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.

Bounded quantifiers preserve primitive recursiveness

Statement

If R(x⃗,u) is primitive recursive and t(x⃗) is primitive recursive, then each relation ∃u<t(x⃗) R(x⃗,u),∀u<t(x⃗) R(x⃗,u) is primitive recursive.

Facts & Assumptions

Given: primitive-recursive R and t.

Proof

technique · direct
1.1

Let r(x⃗,u) be the 0-1 characteristic function of R. The bounded sum S(x⃗)=∑u<t(x⃗)r(x⃗,u) is primitive recursive by primitive recursion and composition.

givenconstruct
2.1

The existential relation has characteristic test S(x⃗)>0. Likewise P(x⃗)=∏u<t(x⃗)r(x⃗,u) is primitive recursive and equals 1 exactly for the universal relation, including the empty product case.

step 1.1algebra∎

Depends on

Used by

Dependency tree · one level

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Sources