ExampleConstruction: AI-generatedVerification: AI-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.

A matching shape-regular MINI mesh family on the unit square

Example

For each positive integer $n$ , divide $D = (0, 1)^{2}$ into $n^{2}$ equal squares and split every square along its lower-left to upper-right diagonal.

\begin{center} \begin{tikzpicture}[scale=1.25] \draw[thick] (0,0) rectangle (2,2); \draw (1,0)--(1,2); \draw (0,1)--(2,1); \draw (0,0)--(1,1); \draw (1,0)--(2,1); \draw (0,1)--(1,2); \draw (1,1)--(2,2); \node at (0.25,1.75) { $K_{1}$ }; \node at (1.75,0.25) { $K_{2}$ }; \end{tikzpicture} \end{center}

The figure shows the $n = 2$ case only; it draws all grid edges and the same diagonal in each square. The cell diameter is $h_{n} = \sqrt{2} / n$ , so this family refines to zero mesh size.

Facts & Assumptions

Given: Countable Choice, the open unit square and the family of square grids with the stated diagonal convention.

[F1]

A matching affine simplicial mesh consists of nondegenerate simplices meeting in common full faces, with conforming affine maps and bounded shape ratio. (Matching affine simplicial meshes and shape regularity, Shape regularity controls affine simplex geometry)

[F2]

Under Countable Choice, on a bounded connected Lipschitz polygonal/polyhedral domain in dimension two or three, the MINI velocity-pressure pair on a shape-regular matching simplicial mesh has a uniformly bounded Fortin map and a positive inf-sup lower bound independent of the mesh size. (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The conforming P1-bubble/P1 mini element, Uniform Fortin stability of the mini element)

Verification

1.1F1givenalgebra

Every small triangle is a right isosceles triangle with legs $1 / n$ and diameter $\sqrt{2} / n$ . Two triangles in one square share the full diagonal, and triangles in adjacent squares meet along either a full grid edge or a grid vertex; hence the family is matching and conforming. All cells are scaled copies of the same two reference triangles, so their shape ratios are constant and the family is shape-regular and quasi-uniform.

2.1F2step 1.1algebra∎

The unit square is a bounded connected Lipschitz polygon in dimension two. Since $h_{n} = \sqrt{2} / n \to 0$ , each mesh lies in the MINI mesh class and the Fortin stability result [F2] applies uniformly in $n$ . The displayed mesh picture fixes the diagonal convention; the stability conclusion follows from the stated geometry and the cited theorem, not from the drawing.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 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.