ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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 equation e^z = 3z has exactly one solution in the unit disc

Example

The equation

$e^{z} = 3 z$

has exactly one solution in the unit disc.

Facts & Assumptions

Given: The function $f (z) = e^{z} - 3 z$ and the unit circle $∣ z ∣ = 1$ .

[L1]

Rouché's theorem preserves the zero count under the strict boundary inequality (Rouche's theorem in the classical strict-inequality form).

Verification

technique · direct

1.1

If $∣ z ∣ = 1$ , then $Re z \leq 1$ , so $∣ e^{z} ∣ = e^{Re z} \leq e < 3 = ∣ 3 z ∣ .$

givenalgebra

2.1

Apply [L1] to $f (z) = e^{z} - 3 z$ and $g (z) = - 3 z$ . Since $- 3 z$ has exactly one zero in $∣ z ∣ < 1$ , so does $f$ .

step 1.1L1∎

Depends on

Rouche's theorem in the classical strict-inequality form

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

R. W. Howell and J. H. Mathews, Complex Analysis, §8.7 (standard reference, not scraped)