ExampleConstruction: Literature-sourcedVerification: 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.

Relative homology of an interval and its endpoints

Example

For $I = [0, 1]$ and $E = {0, 1}$ , $H_{1} (I, E; G) ≅ G$ and all other relative groups vanish. For each $g \in G$ , let $c_{g}$ be the oriented identity simplex of $I$ with coefficient $g$ . Under the canonical isomorphism $G ≅ H_{1} (I, E; G)$ , it represents the class corresponding to $g$ , and $δ [c_{g}] = g [1] - g [0] .$

Facts & Assumptions

Given: The pair $(I, E)$ .

Verification

technique · direct

1.1givenconstruct

$I$ is contractible and $H_{0} (E; G) ≅ G \oplus G$ , while the map to $H_{0} (I; G) ≅ G$ adds the two coordinates.

2.1step 1.1algebra∎

The kernel is ${(- g, g) : g \in G}$ , and $g \mapsto (- g, g)$ is an isomorphism from $G$ onto it. Exactness identifies this kernel with $H_{1} (I, E; G)$ and makes all remaining relative groups vanish. The boundary of $c_{g}$ is $g [1] - g [0]$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)