Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedaudited 2026-09-07
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.

Partial choice graphs have finite character

Example

For a family (Xi)i∈I of nonempty sets, graphs of partial choice functions form a family of finite character in I×⋃iXi. Its maximal members are exactly total choice functions.

Facts & Assumptions

[F1]

Tukey finite character is equivalent to AC: Partial choice graphs are the finite-character family used to derive AC from Tukey.

Verification

Given: The objects and hypotheses in the statement.

1.1

Every finite subset of a partial choice graph is such a graph. Conversely, if a set of pairs is not a permissible graph, one pair has a value outside its prescribed set, or two pairs share an index with different values. A subset of size one or two witnesses the defect. The empty graph passes every finite test.

F1
2.1

If a partial graph omits i, one x∈Xi extends it by (i,x); hence it is not maximal. A total choice graph cannot be extended permissibly, since every index already has its unique value. For empty I the empty graph is already total and maximal.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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