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.
Under the ultrafilter lemma, finite groups and locally finite groups are amenable
Example
Assume the ultrafilter lemma. Every finite group is amenable, and the direct sum is an infinite amenable group.
Facts & Assumptions
Given: The finite-group case, the countable direct sum , and the ultrafilter lemma.
Finite groups are amenable (Finite groups are amenable).
Under the ultrafilter lemma, locally finite groups are amenable (Under the ultrafilter lemma, solvable groups and locally finite groups are amenable).
Verification
The finite case is exactly [L1].
Every finitely generated subgroup of is supported on finitely many coordinates and is therefore finite. Thus is locally finite, and [L2] makes it amenable.
Depends on
Used by
Nothing in the library uses this result yet.
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.