ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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.

A Möbius map $(a z + b) / (c z + d)$ with $a d - b c \neq 0$ is conformal wherever $c z + d \neq 0$

Example

Let $a, b, c, d \in C$ with $a d - b c \neq 0$ , and define $F (z) = \frac{a z + b}{c z + d}$ wherever $c z + d \neq 0$ . Then $F$ is holomorphic and orientation-preserving conformal at every point of its domain.

Facts & Assumptions

Given: Complex coefficients $a, b, c, d$ with $a d - b c \neq 0$ .

[L1]

The linearity and quotient rules hold for complex derivatives wherever the denominator is nonzero (Linearity, product, reciprocal, and quotient rules for complex derivatives).

[L2]

A real-differentiable complex map is orientation-preserving conformal at a point exactly when it is complex differentiable there with nonzero derivative (A real-differentiable complex map is orientation-preserving conformal at a point exactly when it is complex differentiable there with nonzero derivative).

Verification

technique · direct computation

1.1

On the set $D : = {z : c z + d \neq 0}$ , [L1] gives $F^{'} (z) = \frac{a (c z + d) - c (a z + b)}{(c z + d)^{2}} = \frac{a d - b c}{(c z + d)^{2}} .$

L1algebra

2.1

Both numerator and denominator in step 1.1 are nonzero on $D$ , so $F^{'} (z) \neq 0$ throughout $D$ .

step 1.1given

3.1

Thus [L2] makes $F$ orientation-preserving conformal at every point of $D$ .

step 1.1step 2.1L2

4.1

If $c = 0$ , then $a d \neq 0$ , hence $d \neq 0$ and $D = C$ . If $c \neq 0$ , the excluded equation $c z + d = 0$ has the unique solution $z = - d / c$ , so $D = C ∖ {- d / c}$ . No global injectivity claim on $C$ is needed.

givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Howell and Mathews, Complex Analysis, §3.6 (standard reference, not scraped)

A Möbius map (az+b)/(cz+d)(az+b)/(cz+d) with ad−bc≠0ad−bc=0 is conformal wherever cz+d≠0cz+d=0