ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-01

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 minimal diagram exhibits the Greendlinger face covering more than half its boundary

Example

A one-face van Kampen diagram for a defining relator is already a Greendlinger example: that unique face contributes all of its boundary to the outer boundary.

Facts & Assumptions

Given: A one-face van Kampen diagram whose unique face is labelled by a relator $r$ .

[L1]

Greendlinger's conclusion asks for a face contributing more than half of its boundary to the outer boundary (In a reduced C prime(1/6) null diagram, some face contributes more than half of its boundary to the outer boundary).

Verification

technique · direct

1.1

In a one-face disc diagram, every edge of the unique face lies on the outer boundary of the whole diagram. So the distinguished face contributes exactly $∣ r ∣$ boundary edges to the outer boundary.

given

2.1

Since $∣ r ∣ > ∣ r ∣ / 2$ , the unique face satisfies the conclusion of [L1]. Thus this one-cell disc is a concrete Greendlinger face.

L1step 1.1algebra∎

Depends on

In a reduced C prime(1/6) null diagram, some face contributes more than half of its boundary to the outer boundary

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

GAP SmallCancellation manual, Chapter 1: Small Cancellation Theory — the classical conditions (standard reference, not scraped)
Jay Williams, Universal Countable Borel Quasi-Orders (standard reference, not scraped)
Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, Section 3.5 (standard reference, not scraped)
Clara Löh, Geometric Group Theory: An Introduction, Section 7.4.1 (standard reference, not scraped)