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.

The bidisc is holomorphically convex

Example

The bidisc

$D^{2} = {(z_{1}, z_{2}) \in C^{2} : ∣ z_{1} ∣ < 1, ∣ z_{2} ∣ < 1}$

is holomorphically convex.

Facts & Assumptions

Given: The bidisc $D^{2}$ .

[L1]

Polydiscs are convex (Balls, polydiscs and the distinguished boundary in $C^{m}$ ).

[L2]

Every convex domain is holomorphically convex (Convex domains are holomorphically convex).

Verification

technique · direct

1.1

By [L1], the bidisc $D^{2}$ is a convex domain in $C^{2}$ .

L1given

2.1

Applying [L2] to step 1.1 shows that $D^{2}$ is holomorphically convex.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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, §1.1 and Exercise 2.1.7 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, Example 12 (standard reference, not scraped)