ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck 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.

The covering criterion checked in abelian groups

Example

For the short exact sequence $0 \to 2 Z \overset{i}{\to} Z \overset{q}{\to} Z / 2 \to 0,$ the covering criterion says that every homomorphism $h : W \to Z$ with $q h = 0$ factors through $i$ after the trivial epic cover $1_{W} : W \to W$ .

Facts & Assumptions

Given: The displayed short exact sequence in $A b$ .

[L1]

The covering criterion is equivalent to exactness (The covering criterion for exactness).

[L2]

The category $A b$ is abelian (Abelian groups form an abelian category).

Verification

technique · direct

1.1L2algebraconstruct

If $q h = 0$ , then every value of $h$ is even, so $h (W) \subseteq 2 Z$ . Therefore there is a unique homomorphism $l : W \to 2 Z$ with $h = i l$ .

2.1L1step 1.1∎

This realizes the cover in [L1] with $V = W$ and $k = 1_{W}$ . So the exactness criterion is visible here without any nontrivial refinement of domains.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

The Stacks Project, Section 12.5, Lemma 12.5.15 (standard reference, not scraped)