PropositionStatement: Literature-sourcedProof: 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.

Solutions compose under a change of initial time

Statement

Let $Φ (t, x_{0})$ denote the maximal solution map of a smooth autonomous vector field. Whenever both sides are defined,

$Φ (t, Φ (s, x_{0})) = Φ (t + s, x_{0}) .$

Equivalently, changing the initial time along a trajectory composes solutions.

Facts & Assumptions

Given: A smooth autonomous vector field, its maximal solution map $Φ$ , a state $x_{0}$ , and times $s, t$ such that both sides of the displayed equation are defined.

[F1]

The maximal solution domain is open (The maximal solution domain is open).

[L1]

Two locally unique solutions of the same ODE that agree at a time agree on their common interval of definition (Locally unique ODE solutions agree on overlaps and glue across a common endpoint).

Proof

technique · direct

1.1

The curve $u \mapsto Φ (u + s, x_{0})$ solves the ODE and at time $u = 0$ has value [F1, L1] $Φ (s, x_{0})$ . The curve $u \mapsto Φ (u, Φ (s, x_{0}))$ also solves the same ODE and has the same value at time $0$ . Because [F1] makes both curves defined on an open interval about every common time where they exist, [L1] applies and forces them to agree on their common domain.

F1L1

2.1

Evaluating the equality from step 1.1 at the time $u = t$ gives [step 1.1] $Φ (t, Φ (s, x_{0})) = Φ (t + s, x_{0})$ whenever both sides are defined.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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, §4.4 (standard reference, not scraped)