Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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.

Prenex normalization preserves the arithmetical level

Statement

Negating a Σn0 formula produces a Πn0 formula, and finite conjunctions and disjunctions of formulas in one of these classes can be put in prenex form in that same class (after harmless repeated-quantifier padding). Thus negation interchanges the two classes, while positive Boolean combinations preserve either class.

Facts & Assumptions

Given: finitely many Σn0 formulas, or finitely many Πn0 formulas.

Proof

technique · direct
1.1

Push negations through the displayed prefix. Each ∃ changes to ∀ and conversely; the negated primitive-recursive matrix is again primitive recursive.

givenalgebra
2.1

Rename bound variables and merge equal-polarity blocks. Conjunction and disjunction distribute through adjacent like quantifiers, and shorter prefixes are padded by vacuous like-polarity quantifiers. Thus the leading polarity and at most n alternations are retained.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

3 results within one dependency step 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