ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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.

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A coordinate computation of a nonzero Lie bracket

Example

On $R$ , let $X = d / d x$ and $Y = x d / d x$ . Then

$[X, Y] = \frac{d}{d x} .$

Facts & Assumptions

Given: The vector fields $X = d / d x$ and $Y = x d / d x$ on $R$ .

[L1]

The Lie bracket is the commutator of vector-field actions on smooth functions (The Lie bracket of smooth vector fields).

Verification

technique · direct

1.1L1givenalgebra

For every smooth $h : R \to R$ , the commutator definition [L1] gives $[X, Y] h = X (x h^{'}) - Y (h^{'}) = (x h^{'})^{'} - x h^{''} = h^{'} .$ Thus $[X, Y]$ acts exactly as $d / d x$ on all smooth functions.

2.1step 1.1∎

Hence $[X, Y] = d / d x$ , so the Lie bracket is nonzero even though $X$ is constant.

Depends on

The Lie bracket of smooth vector fields

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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)