Direct Proof as a Logical Pipeline — Hypothesis Pellets Become Conclusion Pellets

A direct proof is not a magic trick — it is an assembly line. The hypothesis enters on one end, gets processed by a sequence of logically licensed steps, and exits on the other end as the conclusion. If any stage blocks or leaks, the proof fails. The animation below shows the "pellets" — tokens of information — flowing through each stage of a classical direct proof: "If n is odd, then n^2 is odd."

The pipeline you are about to see

Four stages. Each one has a rule of its own.

Unpack definitions. Replace every defined term with what it actually means. (n odd → n = 2k + 1.)
Apply algebra. Perform legal manipulations on the expressions that the definitions gave you. (n^2 = (2k+1)^2 = 4k^2 + 4k + 1.)
Factor toward the target. Rewrite the expression so the target definition becomes visible. (4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1.)
Re-pack into the target definition. Set m = 2k^2 + 2k, recognise the form n^2 = 2m + 1, conclude n^2 is odd.

Drag the slider to move the pellet through the pipeline. Each stage lights up, and the expression transforms.

stage t:

Drag the slider (or hit Play) to ferry the pellet through four stages: Unpack → Algebra → Factor → Repack. The pellet carries the current expression; each stage rewrites it by a single, clearly-labelled move. The flow is strictly left-to-right — that is the definition of a direct proof.

Why this shape: a direct proof must start from the hypothesis and end at the conclusion, using only steps whose validity is already established. Each arrow is a forward implication. The pipeline imagery forbids moving backwards — you cannot reason "from the conclusion" in a direct proof, because that would mean assuming what you are trying to show. The pellet is a reminder that only one direction of flow is legal.

Why each stage exists

Stage 1 — Unpack definitions. Every named concept (odd, divisible, rational, continuous, open) is a shortcut. The pipeline cannot process shortcuts; it can only process concrete algebraic or set-theoretic expressions. The unpack stage swaps the shortcut name for its defining expression. "n is odd" becomes "n = 2k + 1 for some integer k." Without this swap, there is nothing for algebra to grip.

Stage 2 — Apply algebra. With concrete expressions in hand, legal manipulations take over: expanding (2k+1)^2, combining like terms, substituting. These are steps that would survive in any algebra class. No logic magic — just arithmetic.

Stage 3 — Factor toward the target. This is the creative stage of most direct proofs. The algebraic expression you produced has to be rearranged so that the definition of the conclusion starts peeking through. Here the target is odd, defined as "2m + 1 for some integer m." So you rewrite 4k^2 + 4k + 1 as 2(2k^2 + 2k) + 1, exposing the outer 2(\text{integer}) + 1 shape.

Stage 4 — Re-pack into the target definition. You name the integer you just produced (m = 2k^2 + 2k) and verify that the expression now literally matches the target's definition (n^2 = 2m + 1). The conclusion is now true by definition. Proof complete.

The pipeline fails if a stage is skipped

Every common mistake in direct proofs maps to a missing or corrupted stage.

Skip stage 1 (forget to unpack): you wave hands like "n odd means n^2 is odd because squaring preserves oddness." No algebra to hold onto, no chain, not a proof.
Skip stage 2 (no algebra): you write down n = 2k + 1 and then immediately conclude n^2 = 2m + 1. Where did m come from? Nothing in your text connects the two lines.
Skip stage 3 (no factoring): you compute n^2 = 4k^2 + 4k + 1 and stop. You have an expression but not the target form.
Skip stage 4 (no re-pack): you reach 2(2k^2 + 2k) + 1 but do not name m or state "therefore n^2 is odd." You have done all the work and left the last sentence off.

A full direct proof has all four stages. Most proofs you will write have a sentence or two dedicated to each.

Prove: If a and b are both even, then a + b is even.

Pipeline execution:

Unpack. a even \Rightarrow a = 2p. b even \Rightarrow b = 2q.
Algebra. a + b = 2p + 2q.
Factor. 2p + 2q = 2(p + q).
Repack. Let m = p + q (an integer, since integers are closed under addition). Then a + b = 2m, matching the definition of even. \square

Every sentence of the proof lives in exactly one stage. No stage is skipped, no stage is duplicated.

Reading this pattern in every direct proof

Pick any direct proof in a textbook and you will find the four stages. The names may differ ("by definition," "expanding," "grouping," "therefore") but the function is the same. When a proof feels confusing, it is almost always because one stage is compressed into zero lines and you cannot see where the information actually flows.

When writing a direct proof, use the pipeline as a checklist: did I unpack every defined term? Did I do any algebra at all? Did I manipulate the expression toward the target form? Did I finish by re-packing into the target definition? Four yeses means the proof is complete.