TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-08-29

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.

Conformal equivalence is an equivalence relation

Statement

Conformal equivalence of complex domains is an equivalence relation: every complex domain $U$ is conformally equivalent to itself; if $U$ is conformally equivalent to $V$ then $V$ is conformally equivalent to $U$ ; and if $U$ is conformally equivalent to $V$ and $V$ to $W$ , then $U$ is conformally equivalent to $W$ .

Facts & Assumptions

Given: Complex domains $U, V, W$ , and the conformal-equivalence notion of Conformal equivalence and the automorphism group of a domain.

[F1]

$U$ and $V$ are conformally equivalent when there exists a biholomorphism $f : U \to V$ . The identity map ${id}_{U}$ is biholomorphic; the inverse of a biholomorphism is biholomorphic; and the composite of two biholomorphisms $U \to U$ is biholomorphic (Conformal equivalence and the automorphism group of a domain).

Proof

technique · direct

1.1F1given

Reflexivity: [F1] makes ${id}_{U}$ biholomorphic, witnessing $U \sim U$ .

1.2F1assume-hyp

Symmetry: for biholomorphic $f : U \to V$ , [F1] makes $f^{- 1} : V \to U$ biholomorphic.

1.3F1assume-hyp

Transitivity: for biholomorphic $f : U \to V$ , $g : V \to W$ , [F1] makes $g \circ f$ biholomorphic.

2.1step 1.1step 1.2step 1.3∎

Steps 1.1-1.3 are the three clauses of an equivalence relation.

Depends on

Conformal equivalence and the automorphism group of a domain

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 8 (standard reference, not scraped)