DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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.

The germ of a smooth function at a point

Definition

Let $M$ be a smooth manifold and $p \in M$ . Two pairs $(U, f)$ and $(V, g)$ , where $U$ and $V$ are open neighbourhoods of $p$ and $f : U \to R$ , $g : V \to R$ are smooth, are equivalent at $p$ when there is an open neighbourhood $W \subseteq U \cap V$ of $p$ such that $f ∣_{W} = g ∣_{W}$ . An equivalence class is a germ of a smooth function at $p$ .

Depends on

Smooth manifolds and their smooth charts

The germ of a smooth function at a point

Definition

Depends on

Dependency tree · 2 levels

Sources