ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

The holomorphic hull of a circle in C is the filled disc

Example

In the domain $Ω = C$ , the holomorphic hull of the unit circle

$E = {z \in C : ∣ z ∣ = 1}$

is the closed unit disc

${\hat{E}}_{C} = {z \in C : ∣ z ∣ \leq 1} .$

Facts & Assumptions

Given: The unit circle $E \subseteq C$ .

[L1]

Hull membership is tested by comparison with the boundary suprema of all holomorphic functions on the ambient domain (Holomorphic hulls and holomorphic convexity).

[L2]

A holomorphic function on the unit disc is bounded on the interior by its boundary maximum (Boundary maximum modulus principle on a bounded domain).

Verification

technique · direct

1.1L1L2given

Let $a$ satisfy $∣ a ∣ \leq 1$ , and let $f$ be entire. Then $f$ is holomorphic on the unit disc and continuous on its closure, so [L2] gives $∣ f (a) ∣ \leq \sup_{∣ z ∣ = 1} ∣ f (z) ∣$ . By [L1], this shows $a \in {\hat{E}}_{C}$ . Hence the closed unit disc lies in the hull.

2.1L1step 1.1algebra∎

If $∣ a ∣ > 1$ , take the entire function $f (z) = z$ . Then $∣ f (a) ∣ = ∣ a ∣ > 1 = \sup_{z \in E} ∣ f (z) ∣$ , so [L1] excludes $a$ from the hull. Therefore no point outside the closed unit disc lies in ${\hat{E}}_{C}$ . Together with step 1.1, this identifies the hull exactly.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Jiří Lebl, Tasty Bits of Several Complex Variables, §2.6 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, §3.2.3 (standard reference, not scraped)