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.

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de Boor evaluation with a double knot

Example

Take degree $p = 2$ , knots $t_{0} = t_{1} = t_{2} = 0$ , $t_{3} = t_{4} = 1$ , and $t_{5} = t_{6} = t_{7} = 2$ , and coefficients $(c_{0}, c_{1}, c_{2}, c_{3}, c_{4}) = (0, 0, 4, 8, 0)$ . Evaluate at $x = 3 / 2$ on the nonempty span $[t_{4}, t_{5}) = [1, 2)$ . This span is adjacent to the double knot at $1$ .

Facts & Assumptions

Given: The degree, knot vector, coefficient list, span $[1, 2)$ , and query $x = 3 / 2$ from the Example.

[F1]

de Boor coefficient evaluation on a specified knot span initializes $d_{j}^{(0)} = c_{k - p + j}$ and uses the stated $α_{j, r}$ blend for $j = p, \dots, r$ .

[F2]

de Boor evaluation equals the B-spline coefficient expansion identifies the exact output with the B-spline coefficient expansion.

[F3]

B-spline spaces and knot conventions uses a clamped knot vector with half-open cells, right-side values at interior knots, and left limits at $b$ .

[A1]

$p = 2$ , $k = 4$ , $t_{0} = t_{1} = t_{2} = 0$ , $t_{3} = t_{4} = 1$ , $t_{5} = t_{6} = t_{7} = 2$ , and $(c_{0}, c_{1}, c_{2}, c_{3}, c_{4}) = (0, 0, 4, 8, 0)$ .

Verification

1.1givenA1F1F3

The input is admissible: [F3] allows the displayed clamped knot vector, including the double interior knot; the knots are nondecreasing and clamped for degree two; $t_{4} = 1 < t_{5} = 2$ ; and $x = 3 / 2 \in [1, 2)$ . By [F1], the initial active list is $(d_{0}^{(0)}, d_{1}^{(0)}, d_{2}^{(0)}) = (4, 8, 0)$ .

2.1A1F1step 1.1algebra

For $r = 1, j = 2$ , the indices are $L = 4, U = 6$ , so $α = (3 / 2 - t_{4}) / (t_{6} - t_{4}) = 1 / 2$ and $d_{2}^{(1)} = (8 + 0) / 2 = 4$ . For $r = 1, j = 1$ , $L = 3, U = 5$ , again $α = 1 / 2$ , so $d_{1}^{(1)} = (4 + 8) / 2 = 6$ .

3.1A1F1step 2.1algebra

For $r = 2, j = 2$ , $L = 4, U = 5$ and $α = 1 / 2$ . Therefore $d_{2}^{(2)} = (6 + 4) / 2 = 5$ , which is the algorithm's exact returned value.

4.1A1F2step 3.1algebra∎

By [F2], the B-spline curve value on the specified span is also $5$ . The example exercises a repeated knot without evaluating a basis function outside the active span.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Carl de Boor, A Practical Guide to Splines, revised edition, Chapter X, pp. 123–126 (standard reference, not scraped)