ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-01

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 nonzero smooth compactly supported bump

Example

Let $ϕ$ be The one-sided flat function is $C^{\infty}$ with identically zero Taylor series and define $b (x) = ϕ (1 - x^{2})$ . Then $b$ is smooth, positive on $(- 1, 1)$ , and zero outside $[- 1, 1]$ ; its support is $[- 1, 1]$ .

Facts & Assumptions

Given: The displayed composite.

[L1]

$ϕ$ is smooth and flat at $0$ (The one-sided flat function is $C^{\infty}$ with identically zero Taylor series).

[L2]

Products and composites preserve smoothness (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , A power-series sum is infinitely differentiable inside its radius and satisfies $a_{n} = f^{(n)} (c) / ι (n!)$ at its centre).

[L3]

Closure, boundary, and interior are Interior, closure, boundary, limit point, isolated point and dense subset of a metric space.

Verification

technique · direct

1.1

On $∣ x ∣ < 1$ , $1 - x^{2} > 0$ , so $b (x) > 0$ ; on $∣ x ∣ \geq 1$ , $1 - x^{2} \leq 0$ , so $b (x) = 0$ .

givenL1

1.2

Away from $\pm 1$ , smoothness follows from [L2]. At $\pm 1$ , every one-sided derivative from inside is a finite sum of derivatives of $ϕ$ at $0$ , all zero by [L1], matching the zero function outside.

L1L2

2.1

Thus $b$ is smooth and its nonzero set is $(- 1, 1)$ , whose closure is $[- 1, 1]$ ; this is its support.

step 1.1step 1.2L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

30 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

MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)
J. Lebl, Basic Analysis, Analytic Functions (standard reference, not scraped)
S. Dyatlov, MIT 18.155 lecture notes (standard reference, not scraped)
MIT 18.102, Chapter 4 notes (standard reference, not scraped)