ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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.

A power map sends a sector to a half-plane

Example

Let

$S : = {r e^{i θ} : r > 0, - π / 4 < θ < π / 4} .$

Then the square map $z \mapsto z^{2}$ biholomorphically sends $S$ onto the right half-plane

${w \in C : Re w > 0} .$

Facts & Assumptions

Given: The sector $S$ above.

[F1]

On sectors of angular width less than $π$ , the square map is a biholomorphism onto the angle-doubled sector (Power maps are biholomorphisms on sectors of width less than $2 π / n$ ).

Verification

1.1F1given

The argument interval of $S$ is $(- π / 4, π / 4)$ , which has width $π / 2 < π$ , so [F1] applies to $z \mapsto z^{2}$ .

2.1F1step 1.1algebra∎

Doubling the argument interval gives $(- π / 2, π / 2)$ , hence the image is ${ρ e^{i ϕ} : ρ > 0, - π / 2 < ϕ < π / 2}$ , which is exactly the right half-plane. Therefore $z \mapsto z^{2}$ is a biholomorphism from $S$ onto ${Re w > 0}$ .

Depends on

Power maps are biholomorphisms on sectors of width less than $2\pi/n$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 3 §4.2 (standard reference, not scraped)