ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29‡ rests on unproved material (inherited)

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.

‡ Rests on 1 statement not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Finitely generated abelian groups admit invariant-factor normal form. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The word problem in a finite cyclic presentation

Example

In the cyclic presentation $⟨ x ∣ x^{6} = e ⟩$ , the word $x^{14}$ is equal to $x^{2}$ and therefore is not trivial.

Facts & Assumptions

Given: The word problem is solvable in finitely generated abelian groups.

[L1]

Every finitely generated abelian group has solvable word problem. (The word problem for finitely generated abelian groups is solvable)

Verification

technique · direct

1.1

In the cyclic group of order $6$ , exponents are taken modulo $6$ , so $x^{14} = x^{2}$ .

given

2.1

Because $2 \equiv̸ 0 (m o d 6)$ , the element $x^{2}$ is not the identity. This is the abelian coordinate check promised by [L1].

L1step 1.1∎

Depends on

The word problem for finitely generated abelian groups is solvable

Used by

Nothing in the library uses this result yet.

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

Jack Jeffries, Math 817: Introduction to Modern Algebra I (Fall 2025) (standard reference, not scraped)