ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Computing an algebraic relator area

Example

In the presentation $⟨ a ∣ a^{3} = e ⟩$ , the null word $a^{6}$ has algebraic relator area $2$ .

Facts & Assumptions

Given: The relator-area definition for a finite presentation.

[L1]

The algebraic relator area of a null word is the least number of conjugates of defining relators and their inverses whose product equals that word. (Algebraic relator area and the Dehn function of a finite presentation)

[L2]

Every null word has a minimal algebraic relator area. (Every null word has a minimal algebraic relator area)

Verification

technique · direct

1.1

The word $a^{6}$ is the product $(a^{3}) (a^{3})$ , so [L1] gives $Area (a^{6}) \leq 2$ .

L1given

2.1

It is not $1$ , because a single conjugate of the only relator or its inverse is just $a^{3}$ or $a^{- 3}$ in this cyclic presentation. Thus [L2] forces the minimum to be exactly $2$ .

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)