LemmaStatement: 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.

Turing reducibility is a preorder

Statement

For all $A, B, C \subseteq N$ , $A \leq_{T} A$ , and $A \leq_{T} B \leq_{T} C$ implies $A \leq_{T} C$ .

Facts & Assumptions

Given: sets $A, B, C \subseteq N$ and, in the second assertion, oracle deciders witnessing $A \leq_{T} B$ and $B \leq_{T} C$ .

Proof

technique · direct

1.1

The machine which asks its oracle whether $n$ belongs to it and returns that answer is an $A$ -oracle decider for $A$ .

givenconstruct

2.1

For transitivity, simulate the $B$ -oracle decider for $A$ ; answer each query $q \in B$ by the total $C$ -oracle decider for $B$ . Every inner query terminates and the outer run makes finitely many, so this is a total $C$ -oracle decider for $A$ .

givenconstruct∎

Depends on

Turing reducibility and equivalence

Used by

Turing degrees and their order Definition
The degree order is representative-independent Lemma

Dependency tree · two levels

2 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, §5.1 (standard reference, not scraped)