CorollaryStatement: 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.

Delta_1^0 sets are exactly the decidable sets

Statement

A set of natural numbers is $Δ_{1}^{0}$ if and only if it is decidable.

Facts & Assumptions

Given: a set $A \subseteq N$ .

Proof

technique · direct

1.1

If $A$ is $Δ_{1}^{0}$ , both $A$ and its complement are $Σ_{1}^{0}$ and hence recognizable. Dovetail their recognizers; exactly one accepts, yielding a decider for $A$ .

givenconstruct

2.1

If $A$ is decidable, both $A$ and its complement are recognizable, hence both are $Σ_{1}^{0}$ . Therefore $A \in Δ_{1}^{0}$ .

step 1.1∎

Depends on

Used by

A set lying in both Sigmaₙ⁰ and Piₙ⁰ Counterexample
False: Sigmaₙ⁰ and Piₙ⁰ are disjoint False statement

Dependency tree · two levels

6 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

Ludovic Patey, Computability Theory, Proposition 2.2 (standard reference, not scraped)