ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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 exhaustion metric is explicit on the unit disc

Example

For the unit disc $D$ , the canonical exhaustion is $K_{n} = {z \in D : ∣ z ∣ \leq 1 - 1 / n} (n \geq 1),$ so $K_{1} = {0}$ and for the functions $f (z) = z$ , $g (z) = 0$ the exhaustion metric is $d_{K} (f, g) = \sum_{n \geq 1} 2^{- n} (1 - 1 / n) .$

Facts & Assumptions

Given: The unit disc $D$ and the functions $f (z) = z$ and $g (z) = 0$ .

[L1]

The exhaustion metric is defined from the canonical compact exhaustion (The exhaustion metric on a function space over a plane domain, Every plane domain has the canonical nested compact exhaustion by distance and radius cutoffs).

Verification

technique · direct

1.1

In $D$ , the boundary distance is $1 - ∣ z ∣$ , so the canonical condition $dist (z, \partial D) \geq 1 / n$ is exactly $∣ z ∣ \leq 1 - 1 / n$ . Thus $K_{n} = {∣ z ∣ \leq 1 - 1 / n}$ and in particular $K_{1} = {0}$ .

L1givenalgebra

2.1

On $K_{n}$ , the difference $∣ f (z) - g (z) ∣ = ∣ z ∣$ has supremum $1 - 1 / n$ , which is already at most $1$ . Substituting into the definition from [L1] gives $d_{K} (f, g) = \sum_{n \geq 1} 2^{- n} (1 - 1 / n)$ .

L1givenalgebra∎

Depends on

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

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)