CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 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 real-valued holomorphic function on a domain is constant

If $f : U \to C$ is holomorphic on a domain $U$ and $f (U) \subseteq R$ , then $f$ is constant.

Given: A domain $U$ and a holomorphic $f = u + i v : U \to C$ with $v = 0$ .

[L1]

[L2]

A holomorphic function with zero derivative on a domain is constant (A holomorphic function with zero derivative on a domain is constant).

technique · direct

1.1givenL1algebra

Since $v = 0$ , both $v_{x}$ and $v_{y}$ vanish. The Cauchy–Riemann equations [L1] then give $u_{x} = u_{y} = 0$ , so $f^{'} = 0$ throughout $U$ .

2.1step 1.1L2∎

Apply [L2] to conclude that $f$ is constant on $U$ .

Depends on