ExampleConstruction: Literature-sourcedVerification: AI-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.

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

1.1F1F2

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].

2.1F1F2∎

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.

Euclidean spaces and Euclidean open subsets as smooth manifolds

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources