CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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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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Finite C prime(1/6) presentations have solvable word problem

Statement

Every finite $C^{'} (1 / 6)$ presentation has solvable word problem.

Facts & Assumptions

Given: A finite presentation satisfying $C^{'} (1 / 6)$ .

[L1]

Greendlinger's lemma provides, for every nonempty freely reduced null word, a relator subword longer than half of a defining relator (In a reduced C prime(1/6) null diagram, some face contributes more than half of its boundary to the outer boundary).

[L2]

Every finite Dehn presentation has a terminating decision procedure for the word problem (Dehn's algorithm terminates and decides the word problem for a Dehn presentation).

Proof

technique · direct

1.1

By [L1], the given finite $C^{'} (1 / 6)$ presentation is a Dehn presentation: every nonempty freely reduced trivial word contains the required long relator subword.

L1given

2.1

Apply [L2] to that Dehn presentation. The resulting Dehn algorithm decides triviality of words.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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)