ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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.
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2xy is a harmonic conjugate of x^2-y^2

Example

The function $v (x, y) = 2 x y$ is a harmonic conjugate of $u (x, y) = x^{2} - y^{2}$ on all of $C$ .

Facts & Assumptions

Given: The polynomial $f (z) = z^{2}$ .

[L1]

Complex polynomials are entire (Complex polynomials are entire with the power-rule derivative, and rational functions are holomorphic wherever their denominator is nonzero).

[L2]

If a holomorphic function is written $u + i v$ , then its imaginary part is a harmonic conjugate of its real part (The $C^{2}$ real and imaginary parts of a holomorphic function satisfy Laplace's equation and form a harmonic-conjugate pair, Harmonic conjugates).

Verification

technique · direct

1.1

Since $z^{2} = (x + i y)^{2} = (x^{2} - y^{2}) + 2 i x y$ , the real part of $z^{2}$ is $x^{2} - y^{2}$ and the imaginary part is $2 x y$ .

givenalgebra

2.1

The polynomial $z^{2}$ is entire by [L1], so [L2] makes $2 x y$ a harmonic conjugate of $x^{2} - y^{2}$ .

step 1.1L1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Jeremy Orloff, MIT 18.04 Topic 5: Introduction to Harmonic Functions (standard reference, not scraped)