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.

Lagrangian submanifolds have half dimension

Statement

A Lagrangian submanifold of a symplectic $2 n$ -manifold has dimension $n$ .

Facts & Assumptions

Given: A Lagrangian embedded submanifold $L$ of $(M^{2 n}, ω)$ .

[F1]

Each $T_{p} L$ is a Lagrangian subspace of $T_{p} M$ . Isotropic, coisotropic, symplectic, and Lagrangian submanifolds.

[F2]

A Lagrangian subspace of a $2 n$ -dimensional symplectic space has dimension $n$ . Equivalent characterizations of Lagrangian subspaces.

Proof

technique · direct

1.1

For every $p \in L$ , [F1] and [F2] give $\dim T_{p} L = n$ .

F1F2

2.1

Since $L$ is an embedded constant-dimensional submanifold, its dimension equals the dimension of any tangent space, hence $\dim L = n$ . This includes $n = 0$ .

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)