LemmaStatement: Literature-sourcedProof: 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.

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A Dehn replacement shortens the word strictly

Statement

A Dehn replacement shortens the word strictly.

Facts & Assumptions

Given: A factorisation $r = u v$ of a symmetrised relator with $∣ u ∣ > ∣ v ∣$ , and a word $w = a u s b$ containing $u$ as a subword.

[L1]

Replacing $u$ by $v^{- 1}$ is the Dehn move associated to the relator $r = u v$ (Dehn-reduced words and Dehn presentations).

Proof

technique · direct

1.1L1givenalgebra

By [L1], the Dehn replacement sends $w = a u s b$ to $w^{'} = a v^{- 1} s b$ before free reduction. Because $∣ u ∣ > ∣ v ∣$ , one has $∣ w^{'} ∣ = ∣ w ∣ - ∣ u ∣ + ∣ v ∣ < ∣ w ∣$ .

2.1step 1.1algebra∎

Free reduction can only delete inverse pairs, never add letters. So the freely reduced form of $w^{'}$ is no longer than $w^{'}$ , hence still strictly shorter than $w$ .

Depends on

Dehn-reduced words and Dehn presentations

Used by

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)