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 zero-differential complex has homology equal to each term

Example

Let $C_{∙}$ be a chain complex whose every differential is zero. Then $H_{n} (C) ≅ C_{n}$ for every $n$ .

Facts & Assumptions

Given: A chain complex $C_{∙}$ with $d_{n} = 0$ for all $n$ .

[L1]

The zero complex and zero differentials are legitimate chain-complex data (Zero complex and stalk complex).

[L2]

Homology is the quotient $Z_{n} (C) / B_{n} (C)$ (Homology object of a chain complex).

[L3]

The identity of an object is a kernel of the zero map out of it, and also is a cokernel of the zero map into it (The cokernel of the zero map out of the zero object is the target, and dually for kernels).

Verification

technique · direct

1.1

Since $d_{n} = 0$ , [L3] identifies the cycle inclusion with $1_{C_{n}} : C_{n} \to C_{n}$ , so $Z_{n} (C) ≅ C_{n}$ . Likewise the image of $d_{n + 1} = 0$ is the zero object, so the boundary object is $B_{n} (C) ≅ 0$ .

L1L3givenalgebra

2.1

Substituting those identities into [L2] gives $H_{n} (C) = C_{n} / 0 ≅ C_{n} .$

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)