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.

PCHIP preserves a flat part of monotone data

Example

For nodes $0, 1, 2, 3$ and values $0, 1, 1, 2$ , the PCHIP data are monotone with one flat interval. The algorithm selects slopes $(3 / 2, 0, 0, 3 / 2)$ , and the interpolant is constant on $[1, 2]$ while remaining monotone on the adjoining cells.

Facts & Assumptions

Given: The nodes $x = (0, 1, 2, 3)$ , values $y = (0, 1, 1, 2)$ , and exact real arithmetic.

[F1]

PCHIP slope selection: the secant slopes determine the zero, harmonic-mean, and one-sided endpoint rules, and the output pieces are cubic Hermite pieces.

[F2]

PCHIP is C1, monotone on each data segment, and has no interval overshoot: each segment is monotone between its data endpoints, and a zero-secant segment is constant.

Verification

1.1givenF1algebra

The secants are $d_{0} = 1$ , $d_{1} = 0$ , and $d_{2} = 1$ . The interior rule in [F1] gives $m_{1} = m_{2} = 0$ . The endpoint rule gives ${\hat{m}}_{0} = ((3) (1) - 0) / 2 = 3 / 2$ and, by reflection, ${\hat{m}}_{3} = 3 / 2$ ; neither clip nor cap changes these values.

2.1F1step 1.1algebra

On the middle cell both endpoint values are $1$ and both slopes are zero, so the Hermite formula is the constant polynomial $1$ . On the first cell it is $- t^{3} / 2 + 3 t / 2$ , and on the last cell it is $1 + 3 t^{2} / 2 - t^{3} / 2$ , for normalized local coordinate $t \in [0, 1]$ .

3.1F2step 2.1given∎

The shape theorem [F2] makes the first and last pieces monotone from $0$ to $1$ and from $1$ to $2$ and makes the middle piece constant. Thus the interpolant is globally nondecreasing, has no overshoot, and retains the flat interval exactly.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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