ExampleConstruction: AI-adaptedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Two contiguous maps of a subdivided interval

Example

Let $K$ be the subdivided interval with vertices $x_{0}, x_{1}, x_{2}$ and edges ${x_{0}, x_{1}}$ and ${x_{1}, x_{2}}$ . Let $L$ be the full $2$ -simplex on vertices $a, b, c$ . Define simplicial maps $f, g : K \to L$ by $f (x_{0}) = a, f (x_{1}) = b, f (x_{2}) = c,$ $g (x_{0}) = a, g (x_{1}) = a, g (x_{2}) = c .$

For the edge ${x_{0}, x_{1}}$ , the union of the image vertex sets is ${a, b}$ ; for ${x_{1}, x_{2}}$ it is ${a, b, c}$ . Both are simplices of $L$ , and the same is clear for the vertices, so $f$ and $g$ are contiguous. Their realizations are therefore homotopic, and they induce the same map on every simplicial homology group.

Depends on

Used by

Nothing in the library uses this result yet.

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

Vidit Nanda, Computational Algebraic Topology, Lecture 02: Homotopy (standard reference, not scraped)
Allen Hatcher, Algebraic Topology (standard reference, not scraped)