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 closed half-space as a manifold with boundary

Example

For $n \geq 1$ , the identity chart on $H^{n}$ gives $Int H^{n} = {x^{n} > 0}$ and $\partial H^{n} = {x^{n} = 0}$ ; at the face, inward vectors have positive last component. For $n = 0$ , $H^{0}$ is a point with empty boundary.

Facts & Assumptions

Given: An integer $n \geq 0$ , with $H^{n}$ and $\partial H^{n}$ as in the stated convention.

[L1]

The model half-space is $H^{n} = {x^{n} \geq 0}$ for $n \geq 1$ , with face ${x^{n} = 0}$ ; for $n = 0$ , $H^{0} = R^{0}$ and its face is empty (Euclidean upper half-space and its boundary).

[L2]

Boundary charts define the smooth structure, and face versus relative-interior points is invariant under smooth changes of boundary chart (Topological manifolds with boundary; Smooth charts, atlases, and structures with boundary; Smooth invariance of the manifold boundary).

[L3]

At a face point, the last coordinate classifies tangent vectors as inward, outward, or boundary-tangent according as it is positive, negative, or zero (Inward, outward, and boundary-tangent vectors).

Verification

technique · direct

1.1

For $n \geq 1$ , the identity map is a global boundary chart on $H^{n}$ ; for $n = 0$ , the unique map identifies the point with $R^{0}$ . These charts give the asserted smooth manifolds with boundary.

givenL1L2

2.1

For $n \geq 1$ , [L1] and [L2] identify the intrinsic interior with ${x^{n} > 0}$ and the intrinsic boundary with ${x^{n} = 0}$ . The identity chart identifies every tangent space with $R^{n}$ , and [L3] makes the inward vectors at the face precisely those with positive last component. For $n = 0$ , [L1] makes the unique point interior and the boundary empty.

givenL1L2L3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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)