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 quasi-isomorphism detected by an acyclic cone

Example

For every chain complex $C_{∙}$ , the identity map $1_{C} : C_{∙} \to C_{∙}$ is a quasi-isomorphism because its cone is contractible, hence acyclic.

Facts & Assumptions

Given: A chain complex $C_{∙}$ .

[L1]

The cone of an identity map is contractible (The cone of zero and of the identity).

[L2]

A chain map is a quasi-isomorphism exactly when its cone is acyclic (A chain map is a quasi-isomorphism exactly when its cone is acyclic).

Verification

technique · direct

1.1L1givenalgebra

By [L1], $Cone (1_{C})$ is contractible, so in particular it is acyclic.

2.1L2step 1.1algebra∎

Applying [L2] to the identity map gives that $1_{C}$ is a quasi-isomorphism.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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)