Colour-Coded Direct Proof Template — Hypothesis Yellow, Definitions Green, Algebra Blue, Conclusion Red

Every direct proof, no matter how long, is built from just four types of sentences. Hypothesis sentences restate what you are given. Definition-substitution sentences swap named concepts for their algebraic meanings. Algebraic-move sentences manipulate those expressions. And conclusion sentences close the loop by matching the target definition. Colour-coding each sentence type makes the scaffolding of the proof visible — it is the skeleton of every proof you will write at school, JEE, and early college level.

The four colours and their jobs

Theorem: n odd ⇒ n² odd a even ⇒ 4 ∣ a² 3∣n and 3∣m ⇒ 3 ∣ n+m A⊆B, B⊆C ⇒ A⊆C r, s rational ⇒ r+s rational ◀ ▶ step:

Yellow (hypothesis). This sentence exists to remind you and your reader what you are allowed to assume. It looks trivial and often is — "Let n be odd," "Suppose a and b are rationals," "Assume f is differentiable at x_0." Skip the yellow sentence and your reader has no idea what you are proving.

Green (definition substitution). This is where the work starts. Every named concept in the yellow sentence is swapped for its algebraic or set-theoretic definition. Odd becomes 2k + 1. Rational becomes p/q with integer p, q and q \ne 0. Differentiable becomes \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} exists. Without green, nothing to compute.

Blue (algebraic moves). The blue sentences are the middle of the proof. Here you expand, factor, substitute, apply known theorems, combine like terms. Each blue sentence should be derivable from the previous one by a rule your reader already accepts — nothing controversial, nothing new, just arithmetic and the ambient axioms. A proof typically has between one and a dozen blue sentences depending on the complexity of the manipulation.

Red (conclusion). The last sentence names any new integer or object you introduced, states that the target expression matches the target's definition, and declares the proof complete. A red sentence usually says "Let m = \ldots Then n^2 = 2m + 1, so n^2 is odd. \square" The red sentence makes the implicit jump from "I reached this form" to "therefore the thing I was trying to prove is true." Do not omit it.

Why four categories is the right count: a direct proof proves P \Rightarrow Q. It has to start with P (yellow), unpack P's meaning (green), transform the meaning using legal moves (blue), and recognise Q's meaning in the result (red). Any fewer categories and the proof is missing a structural part; any more and you are subdividing moves that function the same way.

Reading existing proofs through the colour lens

Pick any proof from your textbook and mentally colour each sentence. Three things happen:

1. The skeleton appears. You immediately see the proof is five-to-ten sentences organised by colour bands: yellow → green → blue × several → red.
2. Weak proofs stand out. A proof that jumps from yellow straight to red, or that has green sentences with no blue follow-up, is incomplete. The missing colours are exactly the missing steps.
3. Long proofs become manageable. A forty-line proof is not forty equally-mysterious lines — it is typically one yellow sentence, two green sentences, thirty-five blue sentences doing serious algebra, and two red sentences. The hard part is only in the blue band.

Writing proofs through the colour lens

When composing a proof, the colours are a checklist:

1. Write a yellow sentence that restates the hypothesis in your own words. Do not skip — it anchors the reader.
2. Write green sentences that replace every defined term with its explicit form. Variables you introduce here ("let k be an integer with n = 2k + 1") carry through the rest of the proof.
3. Write blue sentences to carry the expressions from the green state toward the target form. Each blue sentence should be justified by a single legal move — not a chain of several.
4. Write a red sentence that names any new variable you need, states the target equation explicitly, and concludes with the property you were asked to prove. Close with \square or "done" so the reader knows the proof has ended.

What each colour looks like in three domains

The template survives the move from arithmetic into algebra, analysis, and number theory. Here are the four-colour sketches for three flavours of direct proof:

• Divisibility. Yellow: "Suppose 3 \mid n and 3 \mid m." Green: "Then n = 3p and m = 3q for integers p, q." Blue: "So n + m = 3p + 3q = 3(p + q)." Red: "Let r = p + q. Then n + m = 3r, so 3 \mid n + m. \square"
• Set theory. Yellow: "Let A \subseteq B and B \subseteq C." Green: "Take any x \in A." Blue: "Since A \subseteq B, x \in B. Since B \subseteq C, x \in C." Red: "Thus every x \in A is in C, so A \subseteq C. \square"
• Rationals. Yellow: "Suppose r and s are rational." Green: "Write r = a/b and s = c/d with a, b, c, d integers and b, d \ne 0." Blue: "Then r + s = ad/(bd) + bc/(bd) = (ad + bc)/(bd)." Red: "ad + bc and bd are integers, and bd \ne 0, so r + s is rational. \square"

The palette is identical every time. What changes is the meaning of each sentence — not its role.

The shortest useful proofs have all four colours

You will sometimes write proofs that are just two sentences. Even then, if you look closely, each sentence is doing the job of several colours at once:

• "Since n is odd, n = 2k + 1; then n^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1 is odd."

That single sentence compresses yellow ("since n is odd"), green ("n = 2k + 1"), blue ("n^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1"), and red ("is odd"). Compression is fine once you are practised. The colour categories are still there — just bundled.

Related: Mathematical Proof — Direct Proof · Direct Proof as a Logical Pipeline · Logic and Propositions · Proof by Contrapositive