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 constant vector field has translation solutions

Example

Fix $v \in R^{n}$ . The constant vector field $V (x) = v$ has solutions

$x (t) = x_{0} + t v .$

Thus its flow is translation by $t v$ .

Facts & Assumptions

Given: A fixed vector $v \in R^{n}$ and the autonomous equation $x^{'} = v$ .

[L1]

Autonomous smooth ODEs have unique local smooth flows (The fundamental theorem for autonomous smooth ODEs).

Verification

technique · direct

1.1

The curve $x (t) = x_{0} + t v$ satisfies $x (0) = x_{0}$ and $x^{'} (t) = v$ for every $t$ , so [given] it solves the ODE.

given

2.1

By [L1], the local solution through $x_{0}$ is unique, so the displayed [L1, step 1.1] affine curve is the solution and the time- $t$ map is translation by $t v$ .

L1step 1.1∎

Depends on

The fundamental theorem for autonomous smooth ODEs

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, Appendix §10.2 (standard reference, not scraped)
Chin-Lung Wang, Banach Calculus, §4.1 (standard reference, not scraped)