Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01
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.

A free group separates one nontrivial reduced word by a finite quotient

Example

The free group F(a,b) separates the nontrivial word ab by a finite quotient.

Facts & Assumptions

Given: The free group F(a,b) and the word ab.

[L1]

Free groups are residually finite (Free groups are residually finite).

Verification

technique · direct
1.1

The word ab is nontrivial in the free group F(a,b).

given
2.1

By [L1], there is a finite-index normal subgroup omitting ab, equivalently a finite quotient in which the image of ab is nontrivial. This is exactly the residual-finiteness witness promised in the example.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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