ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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.

A two-function smooth partition on the circle

Example

Let $U_{1} : = S^{1} ∖ {(- 1, 0)}$ and $U_{2} : = S^{1} ∖ {(1, 0)}$ . Then there exist smooth functions $ϕ_{1}, ϕ_{2} : S^{1} \to [0, 1]$ such that $ϕ_{1} + ϕ_{2} = 1$ , $supp (ϕ_{1}) \subseteq U_{1}$ , and $supp (ϕ_{2}) \subseteq U_{2}$ .

Facts & Assumptions

Given: The two-set open cover $U_{1}, U_{2}$ of the circle.

[L1]

Every open cover of a smooth manifold admits a subordinate smooth partition of unity (Smooth partitions of unity exist on manifolds).

Verification

technique · direct

1.1

The sets $U_{1}$ and $U_{2}$ are open and cover $S^{1}$ .

given

2.1

Apply [L1] to this cover to obtain the required functions $ϕ_{1}, ϕ_{2}$ .

L1step 1.1

3.1

Thus the circle carries a two-function smooth partition subordinate to the chosen arcs.

step 2.1∎

Depends on

Smooth partitions of unity exist on manifolds

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)