ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

Euclidean spaces and Euclidean open subsets as smooth manifolds

Example

For every $n \geq 0$ , the Euclidean space $R^{n}$ is a smooth $n$ -manifold, with global chart the identity map. More generally, every open subset $U \subseteq R^{n}$ is a smooth $n$ -manifold with its standard restricted smooth structure.

Facts & Assumptions

Given: A natural number $n$ and an open subset $U \subseteq R^{n}$ .

[F1]

Open subsets of Euclidean space carry the standard smooth structure (Open subsets of Euclidean space have the standard smooth structure).

[F2]

A smooth manifold is a topological manifold equipped with a smooth structure (Smooth manifolds and their smooth charts).

Verification

technique · direct

1.1

Taking $U = R^{n}$ , the identity chart exhibits $R^{n}$ as a topological $n$ -manifold and [F1] supplies its smooth structure. Hence $R^{n}$ is a smooth $n$ -manifold by [F2].

F1F2

2.1

For a general open subset $U \subseteq R^{n}$ , [F1] states exactly that $U$ inherits the standard smooth structure, so again [F2] makes $U$ a smooth $n$ -manifold.

F1F2∎

Depends on

Used by

Two noncompatible atlases on the real line Counterexample
A countable disjoint union of lines is a smooth manifold Example

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

Rob van der Vorst, Introduction to differentiable manifolds, §1, Example 1.5 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, §2.3 (standard reference, not scraped)