ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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 canonical iterated free resolution of a module

Example

Take $M = Z / 2 Z$ . The canonical free cover on its underlying set ${0, \overset{ˉ}{1}}$ is $ε_{0} : Z e_{0} \oplus Z e_{1} \to Z / 2 Z,$ with $ε_{0} (e_{0}) = 0$ and $ε_{0} (e_{1}) = \overset{ˉ}{1}$ . Its kernel is generated by $e_{0}$ and $2 e_{1}$ , so the next stage of the canonical construction is the free abelian group on that kernel as a set.

Facts & Assumptions

Given: The module $M = Z / 2 Z$ .

[L1]

The iterated free-cover construction gives a canonical exact free resolution (The iterated free-module resolution is canonical in ZF).

Verification

technique · direct

1.1

A vector $a e_{0} + b e_{1}$ maps to $\overset{ˉ}{b}$ , so it lies in $\ker (ε_{0})$ exactly when $b$ is even. Therefore $\ker (ε_{0}) = Z e_{0} \oplus Z (2 e_{1}) .$

givenalgebra

2.1

The next free object in the canonical construction is therefore $F_{1} = Z^{(\ker (ε_{0}))}$ together with its canonical surjection $F_{1} ↠ \ker (ε_{0})$ . This makes concrete what [L1] does: the construction simply repeats the free-on-the-underlying-set cover on the current kernel, with no arbitrary choices. The same pattern starts even for the zero module.

L1step 1.1discharge-construct∎

Depends on

The iterated free-module resolution is canonical in ZF

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

Charles A. Weibel, An Introduction to Homological Algebra (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 12.28: Projectives (standard reference, not scraped)