ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-05

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.

The homology of an interval from contractibility

Example

For the unit interval $I = [0, 1]$ , $H_{0}^{s i n g} (I; Z) ≅ Z, H_{n}^{s i n g} (I; Z) = 0 for n > 0,$ and all reduced singular homology groups vanish.

Facts & Assumptions

Given: The unit interval $I = [0, 1]$ .

[L1]

Every nonempty convex subset of $R^{n}$ is contractible (Every nonempty convex subset of $R^{n}$ is contractible).

[L2]

A nonempty contractible space has the singular homology of a point (Contractible nonempty spaces have the homology of a point).

[L3]

The one-point space has $H_{0} ≅ Z$ , trivial higher homology, and trivial reduced homology (The singular chain complex of a point).

Verification

technique · direct

1.1

The interval $I$ is a nonempty convex subset of $R$ , so [L1] makes it contractible.

L1given

2.1

By [L2], $I$ has the same singular homology groups as a point. Substituting the computation from [L3] gives the displayed formulas for ordinary and reduced singular homology.

L2L3step 1.1∎

Depends on

Used by

A singular cochain need not have finite support on singular simplices Counterexample

Dependency tree · two levels

9 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 (standard reference, not scraped)