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.

Curl and divergence encoded by the exterior derivative

Statement

For $P, Q, R, A, B, C \in C^{\infty} (R^{3})$ , $d (P d x + Q d y + R d z)$ has the usual curl coefficients, and $d (A d y \land d z + B d z \land d x + C d x \land d y) = (\partial_{x} A + \partial_{y} B + \partial_{z} C) d x \land d y \land d z .$

Facts & Assumptions

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

[F1]

The preceding result states that On a chart, if $ω = \sum_{I} ω_{I} d x^{I}$ , then $d ω = \sum_{I} d ω_{I} \land d x^{I} .$ (The local coordinate formula for the exterior derivative).

Verification

technique · direct

1.1

Apply the coordinate formula to $P d x + Q d y + R d z$ and collect the $d y \land d z$ , $d z \land d x$ , and $d x \land d y$ coefficients.

F1given

2.1

Applying it to $A d y \land d z + B d z \land d x + C d x \land d y$ leaves the coefficient $\partial_{x} A + \partial_{y} B + \partial_{z} C$ of $d x \land d y \land d z$ .

step 1.1∎

Depends on

The local coordinate formula for the exterior derivative

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)