TheoremStatement: 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.

Mayer–Vietoris sequence in singular homology

Statement

For an open cover $X = U \cup V$ , the sequence $\dots \to H_{n} (U \cap V; G) \to H_{n} (U; G) \oplus H_{n} (V; G) \to H_{n} (X; G) \overset{δ}{\to} H_{n - 1} (U \cap V; G) \to \dots$ is exact, with middle map $(u, v) \mapsto u + v$ .

Facts & Assumptions

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

Proof

technique · direct

1.1givenconstruct

Apply the homological long exact sequence to the short exact sequence of the preceding theorem.

2.1step 1.1algebra∎

The cover-small complex has the same homology as $X$ , so replacing its homology term identifies that long exact sequence with the displayed one.

Depends on

Dependency tree · 2 levels

Sources

Allen Hatcher, Algebraic Topology, §2.2