis into a substitution rule. Step 2 plugs it into the conclusion expression. Step 3 rearranges the algebra into the form the conclusion's definition asks for. No other moves are required.

A template proof to memorise

Claim. If n is even, then n^2 is even.

Proof.

Unpack. n even means n = 2k for some integer k.
Substitute. n^2 = (2k)^2 = 4k^2.
Repack. 4k^2 = 2 \cdot (2k^2), and 2k^2 is an integer, so n^2 is even. \blacksquare

Three lines, one per step. Most same-variable direct proofs have exactly this silhouette.

A longer example with the same skeleton

Claim. If n is divisible by 6, then n is divisible by 3.

Unpack. n = 6k for some integer k.

Substitute. Check whether n fits the "divisible by 3" form: n = 6k = 3 \cdot (2k).

Repack. 2k is an integer, so n = 3m with m = 2k, and n is divisible by 3. \blacksquare

The only move that looks "creative" is rewriting 6k as 3 \cdot 2k. That is not creativity — that is recognising which factor the conclusion needs (3) and exposing it in the algebra. Practising with a dozen of these will make the re-exposure automatic.

The failure mode to avoid

Students who have not internalised this pattern often try to prove same-variable claims by case analysis, example checking, or induction — all of which work but take ten times longer than the substitution template. Worse, they sometimes freeze because the opening "unpack" step looks trivial and they assume they are "missing something." They are not. The substitution is the whole proof.

If you find yourself five minutes into a same-variable direct proof with no algebra on the page, stop. Go back to step 1 and ask: what does the hypothesis mean, written as an equation? Then substitute. Then simplify. If the recipe does not yield the conclusion in three lines, that is the signal that the claim is not a same-variable direct proof in the easy sense — and you can start thinking about whether to use contrapositive, contradiction, or a multi-object argument. But the default is the three-line recipe, and the default succeeds most of the time.

The two-second tell

When you read "If P, then Q" in a homework problem, check:

Do P and Q name the same object (one variable, or one tuple of variables acting as a unit)?
Does P admit a definition that converts it to an equation for that object?

Two yeses means: write the unpacking equation, substitute into the conclusion, simplify. Do not reach for a more elaborate technique until the template has been given a chance and failed.

Mastering this single pattern — same variable, substitute, simplify — is the fastest route from "proofs feel impossible" to "proofs feel mechanical" in your first algebra-heavy proof course.