PropositionStatement: AI-adaptedProof: 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 Poisson bracket is bilinear, skew, and a derivation in each entry

Statement

The Poisson bracket is real-bilinear and skew-symmetric, and

${F, G H} = {F, G} H + G {F, H}, {F G, H} = F {G, H} + G {F, H} .$

Facts & Assumptions

Given: Smooth functions $F, G, H$ on $(M, ω)$ .

[F1]

${F, G} = ω (X_{F}, X_{G}) = X_{G} (F)$ . Poisson bracket on a symplectic manifold.

Proof

technique · direct

1.1

Linearity of $d$ and uniqueness of Hamiltonian fields give $X_{a F + b G} = a X_{F} + b X_{G}$ . Bilinearity and alternation of $ω$ now make the bracket bilinear and skew.

F1givenalgebra

2.1

Since a vector field is a derivation, [F1] gives ${F G, H} = X_{H} (F G) = F {G, H} + G {F, H}$ . Skew-symmetry then gives the displayed Leibniz rule in the second entry as well.

F1step 1.1∎

Depends on

Poisson bracket on a symplectic manifold

Used by

Smooth functions form a Poisson algebra Theorem
The Poisson bracket satisfies the Jacobi identity Theorem

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)