Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-30
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.

A proper smooth exhaustion of Euclidean space

Example

The function h(x)=∥x∥2 is a smooth proper function on Rn.

Facts & Assumptions

Given: The function h(x)=∥x∥2.

[L1]

Smooth manifolds admit smooth proper functions (Every smooth manifold admits a smooth proper exhaustion function).

[A1]

For each c≥0, the sublevel set {x:∥x∥2≤c} is the closed ball of radius c.

Verification

technique · direct
1.1

The function h is polynomial in the coordinates, hence smooth.

given
1.2

By [A1], every sublevel set is compact, so h is proper.

A1
2.1

This is an explicit example of [L1].

L1step 1.1step 1.2∎

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