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.

Dividing $z_{1}$ by the Weierstrass polynomial $z_{2}^{2} - z_{1}$

Example

Let $W (z_{1}, z_{2}) = z_{2}^{2} - z_{1}$ . Dividing the germ $f (z_{1}, z_{2}) = z_{1}$ by $W$ gives

$z_{1} = 0 \cdot W + z_{1} .$

So the unique quotient is $q = 0$ and the unique remainder is the degree- $0$ polynomial $r (z_{1}, z_{2}) = z_{1}$ .

Facts & Assumptions

Given: The dividend $f (z_{1}, z_{2}) = z_{1}$ and divisor $W (z_{1}, z_{2}) = z_{2}^{2} - z_{1}$ .

[L1]

Weierstrass division gives a unique quotient and a unique remainder of $z_{2}$ -degree $< 2$ (Weierstrass division theorem).

Verification

technique · direct

1.1

The identity $z_{1} = 0 \cdot (z_{2}^{2} - z_{1}) + z_{1}$ is immediate.

given

2.1

The remainder $z_{1}$ has degree $0$ in $z_{2}$ , hence degree $< 2$ . Therefore [L1] identifies this displayed identity as the unique Weierstrass division of $z_{1}$ by $z_{2}^{2} - z_{1}$ .

step 1.1L1∎

Depends on

Weierstrass division theorem

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Jiří Lebl, Tasty Bits of Several Complex Variables, Section 6.2 (standard reference, not scraped)

Dividing z1z1​ by the Weierstrass polynomial z22−z1z22​−z1​