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 three-cone calculation for a composite

Example

Let $f = \times m$ and $g = \times n$ on the stalk complex $Z [0]$ , with $m, n \neq 0$ . Then the map $α : Cone (m) \to Cone (n m)$ from the three-cone calculation has cone chain-isomorphic to $Cone (n) \oplus Cone (1_{Z [1]}) .$ Since the second summand is contractible, $Cone (α)$ is homotopy equivalent to $Cone (n)$ .

Facts & Assumptions

Given: Nonzero integers $m$ and $n$ .

[L1]

The three-cone calculation identifies $Cone (α)$ with $Cone (g) \oplus Cone (1_{C [1]})$ (The three-cone calculation for a composite chain map).

[L2]

The cone of multiplication by an integer on $Z [0]$ is the two-term complex with that multiplication as differential (The cone of multiplication by m on the integers).

Verification

technique · direct

1.1L1givenalgebra

Apply [L1] to the composable pair $\times m, \times n$ on $Z [0]$ .

2.1L2step 1.1algebra∎

Using [L2], the first summand becomes the two-term complex $0 \to Z \overset{n}{\to} Z \to 0$ , while the second summand is contractible. This is exactly the displayed calculation.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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, Section 13.9: Cones and termwise split sequences (standard reference, not scraped)