Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12
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.

Recognizers for strings containing a 1 and for all-zero strings

Example

Let A⊆{0,1}∗ be the language of binary strings containing at least one 1. Its complement consists of the all-zero strings, including the empty word.

Facts & Assumptions

Given: The language A and its complement Ac.

[L1]

By A language is decidable if and only if it and its complement are recognizable, recognizers for a language and its complement can be interleaved into a decider.

Verification

technique · direct
1.1given

A recognizer for A scans the input from left to right and accepts as soon as it sees a 1; on an all-zero input it may simply continue forever. A recognizer for Ac scans to the end of the input and accepts exactly when every scanned symbol is 0.

2.1L1step 1.1∎

On any input word, exactly one of the two recognizers from step 1.1 eventually accepts. Therefore [L1] interleaves them into a decider for A.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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