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.

Dehn-reduced words and Dehn presentations

Definition

Let $R^{sym}$ be a symmetrised relator set (The symmetrisation of a relator set closes under inverses and cyclic conjugates, Cyclically reduced words). A freely reduced word $w$ is Dehn-reduced when no factorisation

r = u v

with $r \in R^{sym}$ has $u$ appearing as a subword of $w$ and $∣ u ∣ > ∣ v ∣$ .

A finite presentation is a Dehn presentation when every nonempty freely reduced word representing the identity fails to be Dehn-reduced. Equivalently, every such word contains a relator subword longer than half of the relator.

Depends on

Used by

A trivial word is reduced to the empty word by successive Dehn moves Example
FALSE: Dehn reduction is just free reduction under another name False statement
A Dehn replacement shortens the word strictly Lemma

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)