ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-31

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 graph of the sine function is an embedded submanifold

Example

The set

$Γ = {(x, \sin x) : x \in R} \subseteq R^{2}$

is an embedded one-dimensional submanifold of $R^{2}$ .

Facts & Assumptions

Given: The smooth function $f : R \to R$ , $f (x) = \sin x$ .

[L1]

The graph of a smooth map $F : M \to N$ is an embedded submanifold of $M \times N$ of dimension $\dim M$ (The graph of a smooth map is an embedded submanifold).

[L2]

Sine and cosine are differentiable with derivatives cosine and negative sine, respectively (The derivatives of sine and cosine are cosine and minus sine).

Verification

technique · direct

1.1

Repeatedly applying the derivative identities in [L2] shows that every derivative of $f (x) = \sin x$ exists and is again $\pm \sin x$ or $\pm \cos x$ ; these functions are continuous. Hence $f$ is smooth.

L2given

2.1

Therefore its graph is an embedded submanifold of $R \times R ≅ R^{2}$ by [L1].

L1step 1.1

3.1

This is exactly the displayed subset $Γ$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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, Embedded Submanifolds (standard reference, not scraped)