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.

Well-definedness of the relative connector

Statement

The formula $δ [c] = [\partial c]$ defines a homomorphism $H_{n} (X, A; G) \to H_{n - 1} (A; G)$ .

Facts & Assumptions

Given: Relative cycles $c, c^{'}$ representing the same class in $H_{n} (X, A; G)$ .

Proof

technique · direct

1.1givenalgebra

Equality of their relative classes says $c^{'} - c = a + \partial b$ for some $a \in C_{n} (A; G)$ and $b \in C_{n + 1} (X; G)$ .

2.1step 1.1algebra∎

Therefore $\partial c^{'} - \partial c = \partial a$ , an ordinary boundary in $C_{n - 1} (A; G)$ ; the two proposed values agree. Linearity follows from linearity of $\partial$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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, §2.1 (standard reference, not scraped)