ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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.

Pole pushing along an explicit chain of three discs

Example

Take $K = {z : ∣ z ∣ \leq 1}$ and the three discs $D_{1} = D (2, 3 / 4)$ , $D_{2} = D (5 / 2, 3 / 4)$ , $D_{3} = D (3, 3 / 4)$ . Then one may push the pole of $(z - 2)^{- 1}$ successively to $5 / 2$ , to $3$ , to $7 / 2$ , and then to $\infty$ , while keeping the approximation uniform on $K$ .

Facts & Assumptions

Given: The compact set $K$ and the three discs displayed in the Example.

[L1]

Runge's pole-pushing lemma moves a simple pole along any finite disc chain disjoint from the compact set (Runge's pole-pushing lemma).

Verification

technique · direct

1.1

Each closed disc ${\overline{D}}_{j}$ is disjoint from $K$ , and the pairs $(2, 5 / 2)$ , $(5 / 2, 3)$ , and $(3, 7 / 2)$ lie in $D_{1}$ , $D_{2}$ , and $D_{3}$ respectively. Thus the displayed data form a pole-pushing chain from $2$ to $7 / 2$ .

given

2.1

Apply clause 1 of [L1] to that chain to obtain, for any prescribed $ε > 0$ , a rational function with only pole $7 / 2$ that approximates $(z - 2)^{- 1}$ uniformly on $K$ . For the polynomial conclusion, every $z \in K$ satisfies $∣ z ∣ < 3 / 2 < 7 / 2$ , so clause 2 of [L1], with $R = 3 / 2$ , gives a polynomial approximating $(z - 2)^{- 1}$ uniformly on $K$ to within $ε$ .

step 1.1L1algebra∎

Depends on

Runge's pole-pushing lemma

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

J. Lebl, Guide to Cultivating Complex Analysis, Lemma 9.2.2 discussion (standard reference, not scraped)
M. Weber, Complex Analysis, Lemma 4.4.4 (standard reference, not scraped)