ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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.

Tangent basis change between Cartesian and polar coordinates

Example

On every polar-coordinate chart in $R^{2} ∖ {0}$ , the coordinate basis satisfies $\partial_{r} = \cos θ \partial_{x} + \sin θ \partial_{y}$ and $\partial_{θ} = - r \sin θ \partial_{x} + r \cos θ \partial_{y}$ .

Facts & Assumptions

Given: An open set $U \subseteq R^{2} ∖ {0}$ carrying a smooth polar angle $θ$ , with coordinate change $x = r \cos θ$ , $y = r \sin θ$ .

[L1]

Tangent bases transform by the Jacobian of the coordinate change (Change-of-coordinate formula for tangent bases).

Verification

technique · direct

1.1

The Jacobian of $(r, θ) \mapsto (x, y)$ is $(\begin{array}{cc} \cos θ & - r \sin θ \\ \sin θ & r \cos θ \end{array})$ .

given

2.1

Applying [L1] with this Jacobian gives the displayed formulas for $\partial_{r}$ and $\partial_{θ}$ in the Cartesian basis.

L1step 1.1

3.1

This is the standard basis-change example.

step 2.1∎

Depends on

Change-of-coordinate formula for tangent bases

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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)