ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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 has the coordinate formula (Coordinate formula for the Lie bracket).

Verification

technique · direct

1.1

In the standard coordinate, the coefficients are $X^{1} = 1$ and $Y^{1} = x$ . Therefore [L1] gives $[X, Y]^{1} = X^{1} \partial_{x} Y^{1} - Y^{1} \partial_{x} X^{1} = 1 \cdot 1 - x \cdot 0 = 1.$

L1given

2.1

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

step 1.1∎

Depends on

Coordinate formula for the Lie bracket

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

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