Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06
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.

Separation of disjoint convex sets when one is open

Statement

If U,V⊆X are nonempty disjoint convex sets and U is open, then there is a nonzero f∈X∗ such that

Re⁡f(u)<Re⁡f(v)(u∈U,v∈V).

Facts & Assumptions

Given: Nonempty disjoint convex sets U,V, with U open.

[F1]

An exterior point and a nonempty open convex set admit a strict separating continuous functional (Separate a point from an open convex set).

Proof

technique · direct
1.1

The difference D=U−V is open and convex, and 0∉D because U∩V=∅.

givenalgebra
2.1

Apply [F1] to D and 0. It supplies nonzero f∈X∗ with Re⁡f(d)<0 for every d∈D.

step 1.1F1
3.1

For u∈U and v∈V, u−v∈D, so Re⁡f(u)−Re⁡f(v)<0. This is the required strict separation.

step 2.1algebra∎

Depends on

Used by

Dependency tree · two levels

9 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