ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-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.

Projected data oscillation on a reference triangle

Example

Assume Countable Choice for the cited Sobolev/L2 interface.

Let $K = {(x, y) : x \geq 0, y \geq 0, x + y \leq 1}$ , let $f = 1_{{x + y \leq 1 / 2}}$ , and mark this single cell with $θ_{O} = 1$ . Use the uniform three-generation refinement, any estimator marking fraction, and a per-call budget sufficient for completion.

Facts & Assumptions

[A1]

Countable Choice is the background assumption in the cited Sobolev and $L^{2}$ suppliers (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Given: Countable Choice, The P0 cell oscillation and the three-generation refinement rule.

[F1]

Cell oscillation is $h_{K}^{2} ∥ f - Π_{K}^{0} f ∥_{2}^{2}$ (Residual indicators and data oscillation for P1 AFEM).

[F2]

The marked triangle's edge midpoints and the midpoint on its median occur in the three-generation NVB refinement (Shape regularity and local reduction under 2D newest-vertex bisection).

[F3]

The Dörfler-marked three-generation refinement reduces oscillation by at least the factor $γ_{O} = 1 - \frac{3}{4} θ_{O}$ (Data oscillation reduction under three-generation refinement).

Verification

1.1A1F1givenalgebra

The triangle has area $1 / 2$ , while the subtriangle where $f = 1$ has area $1 / 8$ , so $Π_{K}^{0} f = 1 / 4$ . Therefore $∥ f - Π_{K}^{0} f ∥_{2}^{2} = (1 / 2) (1 / 4) (3 / 4) = 3 / 32$ . Since $diam K = \sqrt{2}$ , the initial oscillation square is $3 / 16$ .

2.1F2F3step 1.1algebra∎

The region $x + y \leq 1 / 2$ is the corner triangle whose other two vertices are the midpoints of the coordinate-axis edges. By [F2] its boundary is a union of refined edges; each refined leaf lies wholly in this region or its complement. Thus $f$ is constant on every child, every child P0 projection is exact, and the refined oscillation is zero. This verifies the strict reduction and agrees with [F3], which gives $γ_{O} = 1 / 4$ at $θ_{O} = 1$ .

Finite check

The parent mean is $1 / 4$ and its variance integral is $3 / 32$ ; the child partition aligns with the indicator boundary, so each child variance is zero.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

Ricardo H. Nochetto, Adaptive Finite Element Methods, Lecture 2 (standard reference, not scraped)
Bonito, Canuto, Nochetto & Veeser, Adaptive Finite Element Methods (2024) (standard reference, not scraped)