ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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 multiplication-by-m complex computes a cyclic group

Example

For a nonzero integer $m$ , the two-term complex $0 \to Z \overset{\times m}{\to} Z \to 0$ has $H_{1} = 0$ and $H_{0} ≅ Z / m Z$ .

Facts & Assumptions

Given: A nonzero integer $m$ .

[L1]

The two-term homology computation is $H_{1} = \ker (f)$ and $H_{0} = coker (f)$ (A two-term complex has kernel and cokernel homology).

[L2]

$A b$ is an abelian category (Abelian groups form an abelian category).

Verification

technique · direct

1.1L2givenalgebra

Multiplication by a nonzero integer on $Z$ has zero kernel, because $m x = 0$ forces $x = 0$ . Its cokernel is the quotient group $Z / m Z$ .

2.1L1step 1.1∎

Applying [L1] to $f = \times m$ yields $H_{1} = 0, H_{0} ≅ Z / m Z .$

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Romyar Sharifi, Homological Algebra, §2.7 (standard reference, not scraped)