ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-28

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.

The family z^n is normal on the unit disc and not normal on the complex plane

Example

The sequence $z^{n}$ is a normal family on the unit disc $D$ , but not on all of $C$ .

Facts & Assumptions

Given: The sequence $f_{n} (z) = z^{n}$ .

[L1]

Locally bounded holomorphic families are normal, and normal holomorphic families are locally bounded (Montel's theorem: every locally bounded holomorphic family is normal, Normal holomorphic families are locally bounded).

Verification

technique · direct

1.1

On every compact subset of $D$ , all points satisfy $∣ z ∣ \leq r < 1$ , so $∣ z^{n} ∣ \leq 1$ there for every $n$ . Hence the family is locally bounded on $D$ , and [L1] makes it normal.

L1given

2.1

On the plane, the closed disc $\overline{D} (2, 1 / 2)$ lies in $C$ and every point of it has modulus at least $3 / 2$ , so $∣ z^{n} ∣ \geq (3 / 2)^{n}$ there. Thus the family is not locally bounded near $2$ , and [L1] shows it is not normal on $C$ .

L1givenalgebra∎

Depends on

Used by

FALSE: a normal family contains every locally uniform sequential limit of its own sequences False statement

Dependency tree · two levels

8 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

Matthias Weber, Complex Analysis, Ch. 5 §§5.1-5.2 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 2 §5.2 and Ch. 8 §3.2 (standard reference, not scraped)
Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic Function Theory, Ch. 2 (standard reference, not scraped)