ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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 field has module length one over itself

Example

Let $K$ be a field. Then the regular $K$ -module $K$ has length $1$ .

Facts & Assumptions

Given: A field $K$ .

Verification

technique · direct

1.1

The only ideals of a field are $0$ and the whole field, so $0 < K$ is a composition series of the regular module $_{K} K$ .

givenalgebra

2.1

By Composition series and length of a module, that composition series has exactly one factor, so $ℓ_{K} (K) = 1$ .

step 1.1givenalgebra∎

Depends on

Composition series and length of a module

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

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Section 19 (standard reference, not scraped)