Alphabeta Math
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.

The degree order is representative-independent

Statement

If A≡TA′ and B≡TB′, then A≤TB  ⟺  A′≤TB′. Consequently the order in Turing degrees and their order is well-defined.

Facts & Assumptions

Given: A≡TA′, B≡TB′, and the preorder laws of Turing reducibility is a preorder.

Proof

technique · direct
1.1

If A≤TB, then A′≤TA≤TB≤TB′ by the two equivalences and transitivity, so A′≤TB′.

givenconstruct
2.1

Interchanging the primed and unprimed representatives gives the reverse implication. Thus either representative pair yields the same comparison.

step 1.1∎

Depends on

Used by

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