Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21
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 sublevel set of a convex function is convex

Statement

If f:C→R is convex and α∈R, then

{x∈C:f(x)≤α}

is a convex subset of Rn. Empty and singleton sublevel sets are included.

Facts & Assumptions

Given: The function, domain, and level in the Statement, with convex subsets interpreted by A convex subset of Rm contains every line segment between two of its points.

[F1]

The function f:C→R is convex when f((1−t)x+ty)≤(1−t)f(x)+tf(y) for all x,y∈C and t∈[0,1] (Convex and strictly convex functions on Euclidean convex sets).

Proof

technique · direct
1.1

If f(x),f(y)≤α, then [F1] gives f((1−t)x+ty)≤(1−t)f(x)+tf(y)≤α for every t∈[0,1].

F1givenalgebra
2.1

Thus every segment between two sublevel points remains in the sublevel set, which is convex. If it has fewer than two points, the same condition is vacuous.

step 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