LemmaStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Every null word has a minimal algebraic relator area

Statement

Let $P$ be a finite presentation and let $w$ be trivial in the group presented by $P$ . Then ${Area}_{P} (w)$ exists.

Facts & Assumptions

Given: A finite presentation $P$ and a word $w$ with $w =_{P} 1$ .

[L1]

The algebraic relator area of a null word is defined as the least length of a relator expression for that word. (Algebraic relator area and the Dehn function of a finite presentation)

Proof

technique · direct

1.1

Because $w$ is null, the admissible lengths in [L1] form a nonempty subset of $N$ : every relator expression for $w$ contributes one such length, and the empty product contributes the value $0$ in the boundary case.

L1given

2.1

Every nonempty subset of $N$ has a least element. Applying this to the set of admissible lengths from step 1.1 gives a least $m$ , and [L1] defines that least number to be ${Area}_{P} (w)$ .

L1step 1.1

3.1

Hence the minimal algebraic relator area exists for every null word.

step 2.1∎

Depends on

Algebraic relator area and the Dehn function of a finite presentation

Used by

Computing an algebraic relator area Example
A recursive Dehn function yields a solution to the word problem Proposition

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Meier, Groups, Graphs and Trees (standard reference, not scraped)