ExampleConstruction: Literature-sourcedVerification: 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.

Every convex plane domain is simply connected

Example

Every convex complex domain is simply connected.

Facts & Assumptions

Given: A convex complex domain $Ω \subseteq C$ .

[L1]

Every nonempty convex subset of $R^{n}$ is simply connected (Every nonempty convex subset of $R^{n}$ is simply connected).

[L2]

Trivial fundamental group is one of the clauses equivalent to simple connectivity 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

Regard $Ω$ as a convex subset of $R^{2}$ . Then [L1] gives trivial fundamental group.

givenL1

2.1

By [L2], clause 3 from step 1.1 implies that $Ω$ is simply connected.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 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

A. Hatcher, Algebraic Topology, Example 1.4 (standard reference, not scraped)