First Move on Every 'If P Then Q' — Write 'Assume P,' Then Unpack the Definition

The biggest obstacle in proof-writing is not the proof — it is the blank page. You see "Prove: if n is even, then n^2 is even," and the cursor blinks and nothing happens. The fix for this paralysis is almost stupidly simple. Before you think about strategy, before you scan for techniques, before you try to spot the clever trick, do two moves:

Write "Assume P."
Stare at P. Ask: "what does P mean by definition?" Write the answer below.

That is your opening gambit for nearly every direct proof you will meet. It is boring, it is mechanical, and it works.

Why it works

A direct proof is, by definition, a chain from P to Q. The first link is P. The second link is some concrete consequence of P. The only way to get from an abstract statement to a concrete consequence is through a definition — that is the bridge from word to algebra.

Every mathematical term in P points to a definition. "n is even" points to the definition of even: n = 2k for some integer k. "f is continuous at a" points to the \varepsilon-\delta definition. "A is a subset of B" points to: every element of A is an element of B. The definition is the raw material the rest of the proof works on.

Why this works almost always: a claim "if P then Q" only has content through the definitions of the terms involved. The logical structure (if... then) is a skeleton; the definitions are the flesh. Until you unpack the definitions, you have no flesh to work with. After you unpack them, you have equations — and equations can be manipulated.

The ritual, step by step

Here is the opening ritual spelled out.

Step A. Read the claim. Identify the hypothesis P and the conclusion Q. Put a mental box around P.

Step B. Write "Assume P." as the first line of your proof. This commits you to the direct-proof approach; if it fails, you will switch methods later. For now, start here.

Step C. Stare at P. Which word or phrase in P is a defined term? If P says "n is divisible by 6," the defined term is divisible by 6. If P says "f is injective," the defined term is injective.

Step D. Write the definition. "n is divisible by 6 means n = 6k for some integer k." This is now the second line of your proof, and it is a concrete statement you can manipulate with algebra.

Step E. Now look at Q. Unpack its definition too. "n is divisible by 3" means "n = 3m for some integer m." The form the conclusion will take is now visible.

Step F. Bridge. You have n = 6k and you want n = 3m for some integer m. Substitute: n = 6k = 3(2k). Let m = 2k; this is an integer; done. \blacksquare

The steps A–D are where the ritual lives. Steps E and F are where the proof assembles itself once the definitions are laid out.

A walking tour of three proofs

Watch the same ritual reveal three different proofs.

Proof 1: if n is odd, then n^2 is odd

Assume P. Assume n is odd.
What does P mean by definition? n = 2k + 1 for some integer k.
Unpack Q. To show n^2 is odd, we need n^2 = 2m + 1 for some integer m.
Bridge. n^2 = (2k+1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1. Let m = 2k^2 + 2k; m \in \mathbb{Z}. So n^2 = 2m + 1. \blacksquare

The ritual did the work. Step 2 (unpack even/odd) is the entire key move.

Proof 2: if a divides b and a divides c, then a divides b + c

Assume P. Assume a \mid b and a \mid c.
What does P mean by definition? b = aq and c = ar for some integers q, r.
Unpack Q. To show a \mid (b+c), we need b + c = a \cdot (\text{integer}).
Bridge. b + c = aq + ar = a(q + r). Since q + r \in \mathbb{Z}, done. \blacksquare

Again: unpacking the definition of divides is the whole move.

Proof 3: if A \subseteq B and B \subseteq C, then A \subseteq C

Assume P. Assume A \subseteq B and B \subseteq C.
What does P mean by definition? Every element of A is an element of B, and every element of B is an element of C.
Unpack Q. To show A \subseteq C, we need: every element of A is an element of C.
Bridge. Let x \in A. Since A \subseteq B, x \in B. Since B \subseteq C, x \in C. So x \in A implies x \in C, i.e., A \subseteq C. \blacksquare

Three completely different claims, one routine. Unpack definitions. Bridge. Done.

A picture of the opening move

Step:

The four-step opening ritual. Read the claim, write *"Assume $P$,"* unpack $P$ by definition, bridge to $Q$. The trick is that steps A and B are cheap, and step C almost always unlocks step D. The blank-page problem disappears.

Why this breaks the blank-page problem

The blank page is intimidating because your brain is trying to do two things at once: figure out the whole proof and write the first line. That is too much cognitive load for most problems. The ritual separates the two. The first line is completely mechanical ("Assume P"). The second line is almost mechanical ("By definition..."). By the time you reach the third line, the problem is no longer abstract — you have concrete algebra in front of you, and concrete algebra is something your brain knows how to push around.

This is the same psychological trick that makes the Feynman technique work, that makes Pólya's "restate the problem" step work, that makes any "just write something down" ritual work. You trade the frozen stare for a mechanical first move, and momentum does the rest.

When the ritual does not finish the proof

Sometimes you complete the first two lines and stare at them and still do not see the bridge. That is fine. The ritual did not fail; it did its job — it gave you a starting platform. From there:

Try unpacking Q as well. Writing out what Q demands often reveals the shape of the bridge.
Try working backwards. Start from Q and ask "what would I need to conclude this?" Meet the forward chain in the middle (see Work Backwards on Scratch Paper).
Consider switching methods. If after unpacking both definitions the direct chain is blocked (especially if Q is a negative statement), consider proof by contrapositive or contradiction.

But the ritual itself — write "Assume P," unpack the definition — almost always gets you off the blank page. Whether that is enough to finish, or whether more moves are needed, depends on the proof. The ritual is the start, not the whole.

The three-word internal mantra

When you sit down to write a direct proof, run three words through your head:

Assume. Unpack. Bridge.

Assume. Write "Assume P."
Unpack. Rewrite P using its definition.
Bridge. Connect the unpacked P to Q via algebra, substitution, or a known theorem.

Three words. They will carry you through roughly three-quarters of the direct proofs you meet at school level, and they will never leave you stuck on the first line.

The short summary

First move on any direct proof: write "Assume P." Then unpack P using its definition.
This breaks the blank-page problem by making the first two lines mechanical.
The ritual works because definitions are the bridge from abstract claims to algebra.
Assume. Unpack. Bridge. — three words that open nearly every direct proof.
If the ritual does not finish the proof, it has still given you a platform — then add backward work or switch techniques.

A common worry: "But this feels too simple. Real proofs must require cleverness." The cleverness, when needed, comes in the bridge step — and even then, most school-level bridges are one substitution, one factoring, or one application of a known theorem. The opening ritual is boring on purpose; that is why it works reliably.