Every line of a direct proof has a justification: the axiom, definition, or previously proved theorem that licenses the step. Strip those justifications out and the proof is a list of assertions. Put them back in and the proof becomes a chain — each link anchored to a specific rule that explains why the next link follows.
The forward-chaining tree below makes the chain visible. You step through the five lines of the proof "even + even = even" and, at each node, the justification flips into view. By the time you reach the leaf, the tree contains the whole scaffolding: assumption → definition → algebra → closure → definition → conclusion.
The walkable proof tree
The justification menu
Every step in a direct proof is licensed by exactly one of these categories. Memorise the shortlist and your proof reading speeds up tenfold.
- Hypothesis. The starting assumption given in the problem statement. "Suppose a and b are even." No further justification needed — you are simply granting the premise.
- Definition. Used in two directions.
- Forwards: unpack the word into its algebraic content. "a is even, so a = 2k."
- Backwards (reverse-direction): pack an algebraic expression into the word. "a + b = 2 \cdot (\text{integer}), so a + b is even."
- Axiom / basic property. Closure of integers under addition and multiplication, commutativity of addition, distributive law, order axioms. The ground floor of arithmetic.
- Previously proved theorem. Any result already in your toolbox. "By the division algorithm..." "By the Pythagorean theorem..."
- Algebraic rearrangement. Factoring, expanding, regrouping. These are consequences of the axioms but deserve their own category because most proof steps are algebraic.
Why labelling justifications matters: a proof without justifications is a sequence of claims. A proof with justifications is a deduction — a chain of steps where each is validated by a named rule. When reading someone else's proof, finding the label lets you challenge it ("is that really what the definition says?") or accept it ("yes, that is the distributive law"). When writing your own, forcing yourself to label each step catches the missing bridge before it turns into an error.
A second tree — odd times odd is odd
Claim. If n and m are odd, then nm is odd.
The tree has the same five-node shape.
- Hypothesis. n and m are odd.
- Definition (forwards). n = 2k + 1, m = 2\ell + 1 for integers k, \ell.
- Algebra. nm = (2k+1)(2\ell+1) = 4k\ell + 2k + 2\ell + 1 = 2(2k\ell + k + \ell) + 1.
- Closure. 2k\ell + k + \ell is an integer (integers are closed under + and \times).
- Definition (backwards). nm = 2 \cdot (\text{integer}) + 1, so nm is odd. \blacksquare
Same scaffolding — hypothesis, definition-forwards, algebra, closure, definition-backwards. Once you have walked one tree, every even/odd proof feels like a familiar shape.
What the tree reveals about structure
The tree is not just decoration — it reveals the structural symmetry of direct proofs:
- Entry (node 1): hypothesis.
- Unpack (node 2): use the definition to turn words into algebra.
- Work (node 3): rearrange algebraically until the target form appears.
- Verify (node 4): check the intermediate objects are of the right kind (integers).
- Exit (node 5): use the definition in reverse to turn algebra back into words.
Almost every direct proof about evenness, divisibility, or rationality has this structure. The content changes; the shape does not.
How to label your own proof
When you write a direct proof, add a one-word justification in square brackets or a marginal note for each line. "By definition of even," "By distributive law," "By closure of integers." This is not optional decoration — it is the check that each step is actually licensed by something.
Two failure modes the labelling catches:
- Missing bridge. You write "a = 2k, so a + b = 2(k+m)" — but you skipped defining b = 2m. The labels force you to include it.
- Wrong justification. You write "a + b is even by algebra" — but algebra only tells you a + b = 2(k+m), not that this means "even." The definition, applied in reverse, is what gives you "even." Mislabelling surfaces the gap.
Labels are the compass of proof-writing. They keep you honest.
From tree to prose
Once the tree is built, the prose writes itself. Each node becomes one sentence; each edge becomes a connective like "therefore," "so," or "by the definition of even." The prose version of the first tree is exactly the five-line proof from 100 Examples vs One Direct Proof. The tree is just the prose with the justifications pulled out and made clickable.
Related: Mathematical Proof — Direct Proof · 100 Examples Checked vs One Direct Proof · Logic and Propositions · Proof by Contrapositive