LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

Every computable many-one reduction induces a Turing reduction

Statement

If $A \leq_{m} B$ , then $A \leq_{T} B$ .

Facts & Assumptions

Given: Languages $A$ and $B$ with $A \leq_{m} B$ .

[L1]

A computable many-one reduction is a total computable function $f$ satisfying $x \in A ⟺ f (x) \in B$ , by Computable many-one reductions between languages.

[L2]

A Turing reduction is a halting oracle decider that answers membership in the source language correctly, by Turing reductions via oracle deciders.

Proof

technique · direct

1.1

Let $f$ witness $A \leq_{m} B$ . Build an oracle machine that, on input $x$ , first computes $f (x)$ and then asks the oracle for $B$ whether $f (x) \in B$ .

L1givenconstruct

2.1

By [L1], the oracle answer to step 1.1 is "yes" exactly when $x \in A$ . The machine makes only that one query and then halts with the same answer, so [L2] shows that it is a Turing reduction from $A$ to $B$ .

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

MIT 6.045J / 18.400J, Lecture 9: Mapping Reducibility and Rice's Theorem (standard reference, not scraped)