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.

One shared face-average degree of freedom

Example

Let $F$ be an interior facet of a finite matching mesh and let $ϕ_{F}$ be the global CR basis function indexed by $F$ . It has face mean one on $F$ and zero on every other facet. It is supported only on the two cells incident to $F$ .

Facts & Assumptions

Given: A finite matching mesh, an interior facet $F$ , and the local and global Crouzeix–Raviart bases of Face averages are unisolvent for the local Crouzeix–Raviart element and Interior face means form the global Crouzeix–Raviart degrees of freedom.

[F1]

On a cell $K$ , the face dual polynomial is $ϕ_{K, F} = 1 - d λ_{F}$ and its mean is one on $F$ and zero on all other cell facets. (Face averages are unisolvent for the local Crouzeix–Raviart element)

[F2]

The global face basis function is this polynomial on cells incident to $F$ and zero elsewhere. (Interior face means form the global Crouzeix–Raviart degrees of freedom)

Verification

1.1F1given

On either cell incident to $F$ , [F1] gives face mean one on $F$ and zero on that cell's other facets.

2.1F1F2step 1.1givenalgebra∎

On every other cell the function is zero by [F2]. Thus the two traces have matching mean on $F$ , and all other interior and boundary face means are zero. This is exactly the homogeneous CR condition, and the only nonzero global degree of freedom is the mean on $F$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Jean-Luc Guermond, MATH 661 Fall 2025, Part VIII, Chapter 36, Crouzeix–Raviart approximation (standard reference, not scraped)
Joachim Schöberl, Numerical Methods for Partial Differential Equations, Chapter 25, Non-conforming Finite Element Methods (standard reference, not scraped)