PropositionStatement: Literature-sourcedProof: 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 vector field is complete if and only if its flow is global

Statement

A smooth vector field $X$ on $M$ is complete if and only if its maximal flow domain is all of $R \times M$ .

Facts & Assumptions

Given: A smooth vector field $X$ with maximal flow $Φ : D \to M$ .

[L1]

The maximal flow domain is $D = {(t, p) : t \in I_{p}},$ where $I_{p}$ is the domain of the maximal integral curve through $p$ (The fundamental theorem on flows).

[L2]

A vector field is complete exactly when every maximal integral curve is defined on all of $R$ (Complete vector fields).

Proof

technique · direct

1.1L1L2given

If $X$ is complete, then [L2] gives $I_{p} = R$ for every $p$ . By [L1], this means $D = R \times M$ .

1.2L1L2given

Conversely, if $D = R \times M$ , then [L1] gives $I_{p} = R$ for every $p$ . Therefore [L2] says that $X$ is complete.

2.1step 1.1step 1.2∎

Hence $X$ is complete if and only if its maximal flow is global.

A vector field is complete if and only if its flow is global

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources