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.
A trivial word is reduced to the empty word by successive Dehn moves
Example
In the one-relator presentation
the word
reduces to the empty word by two successive Dehn moves.
Facts & Assumptions
Given: The presentation and the word displayed above.
A relator subword longer than half the relator may be replaced by the inverse complementary arc, and this shortens the word (Dehn-reduced words and Dehn presentations, A Dehn replacement shortens the word strictly).
Verification
The first seven letters of form the defining relator itself, so [L1] replaces that block by the empty word. The result is the single relator .
Apply the same move again to the remaining relator block. A second Dehn replacement produces the empty word.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
7 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)