TheoremStatement: 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Dehn's algorithm terminates and decides the word problem for a Dehn presentation

Statement

For a finite Dehn presentation, Dehn's algorithm terminates and decides the word problem.

Facts & Assumptions

Given: A finite Dehn presentation $⟨ X ∣ R ⟩$ and an input word $w$ on $X^{\pm 1}$ .

[L1]

Every Dehn replacement strictly shortens the current word (A Dehn replacement shortens the word strictly).

Proof

technique · direct

1.1

Start by freely reducing $w$ . Whenever the current freely reduced word contains a relator subword longer than half of a defining relator, perform the corresponding Dehn replacement and freely reduce again. By [L1], each such cycle strictly decreases word length, so no infinite run is possible. Thus the algorithm terminates.

L1given

2.1

Each replacement uses a relation $u v = 1$ in the group and swaps $u$ for $v^{- 1}$ , so every step preserves the group element represented by the current word. Therefore if the algorithm reaches the empty word, the original input represented the identity.

step 1.1algebra

3.1

Conversely, suppose the input represents the identity. After each iteration the current word is freely reduced and still represents the identity by step 2.1. If the current word were nonempty when the algorithm stopped, it would be Dehn-reduced by construction, contradicting the defining property of a Dehn presentation. Therefore the algorithm cannot stop before reaching the empty word, and termination from step 1.1 forces the final output to be empty.

step 1.1step 2.1givencontradiction: a nonempty trivial word in a Dehn presentation is never Dehn-reduced

4.1

Steps 2.1 and 3.1 prove correctness, and step 1.1 proves termination. Therefore Dehn's algorithm decides the word problem for the presentation.

step 1.1step 2.1step 3.1∎

Depends on

Used by

Dependency tree · two levels

8 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)