DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 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 piece is a common initial segment occurring in two distinct places of a symmetrised relator set

Definition

Let $R^{sym}$ be a symmetrised set of cyclically reduced words (The symmetrisation of a relator set closes under inverses and cyclic conjugates, Cyclically reduced words). A nonempty reduced word $p$ is a piece when there are decompositions

r = p u, s = p v,

with $r, s \in R^{sym}$ , such that the two occurrences are distinct: the ordered pairs $(r, u)$ and $(s, v)$ are not equal.

Equivalently, $p$ is an initial segment shared by two distinct symmetrised occurrences of relators. The empty word is not counted as a piece on this page.

Depends on

Used by

The small-cancellation conditions C(lambda) and C prime(lambda) Definition
A concrete relator set with its pieces and a direct C prime(1/6) check Example
FALSE: every repeated subword of a relator is a piece False statement

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

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)