TheoremStatement: 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.

The jump is monotone

Statement

For all $A, B \subseteq N$ , if $A \leq_{T} B$ , then $A^{'} \leq_{T} B^{'}$ .

Facts & Assumptions

Given: a total $B$ -oracle decider for $A$ .

Proof

technique · direct

1.1

Uniformly compile an $A$ -oracle program $e$ into a $B$ -oracle program $q (e)$ by replacing each $A$ -query with the given total $B$ -oracle computation of that bit. Its compilation is effective.

givenconstruct

2.1

Parameter specialization produces an index $r (e)$ which halts with oracle $B$ on input $r (e)$ exactly when program $e$ halts with oracle $A$ on input $e$ . Hence $e \in A^{'}$ iff $r (e) \in B^{'}$ .

step 1.1construct

3.1

An oracle machine with oracle $B^{'}$ computes $r (e)$ and queries its membership, deciding $A^{'}$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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, Exercise 6.2 (standard reference, not scraped)