ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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 long exact Hom sequence of a cone triangle

Example

Applying $K (Z -Mod) (S^{0} (Z), -)$ to the cone triangle of multiplication by $m$ gives the exact segment $Z \overset{m}{\to} Z \to K (S^{0} Z, [Z \overset{m}{\to} Z]) \to 0.$

Verification

Given: The displayed data.

1.1given

The displayed cone triangle is distinguished.

2.1step 1.1given∎

The representable long exact Hom theorem supplies the exact segment, and $K (S^{0} Z, S^{0} Z) ≅ Z$ identifies its first map with multiplication by $m$ .

Depends on

Long exact Hom sequences of a distinguished triangle

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

The Stacks Project, Lemma 13.4.2 (standard reference, not scraped)