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.

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A nonautonomous equation made autonomous by adjoining time

Example

The nonautonomous scalar equation

$x^{'} (t) = t x (t), x (t_{0}) = x_{0},$

becomes autonomous after adjoining the time variable:

$s^{'} (t) = 1, x^{'} (t) = s (t) x (t), (s (0), x (0)) = (t_{0}, x_{0}) .$

Its explicit solution is

$s (t) = t + t_{0}, x (t) = x_{0} e^{(t + t_{0})^{2} / 2 - t_{0}^{2} / 2} = x_{0} e^{(t^{2} + 2 t_{0} t) / 2} .$

Equivalently, in the original time variable $τ = t + t_{0}$ , $x (τ) = x_{0} e^{(τ^{2} - t_{0}^{2}) / 2}$ .

Facts & Assumptions

Given: The scalar equation $x^{'} = t x$ with initial data $(t_{0}, x_{0})$ .

[L1]

The nonautonomous smooth-ODE theorem is proved by adjoining the time variable as an autonomous one (The fundamental theorem for nonautonomous smooth ODEs).

Verification

technique · direct

1.1

The augmented system has $s (t) = t + t_{0}$ because $s^{'} = 1$ and $s (0) = t_{0}$ . Then [given] $x^{'} = s (t) x = (t + t_{0}) x$ , so solving this linear scalar equation gives the displayed exponential formula.

given

2.1

Writing $τ = t + t_{0}$ turns the displayed solution into [L1, step 1.1] $x (τ) = x_{0} e^{(τ^{2} - t_{0}^{2}) / 2}$ , which is exactly the solution of the original nonautonomous equation. This is the concrete reduction promised by [L1].

L1step 1.1∎

Depends on

The fundamental theorem for nonautonomous smooth ODEs

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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 §10.3 (standard reference, not scraped)
Chin-Lung Wang, Banach Calculus, §4.3 (standard reference, not scraped)