Alphabeta MathHow 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.
FALSE: Dehn reduction is just free reduction under another name
Statement
Dehn reduction is just free reduction under another name.
Facts & Assumptions
Given: The one-relator presentation .
A Dehn move replaces a relator subword longer than half the relator by the inverse complementary arc (Dehn-reduced words and Dehn presentations).
Refutation
The word is freely reduced, because no adjacent inverse letters occur.
Nevertheless [L1] applies to the whole word: it is itself a relator and is longer than half of that relator, so one Dehn move replaces it by the empty word.
Thus a word can admit a Dehn reduction while admitting no free reduction. The two notions are different, so the statement is false.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
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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)