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 cone triangle of multiplication by m

Example

For $m \in Z$ , the map $m : S^{0} (Z) \to S^{0} (Z)$ has cone the two-term complex $Z \overset{m}{\to} Z$ in degrees $1, 0$ . Its standard cone triangle is therefore $S^{0} (Z) \overset{m}{\to} S^{0} (Z) \to [Z \overset{m}{\to} Z] \to S^{0} (Z) [1] .$

Facts & Assumptions

Given: The displayed data.

[F1]

The mapping cone has its stated degreewise direct-sum and differential formulas (The mapping cone of a chain map).

[F2]

Passing a chain-level cone triangle to the homotopy category gives its standard cone triangle (Standard cone triangle in the homotopy category).

Verification

1.1

By [F1], the cone has $D_{n} \oplus C_{n - 1}$ in degree $n$ , so only degrees $1$ and $0$ remain and its differential is multiplication by $m$ .

F1given

2.1

By [F2], passing this explicit cone triangle to $K (Z -Mod)$ makes it a standard cone triangle.

F2step 1.1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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, Definition 13.9.1 (standard reference, not scraped)