Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 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.

A choice-strength implication map

Example

In ZF the proved arrows are AC  ⟺  MC⇒DC⇒ACω⇒ACω,fin⇒ACω,2, together with AC⇒AC2 and DC  ⟺  (DMC and ACω,fin). The branch AC2 concerns arbitrary index sets.

Facts & Assumptions

[F1]

AC implies DC implies countable choice: The AC–DC–countable-choice chain and pair-choice restriction hold.

[F2]

Multiple choice is equivalent to AC in ZF: MC and AC are equivalent in ZF.

[F3]

DC and finite multiple selections: DC is equivalent to DMC together with countable finite choice.

Verification

Given: The objects and hypotheses in the statement.

1.1

Insert the MC equivalence immediately beside AC, then concatenate the proved implications through the countable pair-choice endpoint. Keep arbitrary-family pair choice on its separately proved AC branch.

F1F2
2.1

The finite-level description of DC requires both DMC and countable finite choice. Thus the displayed conjunction represents exactly the proved equivalence.

F3∎

Depends on

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