Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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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.

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The kernel of a submersion as an integrable distribution

Example

Consider F:R3→R,F(x,y,z)=x2+y2+z. Since dF=(2x,2y,1) is never zero, F is a submersion. Therefore ker⁡dF is a rank-2 integrable distribution whose leaves are the connected surfaces x2+y2+z=c.

Facts & Assumptions

Given: The submersion F(x,y,z)=x2+y2+z.

[A1]

Its differential never vanishes.

Verification

technique · direct
1.1

Because the third partial derivative of F is 1, the map is a [given] submersion everywhere.

given
1.2

The kernel distribution is therefore integrable, and its maximal connected [given] integral manifolds are the connected components of the level sets of F. Each level set here is a connected paraboloid.

given
2.1

Thus ker⁡dF is an explicit integrable distribution. [given] ∎

given

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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