ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31

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 split exact complex and its contraction

Example

Consider the complex of abelian groups $0 \to Z \overset{u}{\to} Z \oplus Z \overset{v}{\to} Z \to 0,$ where $u (a) = (a, 0), v (a, b) = b .$ It is split exact, and a contraction is given by $s_{0} (c) = (0, c), s_{1} (a, b) = a,$ with all other components zero.

Facts & Assumptions

Given: The split exact three-term complex displayed above.

[L1]

A compatible degreewise split exact complex is contractible (A degreewise split exact complex with compatible splittings is contractible).

[L2]

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

Verification

technique · direct

1.1L2givenalgebra

The complex is exact because $\ker (v) = {(a, 0)}$ equals $im (u)$ , and $u$ is injective while $v$ is surjective. The displayed formulas for $s_{0}$ and $s_{1}$ are compatible with the splitting $Z \oplus Z ≅ Z_{1} \oplus Z_{0} ≅ Z \oplus Z .$

2.1L1step 1.1algebra∎

A direct calculation gives $u s_{1} + s_{0} v = 1_{Z \oplus Z}, s_{1} u = 1_{Z}, v s_{0} = 1_{Z},$ so the identity map is null-homotopic. Hence [L1] applies and the complex is contractible.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)