CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Finite jumps match arithmetical levels

Statement

Put $0^{(0)} = \emptyset$ and $0^{(n + 1)} = (0^{(n)})^{'}$ . For every $n \geq 0$ and $C \subseteq N$ , $C \in Σ_{n + 1}^{0} ⟺ C is 0^{(n)} -c.e., C \in Δ_{n + 1}^{0} ⟺ C \leq_{T} 0^{(n)} .$

Facts & Assumptions

Given: $n \geq 0$ , a set $C$ , and the jump convention above.

Proof

technique · direct

1.1

The recursively defined $0^{(n + 1)}$ is precisely the relative halting oracle for $0^{(n)}$ , because the local jump convention is diagonal relative halting.

givenconstruct

2.1

Apply Post's theorem at level $n$ with the identification in step 1.1. Its two equivalences give exactly the displayed claims.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Sebastiaan Terwijn, Computability Theory, Proposition 5.2.1 (standard reference, not scraped)