ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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 standard collar of a closed ball

Example

For $n \geq 1$ , $c : S^{n - 1} \times [0, ε) \to B^{n}$ , $c (u, t) = (1 - t) u$ , is a collar for $0 < ε < 1$ .

Facts & Assumptions

Given: An integer $n \geq 1$ , a real number $0 < ε < 1$ , the closed unit ball $B^{n}$ , and the map $c : S^{n - 1} \times [0, ε) \to B^{n}$ defined by $c (u, t) = (1 - t) u$ .

[L1]

The boundary of $B^{n}$ is $S^{n - 1}$ (The closed ball and its sphere boundary).

[L2]

A smooth collar is a boundary-fixing smooth embedding whose image is an open neighbourhood of the boundary (Smooth collars of a manifold boundary).

Verification

technique · direct

1.1givenconstructalgebra

The map $c$ is smooth and satisfies $c (u, 0) = u$ . Since $1 - t > 0$ , its inverse on its image is $x ⟼ (\frac{x}{∣ x ∣}, 1 - ∣ x ∣),$ which is smooth; hence $c$ is a smooth embedding.

2.1givenL1L2step 1.1algebra∎

Its image is ${x \in B^{n} : ∣ x ∣ > 1 - ε}$ , which is open in $B^{n}$ and contains $S^{n - 1} = \partial B^{n}$ by [L1]. Consequently $c$ satisfies every clause of [L2] and is a collar.

Depends on

Used by

The double of a disk is a sphere Example

Dependency tree · two levels

7 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

Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)