Alphabeta Math
ExampleConstruction: AI-generatedVerification: 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.

Commuting coordinate fields have commuting flows

Example

On R2, the coordinate vector fields X=∂/∂x and Y=∂/∂y have flows

ΦtX(x,y)=(x+t,y),ΦsY(x,y)=(x,y+s),

and these flows commute.

Facts & Assumptions

Given: The coordinate vector fields X=∂/∂x and Y=∂/∂y on R2.

[L1]

Coordinate vector fields commute (Coordinate vector fields commute).

[L2]

Vanishing Lie bracket is equivalent to commuting local flows (Two vector fields commute if and only if their local flows commute).

Verification

technique · direct
1.1

The explicit integral curves give the flows ΦtX(x,y)=(x+t,y) and ΦsY(x,y)=(x,y+s).

given
2.1

Their compositions satisfy ΦtX(ΦsY(x,y))=(x+t,y+s)=ΦsY(ΦtX(x,y)). This agrees with [L1] and [L2].

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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