ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Lie derivative of the Euclidean metric under dilations

Statement

For $X = x \partial_{x} + y \partial_{y}$ and $g = d x^{2} + d y^{2}$ on $R^{2}$ , $L_{X} g = 2 g$ .

Facts & Assumptions

Given: The manifolds, forms, vector fields, maps, and coordinates explicitly named in the statement.

[F1]

The preceding result states that For a covariant $k$ -tensor $T = T_{i_{1} \dots i_{k}} d x^{i_{1}} \otimes \dots \otimes d x^{i_{k}}$ , $(L_{X} T)_{i_{1} \dots i_{k}} = X^{j} \partial_{j} T_{i_{1} \dots i_{k}} + \sum_{a} (\partial_{i_{a}} X^{j}) T_{i_{1} \dots j \dots i_{k}} .$ (The coordinate formula for the Lie derivative of a covariant tensor).

Verification

technique · direct

1.1

The coefficients of $g$ are constant and $\partial_{i} X^{j} = δ_{i}^{j}$ .

F1given

2.1

The covariant tensor formula therefore adds two copies of each metric component, giving $L_{X} g = 2 g$ .

step 1.1∎

Depends on

The coordinate formula for the Lie derivative of a covariant tensor

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)