DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Distance from a point to a subset

Definition

For $A \subseteq M$ , define the distance to the subset by $d_{g} (x, A) = \inf_{a \in A} d_{g} (x, a)$ , with $\inf \emptyset = + \infty$ .

Use Extended riemannian distance on a disconnected manifold. If the component $C$ of $x$ meets $A$ , all cross-component terms are infinite and may be discarded, so $d_{g} (x, A) = \inf_{a \in A \cap C} d_{g} (x, a) < \infty$ . If $A \cap C = \emptyset$ , every term is infinite and the value is $+ \infty$ . In particular $d_{g} (x, A) = 0$ for $x \in A$ , and $d_{g} (x, {a}) = d_{g} (x, a)$ .

Source locator

Lee, Chapter 13, pp.337–340, Proposition 13.25, Lemma 13.28 and Theorem 13.29; finite piecewise $C^{1}$ refinements and pauses are treated explicitly here.

Depends on

Extended riemannian distance on a disconnected manifold

Used by

Distance to a nonempty subset is one lipschitz Proposition

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

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)