ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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 flat nonprojective module

Example

The $Z$ -module $Q$ is flat but not projective.

Verification

Given: the inclusion $Z \subset Q$ and the PID $Z$ .

1.1

No nonzero integer annihilates a nonzero rational number, so $Q$ is torsion-free.

given

2.1

Torsion-free modules over a PID are flat, hence $Q$ is flat.

step 1.1algebra

3.1

A projective abelian group is free, but a nonzero free abelian group has a nonzero homomorphism to $Z$ , whereas every homomorphism $Q \to Z$ is zero; therefore $Q$ is not projective.

step 2.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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 (standard reference, not scraped)