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.

A graph of finite groups giving a virtually free group

Example

The one-segment graph of groups with vertex groups $C_{2}$ and $C_{3}$ and trivial edge group has fundamental group $C_{2} * C_{3}$ , and this group is virtually free.

Facts & Assumptions

Given: The graph of groups with vertex groups $C_{2}$ and $C_{3}$ and trivial edge group.

[L1]

The graph-of-groups fundamental group acts on its Bass-Serre tree, with vertex and edge stabilizers conjugate to the chosen groups. (The fundamental group acts without inversions on its Bass-Serre tree)

[L2]

A group acting freely without inversions on a tree is free. (A group acting freely without inversions on a tree is free)

Verification

technique · direct

1.1

By [L1], the fundamental group $Γ = C_{2} * C_{3}$ acts on its Bass-Serre tree with vertex stabilizers conjugate to $C_{2}$ and $C_{3}$ and trivial edge stabilizers. The canonical surjection $Γ \to C_{2} \times C_{3}$ has finite image of order $6$ , so its kernel $Γ_{0}$ has index $6$ . Because this quotient map is injective on each factor, $Γ_{0}$ meets every conjugate of $C_{2}$ and $C_{3}$ trivially.

L1given

2.1

The subgroup $Γ_{0}$ therefore acts freely on the same tree, so [L2] makes $Γ_{0}$ a free group. Since $Γ_{0}$ has finite index in $Γ$ , the group $C_{2} * C_{3}$ is virtually free.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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)
Jean-Pierre Serre, Trees (standard reference, not scraped)