DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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.

First-order systems, initial value problems, and solutions on intervals

Definition

Let $D \subseteq R \times R^{n}$ be open, let $F : D \to R^{n}$ , and let $n \geq 1$ . The equation

$x^{'} (t) = F (t, x (t))$

is a first-order system. An initial value problem consists of this equation and data $(t_{0}, x_{0}) \in D$ , written $x (t_{0}) = x_{0}$ .

A solution on an interval $I$ is a function $x : I \to R^{n}$ such that $t_{0} \in I$ , $(t, x (t)) \in D$ for every $t \in I$ , $x$ is differentiable on $I$ in the domain-relative sense of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, and

$x (t_{0}) = x_{0}, x^{'} (t) = F (t, x (t)) (t \in I) .$

Endpoints of $I$ use the corresponding one-sided, domain-relative derivative. A local solution is one whose interval contains a nondegenerate neighborhood of $t_{0}$ relative to the time projection of $D$ .

First-order systems, initial value problems, and solutions on intervals

Definition

Depends on

Dependency tree · 2 levels

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