Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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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.

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Turing degrees form an upper semilattice

Statement

For all degrees [A]T,[B]T, the degree [A⊕B]T is their least upper bound. Hence the Turing degrees form an upper semilattice.

Facts & Assumptions

Given: A,B,C⊆N and the tagged-join convention of Tagged join of oracles.

Proof

technique · direct
1.1

An (A⊕B)-oracle machine decides A by querying 2n, and decides B by querying 2n+1. Hence [A]T≤[A⊕B]T and [B]T≤[A⊕B]T.

givenconstruct
1.2

If A≤TC and B≤TC, a C-oracle machine deciding A⊕B tests parity: on 2n it runs the decider for A(n), and on 2n+1 that for B(n). Thus A⊕B≤TC.

givenconstruct
2.1

Step 1.2 says every common upper bound lies above [A⊕B]T; step 1.1 says it is a common upper bound. Representative-independence makes this a statement about degrees.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

6 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