ExampleConstruction: AI-adaptedVerification: AI-generatedSession-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.

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The square map sends the Cartesian grid lines off the coordinate axes to two orthogonal families of parabolas

Example

Under $w = z^{2}$ , the vertical grid lines $x = c$ with $c \neq 0$ and the horizontal grid lines $y = d$ with $d \neq 0$ become two families of parabolic arcs, opening in opposite directions. The two coordinate axes are the exceptions: each maps onto a ray rather than a parabola. At the image of every grid crossing $z \neq 0$ , the tangent directions of the two curves through it remain orthogonal. The origin is the critical point where this conformality conclusion is unavailable.

Facts & Assumptions

Given: $z = x + i y$ , $w = u + i v = z^{2}$ , and real constants $c, d$ specifying the lines $x = c$ and $y = d$ .

[L1]

Complex polynomials are entire, and the derivative of $z^{2}$ is $2 z$ (Complex polynomials are entire with the power-rule derivative, and rational functions are holomorphic wherever their denominator is nonzero).

[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 · cases and direct computation

1.1

Expanding gives $u = x^{2} - y^{2}$ and $v = 2 x y$ . On $x = c$ , $(u, v) = (c^{2} - y^{2}, 2 c y)$ .

algebra

1.2

At a crossing $z = c + i d \neq 0$ , [L1] gives $f^{'} (z) = 2 z \neq 0$ . Hence [L2] says the real derivative is an oriented similarity, so it carries the perpendicular vertical and horizontal tangent directions to perpendicular nonzero tangent directions of the two image curves.

L1L2

2.1

If $c \neq 0$ , eliminating $y = v / (2 c)$ gives $u = c^{2} - v^{2} / (4 c^{2})$ , a left-opening parabola. If $c = 0$ , the image is $(u, v) = (- y^{2}, 0)$ , the nonpositive real ray, which is the degenerate member of that family.

step 1.1assume-case nonzeroassume-case zeroalgebra

2.2

On $y = d$ , $(u, v) = (x^{2} - d^{2}, 2 d x)$ . If $d \neq 0$ , eliminating $x = v / (2 d)$ gives $u = v^{2} / (4 d^{2}) - d^{2}$ , a right-opening parabola. If $d = 0$ , the image is $(u, v) = (x^{2}, 0)$ , the nonnegative real ray.

step 1.1assume-case nonzeroassume-case zeroalgebra

3.1

At $z = 0$ , the derivative is zero, so [L2] gives no conformality conclusion; indeed the two degenerate rays meet there. Steps 2.1 and 2.2 cover zero and nonzero grid parameters, and step 1.2 covers exactly the noncritical crossings.

step 2.1step 2.2step 1.2L1L2cases-exhaustive∎

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: 29 results over 10 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)