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 the open unit ball

Example

On the open unit ball $B_{1} (0) \subseteq R^{n}$ , the function $h (x) : = 1 / (1 - ∥ x ∥^{2})$ is smooth and proper.

Facts & Assumptions

Given: The open unit ball $B_{1} (0)$ and the displayed function $h$ .

[L1]

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

[A1]

For each $c \geq 1$ , one has $h (x) \leq c$ exactly when $∥ x ∥^{2} \leq 1 - 1 / c$ .

Verification

technique · direct

1.1

The denominator is positive on $B_{1} (0)$ , so $h$ is smooth there.

given

1.2

By [A1], every sublevel set is a closed ball of radius strictly less than $1$ , hence compact in $B_{1} (0)$ .

2.1

Therefore $h$ is a proper smooth function on the open ball, as predicted by [L1].

L1step 1.1step 1.2∎

Depends on

Every smooth manifold admits a smooth proper exhaustion function

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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)