ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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.

Universal-coefficient homology with cyclic coefficients

Statement

For $C = (0 \to Z \overset{m}{\to} Z \to 0)$ in degrees $1, 0$ and coefficients $Z / r$ , $H_{1} (C \otimes Z / r) = {\overset{ˉ}{x} : m \overset{ˉ}{x} = 0}$ and $H_{0} (C \otimes Z / r) = (Z / r) / m (Z / r)$ .

Verification

Given: the tensor complex $0 \to Z / r \overset{m}{\to} Z / r \to 0$ .

1.1given

Its degree-one homology is the kernel and its degree-zero homology the cokernel of multiplication by $m$ .

2.1step 1.1algebra∎

These are respectively the $m$ -torsion subgroup and quotient displayed in the statement. The calculation follows directly from the two-term tensor complex.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Weibel, An Introduction to Homological Algebra (standard reference, not scraped)