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.

A lake at rest over a bump with a dry crest

Example

Take five periodic cells, $g = 1$ , bed values $(b_{0}, b_{1}, b_{2}, b_{3}, b_{4}) = (0, 1, 2, 1, 0)$ , and free-surface level $η_{0} = 2$ . Set $m_{i} = 0$ and $h_{i} = (2 - b_{i})_{+}$ , so $(h_{i}) = (2, 1, 0, 1, 2)$ ; the crest cell is dry.

Facts & Assumptions

Given: The periodic five-cell bed and states above. Verify the exact one-step hydrostatic HLL update.

[F1]

One-dimensional shallow-water balance law with a dry-state convention: on wet rest cells, $m_{i} = 0$ and $h_{i} + b_{i} = η_{0}$ ; a dry cell has $(h_{i}, m_{i}) = (0, 0)$ with $b_{i} \geq η_{0}$ .

[F2]

Hydrostatic reconstruction with a dry-safe HLL flux and one-sided pressure corrections: each side corrected flux is the common HLL flux plus its cell pressure minus reconstructed pressure.

[F3]

The hydrostatic HLL update preserves lakes at rest, including dry cells: if the reconstructed states at a face agree at rest, the cell-side corrections leave equal hydrostatic pressures on the two faces of each cell.

Verification

1.1givenF1F2algebra

The data are admissible by [F1]. The four face maxima, including the periodic face between cells $4$ and $0$ , are $(1, 2, 2, 1, 0)$ , so the paired reconstructed depths are respectively $(1, 1)$ , $(0, 0)$ , $(0, 0)$ , $(1, 1)$ , and $(2, 2)$ . All reconstructed velocities are zero.

2.1F2step 1.1algebra

At these equal states the common HLL fluxes are $(0, 1 / 2)$ , $(0, 0)$ , $(0, 0)$ , $(0, 1 / 2)$ and $(0, 2)$ . Adding the one-sided pressure corrections in [F2] makes the two corrected fluxes adjacent to each cell both equal $(0, h_{i}^{2} / 2)$ : the five cell pressures are $2, 1 / 2, 0, 1 / 2, 2$ .

3.1F2F3step 2.1algebra∎

Each cell therefore has zero corrected-flux difference in the update, so every state is unchanged, including the dry crest. This verifies the exact lake-at-rest preservation in [F3] for this finite wet/dry example.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

3 results within one dependency step 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

E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein, and B. Perthame, A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows (standard reference, not scraped)
F. Bouchut, III. Shallow water problem – Hydrostatic reconstruction scheme (standard reference, not scraped)