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 split exact complex contracts degree by degree

Example

Consider the split short exact sequence $0 \to Z \overset{i}{\to} Z \oplus Z \overset{p}{\to} Z \to 0,$ where $i (a) = (a, 0)$ and $p (a, b) = b$ . As a three-term chain complex in degrees $2$ , $1$ , and $0$ , it admits explicit maps $h_{1} (a, b) = a, h_{0} (c) = (0, c)$ satisfying $d h + h d = 1.$

Facts & Assumptions

Given: The maps $i (a) = (a, 0)$ and $p (a, b) = b$ .

[L1]

Split short exact sequences are the ones equipped with compatible section and retraction data (Split short exact sequence in an abelian category, Splitting lemma in an abelian category).

[L2]

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

Verification

technique · direct

1.1L1L2givenalgebra

The sequence is split: $r (a, b) = a$ is a retraction of $i$ , and $s (c) = (0, c)$ is a section of $p$ . This is exactly the structure in [L1].

2.1step 1.1algebra∎

With $h_{1} = r$ and $h_{0} = s$ , one computes $h_{1} i = 1_{Z}, i h_{1} + h_{0} p = 1_{Z \oplus Z}, p h_{0} = 1_{Z} .$ So the identity on this complex is written degreewise as $d h + h d$ , which is the promised contraction formula.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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)