ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

The level-set foliation of a regular function

Example

On the punctured plane $R^{2} ∖ {0}$ , the function $f (x, y) = x^{2} + y^{2}$ has no critical points. Its level sets are the circles of positive radius, and they form a regular foliation of the punctured plane.

Facts & Assumptions

Given: The regular function $f (x, y) = x^{2} + y^{2}$ on $R^{2} ∖ {0}$ .

[A1]

Its differential is $d f = 2 x d x + 2 y d y$ .

Verification

technique · direct

1.1

On the punctured plane, the differential $d f$ is never zero, so $f$ is a [given] submersion to $(0, \infty)$ .

given

1.2

Therefore $\ker d f$ is an integrable line distribution, and its leaves are [given] the connected components of the level sets $x^{2} + y^{2} = c$ . Those components are exactly the circles of radius $\sqrt{c}$ .

given

1.3

The correspondence between integrable distributions and regular foliations [given] turns this family of circles into a regular foliation of $R^{2} ∖ {0}$ .

given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Will J. Merry, Differential Geometry (standard reference, not scraped)