ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-30

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 subcomplex and its quotient complex

Example

Let $C_{∙}$ be the two-term complex $0 \to Z \overset{\times 2}{\to} Z \to 0,$ and let $S_{∙}$ be the subcomplex with $S_{1} = 0$ and $S_{0} = 2 Z$ . Then the quotient complex $C / S$ is $0 \to Z \overset{0}{\to} Z / 2 Z \to 0.$

Facts & Assumptions

Given: The complexes $C_{∙}$ and $S_{∙}$ just displayed.

[L1]

A subcomplex is a degreewise subobject stable under the differentials (Subcomplex).

[L2]

A quotient complex is obtained by quotienting degreewise and descending the differential (Quotient complex).

[L3]

$A b$ is an abelian category (Modules over a ring form an abelian category).

Verification

technique · direct

1.1L1L3givenalgebra

The family $S_{∙}$ is a subcomplex: the only nontrivial check is that $d_{1} (S_{1}) = 0$ lands in $2 Z = S_{0}$ . Thus [L1] applies.

2.1L2step 1.1algebra∎

By [L2], the quotient has terms $C_{1} / S_{1} ≅ Z$ and $C_{0} / S_{0} ≅ Z / 2 Z$ . The descended differential is zero because $2 x$ lies in $2 Z$ for every representative $x \in Z$ , so changing representatives does not change the class. Hence the quotient complex is as displayed.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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)