ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Constant vector fields have translation flows

Example

Let $v \in R^{n}$ and consider the constant vector field $X (x) = v$ . Its integral curves are

$γ_{x_{0}} (t) = x_{0} + t v,$

so the flow is the translation family $Φ_{t} (x) = x + t v$ .

Facts & Assumptions

Given: A fixed vector $v \in R^{n}$ and the vector field $X (x) = v$ .

[L1]

The flow of a vector field is global exactly when the field is complete (A vector field is complete if and only if its flow is global).

Verification

technique · direct

1.1given

The curve $γ_{x_{0}} (t) = x_{0} + t v$ satisfies $γ_{x_{0}} (0) = x_{0}$ and $γ_{x_{0}}^{'} (t) = v = X (γ_{x_{0}} (t))$ , so it is an integral curve of $X$ through $x_{0}$ .

2.1L1step 1.1∎

The formula is defined for every $t \in R$ , so the flow exists globally and [L1] shows that $X$ is complete. The time- $t$ map is the translation $Φ_{t} (x) = x + t v$ .

Depends on

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

Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)