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 coordinate-plane distribution and its affine leaves

Example

On $R^{n}$ , fix $1 \leq k \leq n$ and let

$D = span (\partial_{x^{1}}, \dots, \partial_{x^{k}}) .$

Its maximal connected integral manifolds, or leaves, are the affine coordinate $k$ -planes with $x^{k + 1}, \dots, x^{n}$ constant. Connected open subsets of those planes are also integral manifolds.

Facts & Assumptions

Given: The standard coordinates on $R^{n}$ and the span of the first $k$ coordinate fields.

[A1]

The remaining coordinates are constant along the displayed planes.

Verification

technique · direct

1.1

The fields $\partial_{x^{1}}, \dots, \partial_{x^{k}}$ are smooth and pointwise [given] independent, so they define a smooth rank- $k$ distribution.

given

1.2

For fixed constants $c_{k + 1}, \dots, c_{n}$ , the affine plane [given] ${x^{k + 1} = c_{k + 1}, \dots, x^{n} = c_{n}}$ has tangent space spanned by those same coordinate fields at every point. Hence it is an integral manifold.

given

1.3

Every connected integral manifold has constant transverse coordinates, so it [given] lies in one of the affine planes from step 1.2. Those full planes are connected and cannot be enlarged while retaining that property. Therefore they are exactly the maximal leaves.

given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)