LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

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.

Adjacent-window divided-difference recurrence

Statement

Let $n \in N$ , $n \geq 1$ , and $d \in N$ with $d < n$ ; let $x_{0} < \dots < x_{n}$ be real nodes, let $x \in R$ be distinct from these nodes, and let $f : {x_{0}, \dots, x_{n}, x} \to R$ . For every $0 \leq i \leq n - d$ , put $G_{i} : = f [x_{i}, \dots, x_{i + d}, x] .$ Then, for every $0 \leq i < n - d$ , $G_{i} - G_{i + 1} = - (x_{i + d + 1} - x_{i}) f [x_{i}, \dots, x_{i + d + 1}, x] .$

Facts & Assumptions

Given: $n \in N$ , $n \geq 1$ , $d \in N$ with $d < n$ , ordered distinct real nodes $x_{0}, \dots, x_{n}$ , an additional distinct real point $x$ , and real-valued data $f$ on these points.

[F1]

Divided differences on distinct nodes: the divided difference satisfies the recursive endpoint rule.

[F2]

Divided differences are symmetric leading coefficients: divided differences are symmetric in all listed nodes.

[F3]

The natural numbers $N$ (von Neumann): $N$ contains $0$ and is closed under successor, so $d + 2 \in N$ .

Proof

1.1F1F2F3algebra

Apply the recursive endpoint rule [F1] to the order- $(d + 2)$ divided difference on the ordered endpoints $x_{i}, x_{i + d + 1}$ and the intermediate list $x_{i + 1}, \dots, x_{i + d}, x$ . Symmetry [F2] identifies its two numerator terms as $G_{i + 1}$ and $G_{i}$ , respectively, so $f [x_{i}, \dots, x_{i + d + 1}, x] = \frac{G_{i + 1} - G_{i}}{x_{i + d + 1} - x_{i}} .$

2.1givenstep 1.1algebra∎

The denominator in step 1.1 is positive because the nodes are strictly ordered. Rearranging that identity gives the stated identity.

Depends on

Used by

Floater–Hormann C^(d+2) parity error bounds Theorem

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

Michael S. Floater and Kai Hormann, Barycentric rational interpolation with no poles and high rates of approximation, proof of Theorem 2 (standard reference, not scraped)