ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

A convex domain is a domain of holomorphy

Example

The half-space

$H = {z \in C^{m} : Re z_{1} > 0}$

is a domain of holomorphy.

Facts & Assumptions

Given: The half-space $H = {Re z_{1} > 0}$ .

[L1]

Every convex domain is a domain of holomorphy (Convex domains are domains of holomorphy).

Verification

technique · direct

1.1

The half-space $H$ is convex because if $Re z_{1} > 0$ and $Re w_{1} > 0$ , then $Re ((1 - t) z_{1} + t w_{1}) > 0$ for every $t \in [0, 1]$ .

givenalgebra

2.1

Applying [L1] to step 1.1 shows that $H$ is a domain of holomorphy.

L1step 1.1∎

Depends on

Convex domains are domains of holomorphy

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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, Exercise 2.1.7 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, Example 12 (standard reference, not scraped)