ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-31

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 unit disc satisfies all of the grand-equivalent simple connectivity clauses

Example

The unit disc

$D = {z \in C : ∣ z ∣ < 1}$

satisfies every clause of For a plane domain, the complement, homology, primitive, logarithm, conjugate, conformal, homotopy, and contractibility conditions are equivalent.

Facts & Assumptions

Given: The unit disc $D$ .

[L1]

The grand theorem makes connected spherical complement, homological simple connectivity, trivial fundamental group, primitives, holomorphic logarithms, harmonic conjugates, conformal equivalence to the disc, and contractibility equivalent for plane domains (For a plane domain, the complement, homology, primitive, logarithm, conjugate, conformal, homotopy, and contractibility conditions are equivalent).

Verification

technique · direct

1.1

The complement $\hat{C} ∖ D = {z : ∣ z ∣ \geq 1} \cup {\infty}$ is connected: the exterior ${z : ∣ z ∣ > 1}$ is path-connected, and $\infty$ is its point at infinity.

given

2.1

By [L1], condition 1 from step 1.1 forces every other clause on the grand-equivalence list. So $D$ satisfies them all.

step 1.1L1∎

Depends on

For a plane domain, the complement, homology, primitive, logarithm, conjugate, conformal, homotopy, and contractibility conditions are equivalent

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

E. Stein and R. Shakarchi, Complex Analysis, Ch. 3, §5 (standard reference, not scraped)