LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Cover-small chains for a two-open cover

Statement

For open $U, V \subseteq X$ with $X = U \cup V$ , $C_{∙}^{{U, V}} (X; G) = C_{∙} (U; G) + C_{∙} (V; G)$ and the intersection of the two summands is $C_{∙} (U \cap V; G)$ .

Facts & Assumptions

Given: An open cover $X = U \cup V$ .

Proof

technique · direct

1.1givenalgebra

A cover-small generator has image in $U$ or in $V$ , hence belongs to the displayed sum; conversely each generator from either summand is cover-small.

2.1step 1.1algebra∎

The singular simplices simultaneously belonging to both free-chain subgroups are exactly those with image in $U \cap V$ , so their intersection is the stated chain group.

Depends on

Cover-small singular chains

Used by

Relative cup product for an excisive triad Definition
Normalized rational vanishing cycles have coherent correspondence operations Theorem
Short exact chain Mayer–Vietoris sequence Theorem

Dependency tree · two levels

2 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

Allen Hatcher, Algebraic Topology, §2.2 (standard reference, not scraped)