ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 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.

A countable disjoint union of lines is a smooth manifold

Example

The countable disjoint union

$X : = ⨆_{m \in N} R_{m}$

of countably many copies of the real line is a smooth $1$ -manifold.

Facts & Assumptions

Given: The countable family $(R_{m})_{m \in N}$ of copies of the real line.

[F1]

Each copy of $R$ is a smooth $1$ -manifold (Euclidean spaces and Euclidean open subsets as smooth manifolds).

[F2]

A countable disjoint union of fixed-dimensional smooth manifolds is a smooth manifold (Countable disjoint unions of fixed-dimensional smooth manifolds are smooth manifolds).

Verification

technique · direct

1.1

By [F1], every summand $R_{m}$ is a smooth $1$ -manifold.

2.1

The index set $N$ is countable, so [F2] applies to the family [F2, step 1.1] $(R_{m})_{m \in N}$ and yields a smooth $1$ -manifold structure on $X$ .

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Nigel Hitchin, Differentiable Manifolds, §2.3 (standard reference, not scraped)
Rob van der Vorst, Introduction to differentiable manifolds, §1 (standard reference, not scraped)