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 free action and the quotient-graph basis
Example
Let act on the bi-infinite line by translation . Then the quotient graph is a single loop, and the resulting basis of the acting group has one generator.
Facts & Assumptions
Given: The quotient graph of an action without inversions.
A group acting freely without inversions on a tree is free. (A group acting freely without inversions on a tree is free)
The quotient graph records vertex and edge orbits of the action. (The quotient graph of an action without inversions)
Verification
Translation by one step on the bi-infinite line has one vertex orbit and one edge orbit, so by [L2] the quotient graph is a single loop. The action is free and without inversions.
Therefore [L1] identifies the acting group with a free group on one generator, namely the loop edge of the quotient graph. This recovers .
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
- C. Loh, Geometric Group Theory: An Introduction (2015 course version) (standard reference, not scraped)