Skip to content

Writting formal papers

Core structural elements

Definition

Purpose: Fix meaning precisely.

Use when: You introduce a new object, concept, or condition.

Rules of thumb:

  • No proofs in definitions
  • Should be unambiguous and reusable
  • Everything that follows depends on this
  • Highlight the term

Lemma

Purpose: A technical result used to prove something else.

Use when: The result is important to reach the goal but not the “main point.”

Typical traits

  • Often structural or restrictive
  • Frequently reused
  • Can be very powerful even if “small”

TODO: Lemma includes a proof?

Theorem

Purpose: A main result or milestone.

Use when: You want to signal significance.

Notes

  • “Theorem” is about importance, not difficulty
  • In speculative work, theorems may be conditional (“If…, then…”)

Proposition

Purpose: A result that’s useful but not central.

Use when: It’s stronger than a lemma, weaker than a theorem.

Often interchangeable with “lemma,” but slightly more self-contained.

Corollary

Purpose: A result that follows almost immediately from another.

Use when: Minimal extra reasoning is required.

Good signals

  • Proof is very short
  • Depends directly on one named result

Supporting elements

Example

Purpose: Illustrate, not prove.

Use when: You want to show how something works.

Important

  • Examples do not establish general truth
  • They help intuition and readability

Remark

Purpose: Clarify, contextualize, or warn.

Use when: Something is useful but not part of the proof.

Common uses:

  • Explain intuition
  • Note limitations
  • Connect to known results
  • Justify why something is interesting

Observation

Purpose: State something evident or lightly justified.

Use when: The result is simple and not worth a full lemma.

Often informal but still precise.

Logical flow elements

Proof

Purpose: Establish truth rigorously.

Conventions:

  • Start with “Proof.”
  • End with □ or QED
  • Avoid storytelling; focus on logic
  • Use complete sentences (this matters more than people think)

Claim

Purpose: A sub-result inside a proof.

Use when: You need a temporary lemma.

Useful for long or nested arguments.

Assumption / Hypothesis

Purpose: Fix the logical universe.

Use when: Working conditionally.

Make assumptions explicit early and remind the reader when needed.

Meta-structure elements

Notation

Purpose: Reduce clutter and ambiguity.

Use when: Symbols recur.

Notation. Let ( T(n) ) denote the Collatz map.

Convention

Purpose: Declare default interpretations.

Use when: Avoiding repeated caveats.

Convention. All integers are positive unless stated otherwise.

How these work together (typical pattern)

  1. Definitions fix the setting
  2. Lemmas restrict structure
  3. Corollaries eliminate cases
  4. Remarks explain meaning or direction
  5. Theorem collects the payoff

One guiding principle (very important)

The name of a statement tells the reader how hard to pay attention.

  • Definition → memorize
  • Lemma → trust and reuse
  • Corollary → quick consequence
  • Remark → helpful but skippable