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.

First steps for a scalar fractional relaxation

Example

Consider the scalar fractional relaxation equation $D_{C}^{1 / 2} u (t) + u (t) = 0$ with $u (0) = 1$ . On a uniform first step of width $h > 0$ , the L1 method returns

$U_{L 1}^{1} = \frac{1}{1 + Γ (3 / 2) \sqrt{h}},$

while backward-Euler convolution quadrature returns

$U_{C Q}^{1} = \frac{1}{1 + \sqrt{h}} .$

These are the methods' first-step outputs; this calculation makes no claim about their later-step values or a long-time limit.

Facts & Assumptions

Given: $0 < α = 1 / 2 < 1$ , $λ = 1$ , $u_{0} = 1$ , $F^{1} = 0$ , and $h > 0$ .

[F1]

Uniform-grid and graded-grid L1 discretization: on the first uniform step, $δ_{h}^{α} U^{1} = h^{- α} (U^{1} - U^{0}) / Γ (2 - α)$ ; the returned sequence satisfies the discrete equation by Positive L1 weights, discrete comparison, and residual stability.

[F2]

Backward-Euler convolution quadrature for a Caputo equation: with $V^{0} = 0$ , $(h^{- α} + λ) V^{1} = F^{1} - λ u_{0}$ and $U^{1} = u_{0} + V^{1}$ ; the algorithm's returned sequence satisfies its shifted discrete equation by Scalar resolvent stability for backward-Euler convolution quadrature.

[F3]

The real Gamma function by Euler's integral: $Γ (q) > 0$ for $q > 0$ .

Verification

technique · direct

1.1givenF1algebra

For $α = 1 / 2$ , the first-step discrete equation is $h^{- 1 / 2} (U^{1} - 1) / Γ (3 / 2) + U^{1} = 0$ by [F1]. Multiplying by $Γ (3 / 2) \sqrt{h}$ and solving gives $U^{1} = (1 + Γ (3 / 2) \sqrt{h})^{- 1}$ .

2.1givenF2F3algebra∎

For CQ, [F2] gives $(h^{- 1 / 2} + 1) V^{1} = - 1$ , so $V^{1} = - \sqrt{h} / (1 + \sqrt{h})$ and $U^{1} = 1 + V^{1} = (1 + \sqrt{h})^{- 1}$ . Both values are positive and less than $1$ because $h > 0$ and $Γ (3 / 2) > 0$ .

Depends on

Used by

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Dependency tree · two levels

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Sources

Yumin Lin and Chuanju Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics 225 (2007), §3 (standard reference, not scraped)
Ralf Hiptmair, Advanced Numerical Methods for Computational Science and Engineering, lecture notes (2023), §3.3.2 (standard reference, not scraped)