Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29
How 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.

Free reduction decides a word in a free group

Example

In the free group on generators a,b, the word

w=abb−1a−1aba−1

is nontrivial because its free reduction is aba−1.

Facts & Assumptions

Given: The free-group word-problem criterion.

[L1]

A word in a finitely generated free group is trivial exactly when its free reduction is empty. (The word problem for a finitely generated free group is solvable by free reduction)

Verification

technique · direct
1.1

Cancel the adjacent pair bb−1 in w, then cancel the adjacent pair a−1a. The resulting reduced word is aba−1.

given
2.1

The reduced word aba−1 is not empty, so [L1] implies that the original word w is nontrivial.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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