ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passverified 2026-09-23 (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 smooth function with a prescribed closed zero set

The function $g (x) = \sin^{2} (π x)$ is smooth and nonnegative on $R$ , and its zero set is exactly $Z$ .

Given: The function $g (x) = \sin^{2} (π x)$ .

[A1]

One has $\sin (π x) = 0$ exactly when $x \in Z$ .

technique · direct

1.1given

The function $g$ is smooth and nonnegative.

1.2A1

By [A1], the equation $g (x) = 0$ holds exactly when $x \in Z$ .

2.1step 1.1step 1.2∎

Thus $g$ realizes the closed set $Z$ as a smooth zero set. This explicit construction uses no choice principle.

Used by

Nothing in the library uses this result yet.

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