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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Flow-box coordinates for a nonconstant planar field

Example

For the vector field

$X = \frac{\partial}{\partial x} + y \frac{\partial}{\partial y}$

on $R^{2}$ , the coordinates

$u = x, v = y e^{- x}$

turn $X$ into $\partial / \partial u$ .

Facts & Assumptions

Given: The vector field $X = \partial / \partial x + y \partial / \partial y$ on $R^{2}$ .

[L1]

The flow-box theorem straightens a nonvanishing vector field (The flow-box theorem).

Verification

technique · direct

1.1

Compute $X (u) = 1, X (v) = X (y e^{- x}) = y (- e^{- x}) + e^{- x} X (y) = - y e^{- x} + e^{- x} y = 0.$

given

2.1

Step 1.1 means that in the coordinates $(u, v)$ the field differentiates the first coordinate by $1$ and the second by $0$ , so it is exactly $\partial / \partial u$ . This is an explicit flow-box chart, as predicted by [L1].

L1step 1.1∎

Depends on

The flow-box theorem

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

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