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.

The cone of zero and of the identity

Example

For any chain complex $C_{∙}$ , $Cone (0 : C_{∙} \to C_{∙}) ≅ C_{∙} \oplus C [1]_{∙},$ whereas $Cone (1_{C})$ is contractible.

Facts & Assumptions

Given: A chain complex $C_{∙}$ .

[L1]

The cone of the zero map is the direct sum with a shift (The cone of the zero map is the direct sum with a shift).

[L2]

The cone of an identity map is contractible (The cone of an identity map is contractible).

Verification

technique · direct

1.1L1givenalgebra

Apply [L1] to the zero map $0 : C_{∙} \to C_{∙}$ . This gives the first displayed isomorphism.

2.1L2step 1.1algebra∎

Apply [L2] to the identity map $1_{C} : C_{∙} \to C_{∙}$ . This gives the second displayed conclusion.

Depends on

Used by

A quasi-isomorphism detected by an acyclic cone Example

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)