ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30‡ 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.

The nine lemma verified on a diagram of cyclic groups

Example

In $A b$ , take the commutative $3 \times 3$ diagram whose top row is the zero short exact sequence $0 \to 0 \to 0 \to 0 \to 0,$ whose middle and bottom rows are both $0 \to Z / 2 \to Z / 4 \to Z / 2 \to 0,$ whose vertical maps from the top row to the middle row are zero, and whose vertical maps from the middle row to the bottom row are identities. Then each column is short exact, and the nine lemma says that the top row is short exact if and only if the bottom row is.

Facts & Assumptions

Given: The cyclic-group $3 \times 3$ diagram just described.

[L1]

Abelian groups form an abelian category (Abelian groups form an abelian category).

[L2]

The nine lemma applies to any such $3 \times 3$ diagram (Nine lemma in an abelian category).

Verification

1.1

In this diagram the middle and bottom rows are the standard short exact sequence $0 \to Z / 2 \to Z / 4 \to Z / 2 \to 0,$ and each column is one of the short exact sequences $0 \to 0 \to G \overset{1_{G}}{\to} G \to 0$ with $G \in {Z / 2, Z / 4}$ or the zero sequence. Hence all three columns and the middle row are short exact in $A b$ .

L1algebra

2.1

The top row is the zero short exact sequence and the bottom row is the standard short exact sequence above, so both outer rows are short exact. This agrees with [L2], which predicts that under the hypotheses verified in step 1.1 the two outer rows stand or fall together.

L2step 1.1algebra

3.1

Thus this cyclic-group diagram is a concrete instance of the nine lemma.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

Charles A. Weibel, An Introduction to Homological Algebra, Exercise 1.3.2 (standard reference, not scraped)