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

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.

A relative simplicial approximation fixed on the endpoints

Example

For $f : [0, 1] \to [0, 1]$ , $f (x) = x^{2}$ , fix the endpoint subcomplex $A = {0, 1}$ . On the midpoint subdivision define the simplicial vertex map $g (0) = g (1 / 2) = 0$ , $g (1) = 1$ . Its realization is $∣ g ∣ (x) = 0$ for $0 \leq x \leq 1 / 2$ , and $∣ g ∣ (x) = 2 x - 1$ for $1 / 2 \leq x \leq 1$ . Then $∣ g ∣$ is homotopic to $f$ rel both endpoints.

Source locators

Relative theorem pp.39–42, interval specialization.

Facts & Assumptions

[F1]

Star inclusions certify a simplicial approximation. The open star criterion produces a simplicial map.

[F2]

Relative approximation permits fixed endpoint subcomplexes. Relative simplicial approximation after subdivision.

Verification

Given: The specified interval map and the midpoint triangulation.

1.1

The first source edge maps to the target vertex $0$ and the second to the target edge, so $g$ is simplicial. The two affine formulas agree at $1 / 2$ , both giving $0$ , and $∣ g ∣ (0) = 0$ , $∣ g ∣ (1) = 1$ . The target stars are $[0, 1)$ and $(0, 1]$ . The source stars of $0, 1 / 2, 1$ are $[0, 1 / 2)$ , $(0, 1)$ , $(1 / 2, 1]$ ; under $x^{2}$ they lie respectively in $[0, 1)$ , $[0, 1)$ , $(0, 1]$ . Thus this is even a star approximation to $f$ .

2.1

The explicit homotopy is $H (x, t) = (1 - t) x^{2} + t ∣ g ∣ (x)$ . It is continuous because its piecewise polynomial formulas agree at $x = 1 / 2$ ; its values lie in $[0, 1]$ as convex combinations. It satisfies $H (x, 0) = x^{2}$ , $H (x, 1) = ∣ g ∣ (x)$ and $H (0, t) = 0, H (1, t) = 1$ . At the midpoint, $f (1 / 2) = 1 / 4$ , $∣ g ∣ (1 / 2) = 0$ , and $H (1 / 2, t) = (1 - t) / 4$ , so fixing the endpoints does not mean fixing the entire interval. This explicitly realizes the relative theorem for the pair.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

E. C. Zeeman, Relative simplicial approximation (1964) (standard reference, not scraped)