PropositionStatement: 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.

Truth-table reducibility implies Turing reducibility

Statement

For all $X, Y \subseteq N$ , $Y \leq_{t t} X$ implies $Y \leq_{T} X$ .

Facts & Assumptions

Given: a truth-table functional $Γ$ witnessing $Y \leq_{t t} X$ .

Proof

technique · direct

1.1

By definition, $Γ^{X} (n)$ halts on every $n$ and equals $χ_{Y} (n)$ .

given

2.1

The same machine, run with oracle $X$ , is consequently a total oracle decider for $Y$ . This is exactly $Y \leq_{T} X$ .

step 1.1∎

Depends on

Used by

Some Turing reductions are not truth-table reductions Theorem

Dependency tree · two levels

3 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, Definition 5.9 (standard reference, not scraped)