ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

The standard symplectic vector space

Example

On $R^{2 n}$ with coordinates $(q^{1}, \dots, q^{n}, p_{1}, \dots, p_{n})$ , the form $ω_{0} = \sum_{i} d q^{i} \land d p_{i}$ is symplectic.

Facts & Assumptions

Given: The displayed vector space and alternating form.

[F1]

Nondegeneracy means that $v \mapsto ι_{v} ω_{0}$ has zero kernel. Symplectic vector space.

Verification

technique · direct

1.1givenalgebra

For $v = \sum_{i} (a^{i} \partial_{q^{i}} + b_{i} \partial_{p_{i}})$ , contraction gives $ι_{v} ω_{0} = \sum_{i} (a^{i} d p_{i} - b_{i} d q^{i})$ .

2.1F1step 1.1∎

This covector vanishes only when every $a^{i}$ and $b_{i}$ is zero, so [F1] proves nondegeneracy. Equivalently, $ω_{0}^{n} = n! d q^{1} \land d p_{1} \land \dots \land d q^{n} \land d p_{n} \neq 0$ . For $n = 0$ the zero space is included.

Depends on

Symplectic vector space

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)