Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10
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 clopen game decided by the first move

Example

On T=N<ω let A={x:x(0)=0}. Player I wins by the strategy σ(s)=0 at every even-length position s. The payoff is clopen.

Facts & Assumptions

[F1]

Full-position strategies and their winning condition are in Gale–Stewart games and strategies.

[F2]

Finite-prefix cylinders form the topology of Baire sequence space NN and its cylinder topology.

Verification

Given: The full natural-number tree, payoff A, and the constant-zero I strategy.

1.1

Every s⌢0 belongs to the full tree, so σ is a legal strategy, including at s=∅. We have A=N(0) and N∖A=⋃k≥1N(k), both open by F2. Hence A is clopen.

givenF1F2
2.1

Every branch consistent with σ satisfies x(0)=σ(∅)=0, so belongs to A. For example, if II always plays 1, the unique compatible branch is (0,1,0,1,…) and its first coordinate is 0. The same first-coordinate calculation holds for every sequence of II moves, proving that σ wins. QED.

F1step 1.1

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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