ExampleConstruction: AI-generatedVerification: AI-generatedverified 2026-09-26 (gpt-6-sol)

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 $h$ .

[A1]

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

Verification

technique · direct

1.1given

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

2.1A1step 1.1

For $c < 1$ the sublevel set ${h \leq c}$ is empty. For $c \geq 1$ , [A1] makes it a closed ball of radius $\sqrt{1 - 1 / c} < 1$ , hence compact in $B_{1} (0)$ .

3.1step 1.1step 2.1algebra∎

If $C \subseteq R$ is compact, it is closed and bounded above by some $c$ . Continuity of $h$ makes $h^{- 1} (C)$ closed in $B_{1} (0)$ , and it lies in the compact set ${h \leq c}$ from step 2.1. Thus $h^{- 1} (C)$ is compact, so $h$ is proper and smooth.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

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)