ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Inserting a knot leaves a linear B-spline curve fixed

Example

The degree-one B-spline curve with knots $(0, 0, 1, 1)$ and controls $(0, 1)$ is $C (x) = x$ on $[0, 1]$ . Insert the new knot $u = 1 / 2$ . The new knots are $(0, 0, 1 / 2, 1, 1)$ and the updated controls are $(0, 1 / 2, 1)$ .

In the control diagram, arrows show the copied endpoints and the single midpoint blend. The diagram records the update formulas; the verification below establishes the represented curves agree.

Facts & Assumptions

Given: Degree $p = 1$ , knot vector $(0, 0, 1, 1)$ , controls $(0, 1)$ , and the inserted knot $u = 1 / 2$ .

[F1]

Single new-knot insertion for a B-spline curve uses $Q_{i} = α_{i} P_{i} + (1 - α_{i}) P_{i - 1}$ on the active blend range, with $α_{i} = (u - t_{i}) / (t_{i + p} - t_{i})$ .

[F2]

Single new-knot insertion preserves the B-spline curve proves that the old and refined coefficient expansions define the same function.

[F3]

B-spline spaces and knot conventions defines the degree-one basis by the Cox–de Boor recurrence and half-open knot cells.

[A1]

The original active span is $[0, 1)$ and $P_{0} = 0$ , $P_{1} = 1$ .

Verification

1.1A1F1algebra

For the insertion span $k = 1$ , the only blended index is $i = 1$ and $α_{1} = (1 / 2 - t_{1}) / (t_{2} - t_{1}) = 1 / 2$ . Therefore [F1] returns $Q_{0} = P_{0} = 0$ , $Q_{1} = (P_{0} + P_{1}) / 2 = 1 / 2$ , and $Q_{2} = P_{1} = 1$ .

1.2A1F3algebra

Before insertion, the two degree-one basis functions on $[0, 1)$ are $1 - x$ and $x$ , so the curve is $x$ .

2.1A1F2F3step 1.1step 1.2algebra∎

On $[0, 1 / 2)$ , the refined curve is $0 (1 - 2 x) + (1 / 2) (2 x) = x$ . On $[1 / 2, 1)$ , it is $(1 / 2) 2 (1 - x) + 1 (2 x - 1) = x$ . The endpoint value at $1$ is the left limit. Therefore the piecewise calculation agrees with the general preservation statement [F2].

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

Carl de Boor, A Practical Guide to Splines, revised edition, Chapter XI, pp. 135–138 (standard reference, not scraped)