TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Cartan's magic formula

Statement

For every vector field $X$ and differential form $ω$ , $L_{X} ω = d (ι_{X} ω) + ι_{X} (d ω) .$

Facts & Assumptions

Given: The manifolds, forms, vector fields, maps, and coordinates explicitly named in the statement.

[F1]

The preceding result states that For $f \in C^{\infty} (M) = Ω^{0} (M)$ , $d f (X) = X f$ for every smooth vector field $X$ . (The exterior derivative of a function is its differential).

[F2]

For a one-form $α$ , the invariant formula gives $d α (X, Y) = X α (Y) - Y α (X) - α ([X, Y])$ (The exterior derivative by the invariant vector-field formula).

Proof

technique · direct

1.1F1F2given

For a function $f$ , the right side is $ι_{X} d f = X f = L_{X} f$ ; for a one-form $α$ , evaluate both sides on $Y$ and use $d α (X, Y) = X α (Y) - Y α (X) - α ([X, Y])$ .

2.1step 1.1∎

Both $L_{X}$ and $d ι_{X} + ι_{X} d$ are degree-zero derivations, so equality on functions and exact coordinate one-forms extends to every local coordinate expression and hence globally.

Depends on

Dependency tree · 2 levels

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed.