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.

Long exact sequence of a pair

Statement

For $A \subseteq X$ there is an exact sequence $\dots \to H_{n} (A; G) \to H_{n} (X; G) \to H_{n} (X, A; G) \overset{δ}{\to} H_{n - 1} (A; G) \to H_{n - 1} (X; G) \to \dots .$

Facts & Assumptions

Given: A subspace $A \subseteq X$ .

Proof

technique · direct

1.1givenconstruct

Inclusion and quotient form a degreewise short exact sequence $0 \to C_{∙} (A; G) \to C_{∙} (X; G) \to C_{∙} (X, A; G) \to 0$ .

2.1step 1.1algebra∎

The long-exact-sequence theorem for a short exact sequence of complexes applied to step 1.1 gives precisely the displayed sequence, with the third homology group equal to relative homology by definition.

Depends on

Dependency tree · 2 levels

Sources

Allen Hatcher, Algebraic Topology, Theorem 2.16