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.

Naturality of the pair long exact sequence

Statement

A map of pairs $f : (X, A) \to (Y, B)$ induces a commuting morphism from the long exact sequence of $(X, A)$ to that of $(Y, B)$ , including the connecting maps.

Facts & Assumptions

Given: A map of pairs $f : (X, A) \to (Y, B)$ .

Proof

technique · direct

1.1givenconstruct

The maps on $A$ , $X$ , and quotient chains give a commuting morphism of the short exact sequences used for the two pair sequences.

2.1step 1.1algebra∎

Naturality of the homological long exact sequence makes every resulting square commute. In particular, chainwise $f_{#} (\partial c) = \partial f_{#} (c)$ gives commutation at the connector.

Depends on

Dependency tree · 2 levels

Sources

Allen Hatcher, Algebraic Topology, §2.1