ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Two composition series of Z/12 refine to the same simple factors

Example

In the $Z$ -module $Z / 12$ , the chains

$0 < ⟨ 6 ⟩ < ⟨ 2 ⟩ < Z / 12, 0 < ⟨ 4 ⟩ < ⟨ 2 ⟩ < Z / 12$

are two composition series. Their successive factors are $Z / 2, Z / 3, Z / 2$ and $Z / 3, Z / 2, Z / 2$ , so the same simple factors occur up to permutation.

Facts & Assumptions

Given: The module $Z / 12$ .

[L1]

The butterfly lemma is the cell-by-cell quotient comparison behind refinements (Zassenhaus butterfly lemma in an abelian category).

[L2]

Any two finite subobject chains admit equivalent refinements (Schreier refinement theorem in an abelian category).

[L3]

Jordan-Holder identifies the factor multiset up to permutation (Jordan-Holder theorem in an abelian category).

Verification

technique · direct

1.1

The subgroup $⟨ 6 ⟩$ has order $2$ and $⟨ 2 ⟩ / ⟨ 6 ⟩$ has order $3$ , while $(Z / 12) / ⟨ 2 ⟩$ has order $2$ . Likewise $⟨ 4 ⟩$ has order $3$ , $⟨ 2 ⟩ / ⟨ 4 ⟩$ has order $2$ , and the top quotient is again order $2$ . So both displayed chains are composition series.

L1algebra

2.1

The two series already display the same three simple factors up to permutation, which is the conclusion predicted abstractly by [L2] and [L3]. This is the concrete module calculation that the categorical refinement theorems package.

L2L3step 1.1∎

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.

Sources

Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, Section 1.5 (standard reference, not scraped)