Alphabeta Math
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.

Assuming the Axiom of Choice, the slit plane is simply connected

Example

Assume the Axiom of Choice. Let

Ω=C∖(−∞,0].

Then Ω is simply connected.

Facts & Assumptions

Given: The Axiom of Choice and the slit plane Ω=C∖(−∞,0].

[L1]

Assuming the Axiom of Choice, a plane domain is simply connected exactly when its spherical complement is connected (Assuming the Axiom of Choice, a plane domain is simply connected exactly when its spherical complement is connected).

Verification

technique · direct
1.1

The spherical complement of Ω is C^∖Ω=(−∞,0]∪{∞}, which is connected: it is the closure in C^ of one arc from 0 to ∞.

givenalgebra
2.1

Therefore [L1] makes Ω simply connected.

step 1.1L1∎

Depends on

Used by

Dependency tree · two levels

4 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