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.

A time-dependent translation field and its evolution operator

Example

Let $I \subseteq R$ be an open interval, let $a : I \to R$ be smooth, and consider the time-dependent vector field

$X_{t} = a (t) \frac{\partial}{\partial x}$

on $R$ . Its evolution operator is

$Ψ_{t, s} (x) = x + \int_{s}^{t} a (r) d r .$

Facts & Assumptions

Given: An open interval $I \subseteq R$ , a smooth function $a : I \to R$ , and the field $X_{t} = a (t) d / d x$ on $R$ .

[L1]

Time-dependent vector fields admit local smooth evolution operators (Time-dependent vector fields have local smooth evolution operators).

[L2]

Evolution operators satisfy the cocycle law (Time-dependent evolution satisfies the two-time cocycle law).

Verification

technique · direct

1.1

Differentiating the proposed formula yields $\frac{d}{d t} Ψ_{t, s} (x) = a (t) = X_{t} (Ψ_{t, s} (x)),$ and clearly $Ψ_{s, s} (x) = x$ . So the formula solves the initial-value problem.

given

2.1

Because $I$ is an interval, the segment between any $s, t \in I$ stays in $I$ , so the integral and the displayed map are defined for every $s, t \in I$ . Thus this is the global evolution operator extending the local one from [L1]. Moreover, $Ψ_{r, t} (Ψ_{t, s} (x)) = x + \int_{s}^{t} a + \int_{t}^{r} a = x + \int_{s}^{r} a = Ψ_{r, s} (x),$ matching [L2].

L1L2step 1.1algebra∎

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

Marius Crainic, Rui Loja Fernandes, and Ioan Marcut, Lectures on Poisson Geometry (standard reference, not scraped)