ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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.

Evaluation at four distinct real points gives an inner product on polynomials of degree at most three

Example

On the real vector space $P_{3} (R)$ of polynomials of degree at most $3$ , evaluation at four distinct real points $t_{0}, t_{1}, t_{2}, t_{3}$ defines an inner product by

$⟨ p, q ⟩ = \sum_{k < 4} p (t_{k}) q (t_{k}) .$

For example, the points may be $0, 1, 2, 3$ .

Facts & Assumptions

Given: Four distinct real scalars $t_{0}, t_{1}, t_{2}, t_{3}$ and polynomials $p, q \in P_{3} (R)$ .

[L1]

An inner product is a symmetric bilinear form whose diagonal is nonnegative and vanishes only at zero (Real and complex inner product spaces, with the inner product linear in the first argument).

[L2]

A nonzero polynomial over a domain has at most as many distinct roots as its degree (A nonzero polynomial of degree $n$ over an integral domain has at most $n$ distinct roots).

[L3]

Finite sums are compatible with the additive and multiplicative laws of the scalar field (The product $g_{0} g_{1} \dots g_{n - 1}$ of a finite list in a monoid, by recursion, with the empty product ( $n = 0$ ) equal to the identity).

Verification

technique · direct

1.1

By [L3], the displayed formula is bilinear and symmetric. Also $⟨ p, p ⟩ = \sum_{k < 4} p (t_{k})^{2} \geq 0$ .

L1L3

2.1

If $⟨ p, p ⟩ = 0$ , every real square in the sum is zero, so $p (t_{k}) = 0$ for four distinct $t_{k}$ . If $p$ were nonzero, [L2] would give at most three roots because $\deg p \leq 3$ . Thus $p = 0$ , proving definiteness and hence [L1].

step 1.1L1L2∎

Depends on

Used by

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